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Title: [Graduate Text Book] A Primer in Game Theory
Description: A Primer in Game Theory by T R Gibbons - one of the core textbooks for postgraduate study in economics and game theory.

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A BEAUTIFUL MATH
A BEAUTIFUL MATH
JOHN NASH,
GAME THEORY,
AND THE
MODERN QUEST
FOR A CODE
OF NATURE

TOM SIEGFRIED

JOSEPH HENRY PRESS

Washington, D
...

Joseph Henry Press • 500 Fifth Street, NW • Washington, DC 20001
The Joseph Henry Press, an imprint of the National Academies Press, was
created with the goal of making books on science, technology, and health
more widely available to professionals and the public
...

Any opinions, findings, conclusions, or recommendations expressed in this
volume are those of the author and do not necessarily reflect the views of the
National Academy of Sciences or its affiliated institutions
...
— 1st ed
...
cm
...

ISBN 0-309-10192-1 (hardback) — ISBN 0-309-65928-0 (pdfs) 1
...
I
...

QA269
...
3—dc22
2006012394

Printed in the United States of America
...
He
suggested that the terrorist organization took its name from Isaac
Asimov’s famous 1950s science fiction novels known as the Foundation Trilogy
...
” And the first novel in
Asimov’s trilogy, Foundation, apparently was titled “al-Qaida” in an
Arabic translation
...
The empire was
hopeless, destined to crumble into chaos, leaving civilization in
ruins for 30,000 years
...
His strategy was to establish a
“foundation” of scholars who would preserve human knowledge
for civilization’s eventual rebirth
...

In fact, Asimov’s hero, a mathematician named Hari Seldon,
created a community of scientists devoted to manipulating the future
...
Foundation I participated
openly in the affairs of the galaxy
...

Seldon’s plan for controlling human affairs was based on a
iii

PREFACE

mathematical system that he invented called psychohistory
...

I don’t think Osama bin Laden is Hari Seldon
...
Or that they
anointed themselves as society’s saviors, hoping to manipulate
events in a way that would lead to a new world order more to their
liking
...

(Certainly Osama bin Laden’s occasional taped messages are eerily
similar to Seldon’s video appearances from time to time, prepared
before his death for delivery decades or even centuries later
...
Al Qaeda gains no justification for atrocity from any
connection to science fiction
...
Had the terrorists really studied Foundation, they
would have noticed Asimov’s assertion that “violence is the last
refuge of the incompetent
...
If there is a real-life Hari Seldon, it is
not Osama bin Laden, but John Forbes Nash
...
Nash’s math, for which he won a Nobel Prize, is an entirely
different tale, still unfolding, about science’s struggle to cope with
the complexities of collective human behavior
...
Game theory is the science of
strategy; its formulas tell you what choices to make to get the best
deal you can get when interacting with other people
...
It has begun to establish

PREFACE

links with the physical sciences as well, and ultimately, I suspect, it
will forge a merger of all the sciences in the spirit of Asimov’s
psychohistory
...

Game theory is a rich, profound, and controversial field, and
there is much more to it than you could find in any one book
...
Nor do I
attempt to give any account of its widespread uses in economics,
the realm for which it was invented, or the many variants and
refinements that have been developed to expand its economic applications
...
I
view these efforts in the context of the ancient quest for a “Code
of Nature” describing the “laws” of human behavior, a historical
precursor to Asimov’s notion of psychohistory
...
There are scientists who
regard some of this pioneering work as at best misguided and at
worst a fruitless waste of time
...
Well, maybe so
...
For now, the fact is
that game theory has already established itself as an essential tool
in the behavioral sciences, where it is widely regarded as a unifying
language for investigating human behavior
...
And connections have
been established between game theory and two of the most prominent pillars of physics: statistical mechanics and quantum theory
...
Game theory is showing signs of playing an increasingly important role in that endeavor
...

I owe much gratitude to those who helped make this book
possible, particularly the many scientists who have discussed their
research with me over the years
...
Many other friends and
colleagues have listened patiently while I’ve shaped my thoughts
on this book during conversations with them
...
The one person I want to thank
by name is my wife, Chris, who really made it possible for me to
write this book, because she has a job
...

In one of his early science fiction stories, he introduced pocket
calculators decades before you could buy them at Radio Shack
...
1 He just forgot to mention that
you could also use the same device to make phone calls
...
Psychohistory, as Asimov envisioned
it, was “the science of human behavior reduced to mathematical
equations
...
But there are many research enterprises under way in the world today that share the goal of better
understanding human behavior in order to foresee the future
...
And in the midst of it
all is the work of a mathematician named John Forbes Nash
...
He rattled the
routines at Princeton University and the Rand Corporation in California with both his mental magnificence and his disruptive behavior
...
Yet while book
and movie probed the conflicting complexities in Nash the man,
neither delved deeply into Nash’s math
...
Within the world of science,
though, Nash’s math now touches more disciplines than Newton’s
or Einstein’s
...

Indeed, had mental illness not intervened, Nash’s name might
today be commonly uttered in the same breath with those scientific giants of the past
...
But he achieved his greatest fame in
economics, the field in which he shared the 1994 Nobel Prize
with John Harsanyi and Reinhard Selten for their seminal work on
the theory of games—the math that analyzes how people make
choices in contests of strategy
...
But in principle, game
theory encompasses any situation involving strategic interaction—
from playing tennis to waging war
...
So game theory’s math specifies the formulas for making sound decisions in any competitive
arena
...
But it is much more than a
mere tool
...
“Game theory is about the

INTRODUCTION

emergence, transformation, diffusion and stabilization of forms of
behavior
...
At first, though, the depth of his accomplishment was
little appreciated
...
But within economics, game theory
remained mainly a curiosity
...
“Like a lot of good ideas in economics, it
just fell by the wayside
...
And in the 1980s, economists finally began to use game
theory in various ways, finding it especially helpful in designing
actual experiments to test economic theory
...

Even before then, game theory had already migrated into the
curricula of many scientific disciplines
...
By the opening years of the 21st century, game theory’s uses had spread even
wider, to fields ranging from anthropology to neurobiology
...
Biologists apply it to scenarios
explaining the survival of the fittest or the origin of altruism
...
And neuroscientists
have joined the fun, peering inside the brains of game-playing
people to discover how their strategies reflect different motives
and emotions
...
“We’re

A BEAUTIFUL MATH

quantifying human experience,” says neuroscientist Read
Montague, “in the same way we quantify airflow over the wings of
a Boeing 777
...
In fact, Herbert Gintis contends, game theory has become “a universal language for the unification of the behavioral
sciences
...
Game theory may
become the language not just of the behavioral sciences, but of all
the sciences
...
It might even
be wrong
...
And it is now, in the works of a few
pioneering scientists, forming a powerful alliance with physics
...

That realization hit me in early 2004, when I read a paper by
physicist-mathematician David Wolpert, who works at NASA’s
Ames Research Center in California
...

Physicists have used statistical mechanics for more than a century to describe such things as gases, chemical reactions, and the
properties of magnetic materials—essentially to quantify the behavior of matter in all sorts of circ*mstances
...
You can’t track
every one of the trillion trillion molecules of air zipping around in
a room, for instance, but statistical mechanics can tell you how an
air conditioner will affect the overall temperature
...
As Janov Pelorat, a character in the later novels of the
Foundation series, explained:
Hari Seldon devised psychohistory by modeling it upon the kinetic
theory of gases
...

Nevertheless, using statistics, we can work out the rules governing
their overall behavior with great precision
...
7

In other words, put enough people together and the laws of
human interaction will produce predictable patterns—just as the
interactions and motion of molecules determine the temperature
and pressure of a gas
...

One of the best ways to take that temperature, it turns out, is
to view society in terms of networks
...
Today’s new network math applies statistical
mechanics to all sorts of social phenomena, from fashion trends
and voting behavior to the growth of terrorist cells
...

For the most part, this merger of network math and statistical
mechanics has been exploring human behavior without recourse
to the modern views of game theory built on Nash’s math
...
But the latest research has begun to show ways that game
theory can help make sense out of the intricate pattern of links in
complicated networks
...

A BEAUTIFUL MATH

Wolpert’s insight suggests that game theory itself can be elevated to a new level by exploiting its link to statistical mechanics
...
In other words, at some deep level
statistical mechanics and game theory are, in a sense, two versions
of the same underlying idea
...

This new realization—that game theory and statistical mechanics share a deep mathematical unity—enhances game theory’s status as the preferred tool for merging the life sciences and physical
sciences into a unified description of nature
...
Game theory could someday become the glue that holds all
of science’s puzzle pieces together
...
But pause to consider how much sense it makes
...
“Intelligent” design produces simple, predictable
systems that are easy to understand
...
And such competitive interaction is precisely what game theory is all about
...
Game theory is about competition,
and evolution is the ultimate never-ending Olympic event
...
So it’s perfectly natural that game theory has become
popular today in efforts to understand how the brain works, as
brain scientists explore the neural physiology behind economic
choices
...

All that behavior directs the evolution of all those networks of

INTRODUCTION

personal, social, political, or economic activity
...

So Nash’s math does seem capable of catalyzing a merger of methods for understanding individual behavior, biology, and society
...
But in a way there is, and the connections
between game theory and statistical mechanics promise to reveal
ways in which game theory still applies
...
The “desire” for minimum energy in molecules is not
so different from the “desire” for maximum fitness in organisms
...

True, there’s much more to physics than statistical mechanics
...
But
guess what? In the past few years physicists and mathematicians
have developed quantum versions of game theory
...
8
Furthermore, Wolpert forges the link between statistical mechanics and game theory with help from the mathematical theory
of information
...
Quantum physics itself has been illuminated over the past decade by new insights emerging from
quantum information theory
...
” It’s possible, Wolpert speculates, that game

A BEAUTIFUL MATH

theory is the ingredient that could enhance the prospects for success in finding such a theory
...
As I described in my book
Strange Matters (Joseph Henry, 2002), there is something strange
about the human brain’s ability to produce math that captures deep
and true aspects of reality, enabling scientists to predict the existence of exotic things like antimatter and black holes before any
observer finds them
...
I failed then to realize that game theory offers a
tool for describing how the laws of physics and biology are
related
...
And math
in turn, as Asimov envisioned, can be used to describe the behavior
guided by those brains—including the social collective behavior that creates civilization, culture, economics, and politics
...
And just maybe we’ll see how Nash’s
math can broker the merger of economics and psychology, anthropology and sociology, with biology and physics—producing a
grand synthesis of the sciences of life in general, human behavior
in particular, and maybe even, someday, the entire physical world
...

It would be wrong, though, to suggest that Asimov was the
first to articulate that dream
...
As interpreted much later,
that code supposedly captured the essence of human nature, providing a sort of rule book for behavior
...
With the arrival of the Age of Reason in
the 18th century, philosophers and the forerunners of social scientists sought in earnest to discover that code of codes—the key to
understanding the natural order of human interaction
...

Smith’s Hand
Searching for the Code of Nature
If in the seventeenth century natural philosophers
borrowed notions of law in human affairs and applied them to the study of physical nature, in the
eighteenth century it was the turn of the laws of
physical nature to suggest ways forward for knowledge about human life
...
By age 5, he was reading Time magazine (even though no
one had taught him to read), and at 14 he entered Johns Hopkins
University
...
B
...
and, for good measure, a Ph
...

He joined the faculty at Northwestern University’s graduate school
of management by the age of 22
...
Or more accurately, he likes to analyze the
behavior of other people during various game-playing experiments
...
He studies how game theory reveals the realities of human economic behavior, how people in real life depart from the
purely rational choices assumed by traditional economic theory
...
Even in his prodigy days, he was a
wrestler and a golfer, so he has a broader view of the world than
some of the intellectually exalted scholars who live their lives on
such a higher mental level
...
But in a sense,
Camerer’s views on economic behavior are not so revolutionary
...

Smith’s “invisible hand” is probably the most famous metaphor
in all of economics, and his equally famous book, Wealth of Nations, remains revered by today’s advocates of free-market economies more than two centuries after its publication
...
His
insights foreshadowed much in current attempts to decipher the
code of human conduct, in economics and other social arenas
...
The way I see it,
Adam Smith was the premier player in the origins of this story, as
he inspired belief in the merit of melding the Newtonian physics
of the material world with the science of human behavior
...
Both were
lifelong bachelors
...
Both were born after their fathers had died
...

Newton built the foundation of physics; Smith authored the bible
of economics
...
Just as modern physics de-

SMITH’S HAND

scended from Newton’s codification of what was then known as
natural philosophy, modern economics is the offspring of Adam
Smith’s treatise on political economy
...

While Newton established the notion of natural law in the
physical world, Smith tried to do the same in the social world of
economic intercourse
...
Together, Newton’s and Smith’s works
inspired great thinkers to believe that all aspects of the world—
physical and social—could be understood, and explained, by science
...

Nowadays, of course, physics has moved beyond Newton, and
most economists would say that their science has moved far beyond Adam Smith
...
If you
look closely, you can even find echoes of Smith’s ideas in various
aspects of game theory
...
And it is pursuit of selfinterest that game theory, at its most basic level, attempts to quantify
...
Game theory tries to delimit what
rational behavior is; Smith helped deposit the idea in the modern
mind that minds operate in a rational way
...
It was much
more ambitious for Smith to ascribe similar orderliness to the social behavior of humans engaging in economic activity
...
“In order to discover such a
science as economics,” they wrote, “Smith had to posit a faith in
the orderly structure of nature, underlying appearances and accessible to man’s reason
...
First philosophers, and then later sociologists and psychologists, tried to
articulate a science of human behavior based on principles “underlying appearances” but “accessible to man’s reason
...
“There is no question of importance, whose decision is not comprised in the science
of man,” Hume wrote, “and there is none, which can be decided
with any certainty, before we become acquainted with that science
...
”
Today, game theory’s ubiquitous role in the human sciences
suggests that its ambitions are woven from that same fabric
...

That claim is enhanced, I think, with the realization
that threads of Smith’s thought are entangled not only in physical
and social science, but biological science as well
...
Principles describing competition in the economic world, Darwin realized, made
equal sense when applied to the battle for survival in the biological
arena
...
So it is surely no accident that, today, applying
economic game theory to the study of evolution is a major intellectual industry
...
His influence on today’s world stemmed from a
life spent gathering unusual insights into his own world
...
At the
age of 3, he was kidnapped from his uncle’s front porch by some
gypsylike vagrants known as tinkers
...
Growing up, Adam was a bright kid,
earning a reputation as a bookworm with a spectacular memory
...
At 17 he went to Oxford, at first with the
intention of entering the clergy
...
His interests destined him to the academic world, as he had no acumen for
business and, as one biographer noted, “a strong preference for the
life of learning and literature over the professional or political
life
...
Soon he
was also appointed to a professorship in “moral philosophy,” providing a fitting combination of duties for someone planning to
forge a rational understanding of human behavior
...
And in it he outlined a very different view of life and government than what he is generally known for today
...

Smith left Glasgow for London in 1764 to assume his tutorial task
...

Smith was especially taken with one François Quesnay, a fascinating character who deserves to be much better known than he is
...
He established himself as a physician and became an early advocate for surgery as an
important part of medical practice—not such a popular position
among doctors of his day
...
Quesnay’s even stronger influence
with King Louis XV was later secured when he attained an appointment as personal physician to Madame de Pampadour, the
king’s mistress
...
Once established
among the aristocracy, Quesnay’s brilliance attracted the other
leading intellects of his age, so much so that he was invited to
write articles on agriculture for the famous French Encyclopédie
...

In those days, conventional wisdom conceived of a nation’s
economic strength in terms of trade; favorable trade balances,
therefore, supposedly brought wealth to a nation
...
He further argued that governments imposed
a human-designed impairment to the “natural order” of economic
and social interaction
...

Encountering Quesnay while in Paris, Smith was also entranced
and began to merge the physiocratic philosophy with his own
...

SMITH’S HAND

THE INVISIBLE HAND
Smith’s views differed from Quesnay’s in one major respect: The
source of wealth, Smith argued, was not the land, but labor
...
And the production of wealth was enhanced by dividing the labor into subtasks that could be performed
more efficiently using specialized skills
...
4
Modern caricatures of Wealth of Nations do not do it justice
...
There is no need for any planning or external economic
controls—if everyone simply pursues profits without restraint, the
system as a whole will be most efficient at distributing goods and
services
...
“By directing that industry in such a manner as its
produce may be of the greatest value, he intends only his own
gain, and he is in this, as in many other cases, led by an invisible
hand to promote an end which was no part of his intention
...
(Among other things, he noted that the invisible hand
worked effectively only if the people doing business weren’t crooks
cooking the books
...
By eliminating both preferences (or
“encouragements”) and restraints, “the obvious and simple system

A BEAUTIFUL MATH

of natural liberty establishes itself of its own accord
...
” And he cited three specific roles that
government ought to fulfill: defending the country from invasion,
enforcing the laws so as to protect individuals from injustice, and
providing for the public works and institutions that private individuals would not find profitable (like protecting New Orleans
from hurricanes)
...
“There
can be little doubt that Smith’s faith in the power of an invisible
hand has been exaggerated by modern commentators,” Princeton
economist Alan Krueger wrote in an introduction to a recent reprinting of Wealth of Nations
...
”7
All this is not to say that Smith’s support for free enterprise is
entirely a misreading
...
) But as economists who followed Smith often
observed, his invisible hand does not always guarantee efficient markets or fairness
...
However deep his insights into the world he lived in, Smith was
nevertheless incapable of escaping his own time
...
” Cliffe Leslie
demurred: “I venture to maintain, to the contrary, that political
economy is not a body of natural laws in the true sense, but an
assemblage of speculations and doctrines
...
”8
Cliff Leslie’s account, published in 1870, dismissed the idea—

SMITH’S HAND

promoted by many of Smith’s disciples—that Smith had revealed
“a natural order of things,” an “offshoot of the ancient fiction of a
Code of Nature
...
The Roman legal
system recognized not only Roman civil law (Jus Civile), the specific legal codes of the Romans, but a more general law (Jus Gentium), consisting of laws arising “by natural reason” that are
“common to all mankind,” as described by Gaius, a Roman jurist
of the second century A
...

Apparently some Roman legal philosophers regarded Jus Gentium as the offspring of a forgotten “natural law” (Jus Naturale) or
“Code of Nature”—an assumed primordial “government-free” legal code shared by all nations and peoples
...
” So as near as I can tell, “Code of Nature” is what
people commonly refer to today as the law of the jungle
...
) “The belief gradually prevailed among the Roman lawyers
that the old Jus Gentium was in fact the lost code of Nature,” English legal scholar Henry Maine wrote in an 1861 treatise titled
Ancient Law
...
jurisprudence on the principles of the Jus
Gentium was gradually restoring a type from which law had only
departed to deteriorate
...
” The Code of Nature method
sought to reason out the laws of society by deducing the natural
order of things from innate features of the human mind
...

In fact, Smith’s work did express sentiments favorable to the
Code of Nature view; his statement that eliminating governmental
preferences and restraints allows “the obvious and simple system
of natural liberty” to establish itself clearly resonates with the con-

A BEAUTIFUL MATH

cept of such a code
...
”11
Cliffe Leslie maintained, on the other hand, that Smith actually pursued both methods—some deductive reasoning, to be sure,
but also thorough observations of actual economic conditions of
his day
...

“What he did not see was, that his own system
...
“Had
he lived even two generations later, his general theory of the organization of the economic world
...
”12
If Smith’s Code of Nature was tainted by his times, it was
nevertheless in tune with many similar efforts by others, before
him and after
...
Smith’s two great works, on moral philosophy and the laws
of wealth, were really part of one grander intellectual enterprise
that ultimately produced both economics and the “human sciences”
of sociology and psychology
...

“In the eighteenth century,” Roger Smith writes, “pleasure and
confidence in the design of the created physical world played an
important part in the search for the design of the human world
...
In fact, the forerunner of economics,
political economy, emerged in the last half of the 18th century as
“the study of the link between the natural order and material prosperity,” investigating “the laws, physical and social, that underlie
wealth
...
The mix of math and physics with biology, sociology, and economics is (to use economic terminology) a growth
industry, and game theory is becoming the catalyst accelerating
the trend
...
The cartoon view of
Smith’s story is that human nature is selfish, and that economic
behavior is rooted in that “truth
...
In its original form, game theory math describes
“rational” behavior in a way that essentially synonymizes “rational”
with “selfish
...

Game theory tells you what will happen if people do behave selfishly and rationally
...
In fact, Smith glimpsed many findings of

A BEAUTIFUL MATH

today’s experimental economic science
...

When writing Wealth of Nations, Smith assumed (as do all authors) that its readers would have also read his first book: the Theory
of Moral Sentiments, published in 1759
...
Read together, Smith’s two books show that
he had a kinder and gentler view of human nature than today’s
economics textbooks indicate
...
Camerer’s specialty, “behavioral
game theory,” is a subdivision of the field generally described as
“behavioral economics
...
Some even hit on the bright idea of testing economic
theory by doing experiments, with actual people (and sometimes
even real money)
...
Pursuing such experiments led to some Nobel
prizes and some new insights into the mathematics underlying economic activity
...
In a complicated experiment, it’s not always obvious what the utility-maximizing strategy
really is
...
In any event, Camerer finds it
fascinating that game theory shows, in so many ways, that humans
defy traditional economic ideas
...

During one of our conversations, at a coffee shop on the
Caltech campus, Camerer stressed that Smith never contended that

SMITH’S HAND

all people are inherently selfish, out for themselves with no concern for anyone else
...
“The idea was, if people
want to make a lot of dough, the way to do it is by giving you
what you want, and they don’t care about you per se
...
“I
think Adam Smith has been kind of misread
...
’ What
he conjectured, and later was proved mathematically, was that even
if people didn’t care about each other, markets could do a pretty
good job of producing the right goods
...
”14 So human nature is not necessarily
as adamantly self-serving as some people would like to believe
...

In fact, in Smith’s treatise on moral sentiments, he identified
sympathy as one of the most important of human feelings
...
The
brain’s impartial spectator weighs the costs and consequences of
actions, encouraging rational choices that should control the reactions of the passions
...
“Smith recognized
...
15
Nevertheless, the notion of self-interest and utility was dramatized by Smith in such a way that it formed a central core of
subsequent economic philosophy
...
His books also contributed in a significant
way to the birth of modern biology
...

But he certainly read accounts of it, including Dugald Stewart’s
eulogy-biography of Adam Smith
...
And while Darwin’s Origin of Species does
not mention Smith, its notion of natural selection and survival of
the fittest appears to be intellectually descended from Smith’s ideas
of economic competition
...
I first encountered the connection, though, in the late Stephen Jay Gould’s
massive tome on evolutionary biology
...
Among the most intriguing of those
influences was the work of William Paley, the theologian often
cited today by supporters of creationism and intelligent (sic)
design
...
If you find a
watch on the ground, Paley wrote in 1802, you can see that it’s
nothing like a rock
...
” The inevitable inference, Paley concluded,
was “the watch must have a maker
...
” Paley’s point was that the biological world was so full of orderly complexity, exquisite adaptation to the needs of efficient living, that it must have been the
product of an exquisite design, and hence, a designer
...
Adam Smith, Gould concluded,
supplied that logic
...
“Individual organisms engaged

SMITH’S HAND

in the ‘struggle for existence’ act as the analog of firms in competition
...
”16 In
other words, as Smith argued, there is no need to design an efficient economy (and in fact, a designer would be a bad idea)
...

Darwin saw a similar picture in biology: Organisms pursuing their
own interest (survival and reproduction) can create, over time, complexities of life that mirror the complexities of an economy
...
In his famous example of the
pin factory, Smith described how specialization breeds efficiency
...

“No naturalist doubts the advantage of what has been called
the ‘physiological division of labour’; hence we may believe that it
would be advantageous to a plant to produce stamens alone in one
flower or on one whole plant, and pistils alone in another flower
or on another plant,” Darwin wrote in Origin of Species
...

“We may, I think, assume that the modified descendants of any
one species will succeed by so much the better as they become
more diversified in structure, and are thus enabled to encroach on
places occupied by other beings,” Darwin commented
...
”17
Clearly Darwin’s “general economy” of life reflected sentiments
similar to those expressed in the “political economy” described by
Adam Smith
...
And via
Smith’s insights, Paley’s argument for the necessity of a creator is
refuted
...
‘just happen’ as a consequence of
causes operating at a lower level among struggling individuals,”
Gould asserted
...
Just as Newton had tamed the physical world in the 17th century, and Smith had codified economics
in the 18th, Charles Darwin in the 19th century added life to the
list
...
So by the end of the 19th century, the groundwork was
laid for a comprehensive rational understanding of just about everything
...
20 No volume arrived, for instance, to
articulate the long-sought Code of Nature
...

Von Neumann’s Games
Game theory’s origins
Games combining chance and skill give the best representation of human life, particularly of military
affairs and of the practice of medicine which necessarily depend partly on skill and partly on chance
...

—Gottfried Wilhelm von Leibniz (quoted by
Oskar Morgenstern, Dictionary of the History of Ideas)

It’s no mystery why economics is called the dismal science
...

Mix two known chemicals, and a chemist can tell you ahead of
time what you’ll get
...

But mix people with money, and you generally get madness
...
Yet many economists continue to
believe that they will someday practice a sounder science
...

A BEAUTIFUL MATH

At first glance, building economic science on the mathematical
theory of games seems about as sensible as forecasting real-estate
trends by playing Monopoly
...

Game theory provides precision to the once fuzzy economic
notion about how consumers compare their preferences (a measure
labeled by the deceptively simple term utility)
...

Yet while people have played games for millennia, and have
engaged in economic exchange for probably just as long, nobody
had ever made the connection explicit—mathematically—until the
20th century
...
And most of the credit
for game theory’s invention goes to one of the 20th century’s
most brilliant thinkers, the magical mathman John von Neumann
...
I’m really sorry he died so young
...
And that would have given me a chance to observe
his remarkable genius for myself
...

But he lived long enough to leave a legendary legacy in several
disciplines
...
Imagine what he could have accomplished if
he’d learned to focus himself !

VON NEUMANN’S GAMES

Of course, he accomplished plenty anyway
...
He didn’t exactly invent the modern
digital computer, but he improved it and pioneered its use for scientific research
...

Born in 1903 in Hungary, von Neumann was given the name
Janos but went by the nickname Jancsi
...
As a
child, Jancsi dazzled adults with his mental powers, telling jokes in
Greek and memorizing the numbers in phone books
...
He traveled back to Budapest
for exams, aced them, and continued his chemical education, first
at Berlin and then later at the University of Zurich
...
Rand didn’t need a new computer, von Neumann declared,
after solving the problem in his head
...
Two cyclists start out
20 miles apart, heading for each other at 10 miles an hour
...
How far has the fly flown by the time the bicycles meet? You
can solve it by adding up the fly’s many shorter and shorter paths
between bikes (this would be known in mathematical terms as summing the infinite series)
...

When jokesters posed this question to von Neumann, sure
enough, he answered within a second or two
...
“What trick?” said von Neumann
...
”

A BEAUTIFUL MATH

Before von Neumann first came to America in 1930, he had
established himself in Europe as an exceptionally brilliant mathematician, contributing major insights into such topics as logic and
set theory, and he lectured at the University of Berlin
...
He enjoyed Berlin’s cabaret-style
nightlife, and more important for science, he enjoyed poker
...
More than that, he showed how to apply rigorous
methods to social questions, not unlike Asimov’s Hari Seldon
...
“Accordingly, his work proved crucial in converting mathematics into a key tool to social theory
...
But the roots
of game theory reach much deeper
...
As a branch
of mathematics, though, game theory did not appear in its modern
form until the 20th century, with the merger of two rather simple
ideas
...

Utility is basically a measure of value, or preference
...
One of the more famous expositors of the idea was Jeremy Bentham, the British social
philosopher and legal scholar
...
or
...
” 2 So to

VON NEUMANN’S GAMES

Bentham, utility was roughly identical to happiness or pleasure—
in “maximizing their utility,” individual people would seek to increase pleasure and diminish pain
...
”3
Bentham’s utilitarianism incorporated some of the philosophical
views of David Hume, friend to Adam Smith
...

In economics, utility’s usefulness depends on expressing it
quantitatively
...
Wealth, for example, provides a means of
enhancing happiness, and wealth is easier to measure
...
It’s a convenient medium of exchange for comparing the
value of different things
...
So you need a general definition that makes it possible to express utility in a useful mathematical form
...
In solving a mathematical paradox about
gambling posed by his cousin Nicholas, Daniel realized that utility
does not simply equate to quantity
...
A million-dollar lottery prize has less utility for Bill Gates
than it would for, say, me
...
4
Obviously the idea of utility—what you want to maximize—
can sometimes get pretty complicated
...
If you’re playing basketball, you want
to score the most points
...
In poker, you want to win the pot
...
Game theory is all about figuring out which strategy is best
...
Waldegrave was analyzing a twoperson card game called “le Her,” and he described a way to find
the best strategy, using what today is known as the “minimax” (or
sometimes “minmax”) approach
...
Other mathematicians also occasionally dabbled
in what is now recognized to be game theory math, but there was
no one coherent approach or clear chain of intellectual influence
...
First
was Ernst Zermelo, a German mathematician, whose 1913 paper
examining the game of chess is sometimes cited as the beginning
of real game theory mathematics
...
And that is an important distinction, by the way
...
If
you get a bum hand, you’re likely to lose no matter how clever
your strategy
...
Zermelo limited himself to games of pure strategy, games without the complications
of random factors
...
6 But it seems he tried to show that if the White player
managed to create an advantageous arrangement of pieces—a
“winning configuration”—it would then be possible to end the
game within fewer moves than the number of possible chessboard
arrangements
...
)
Using principles of set theory (one of von Neumann’s mathematical specialties, by the way), Zermelo proved that proposition
...
But the main lesson from it all
was not so important for strategy in chess as it was to show that
math could be used to analyze important features of any such
game of strategy
...
It’s called “zero-sum” because whatever one player wins, the other loses
...
(Chess is also a
game where the players have “perfect information
...
)
Zermelo did not address the question of exactly what the best
strategy is to play in chess, or even whether there actually is a
surefire best strategy
...
In the early 1920s,
Borel showed that there is a demonstrable best strategy in twoperson zero-sum games—in some special cases
...

But that’s exactly what von Neumann did
...
That’s the modern
minimax7 theorem, which von Neumann first presented in December 1926 to the Göttingen Mathematical Society and then developed fully in his 1928 paper called “Zur Theorie der
Gesellshaftsspiele” (Theory of Parlor Games), laying the foundation for von Neumann’s economics revolution
...

Only years later did he merge game theory with economics, with
the assistance of an economist named Oskar Morgenstern
...
In a book published
in 1928, the same year as von Neumann’s minimax paper,
Morgenstern discussed problems of economic forecasting
...
” This, Morgenstern knew, was a problem peculiar
to the social sciences, including economics
...
They do what they do the same way whether a chemist
correctly predicts it or not
...
In particular, if
people know what you’re predicting they will do, they might do
something else just to annoy you
...
(By the way, in
the Foundation Trilogy, that’s why Seldon’s Plan had to be so secret
...
)
Anyway, Morgenstern illustrated the problem with a scenario
from The Adventures of Sherlock Holmes
...
It wasn’t obvious that Holmes could
simply outthink Moriarty
...
But then Holmes could anticipate Moriarty’s anticipation, and so on: I think that he thinks that I think that he thinks,
ad infinitum, or at least nauseum
...
He returned to the
Holmes–Moriarty issue in a 1935 paper exploring the paradoxes
of perfect future knowledge
...

Morgenstern was entranced, and he awaited an opportunity to meet
von Neumann and discuss the relevance of the 1928 paper to
Morgenstern’s views on economics
...
(Von Neumann
had by then taken up his position at the nearby Institute for Advanced Study
...
”11 As Morgenstern told the
story, he soon revived von Neumann’s interest in game theory and
began writing a paper to show its relevance to economics
...
By this
time—it was now 1940—the paper had grown substantially, and
it kept growing, ultimately into a book published by the Princeton
University Press in 1944
...
12 )
Theory of Games and Economic Behavior instantly became the
game theory bible
...
It was a sort of
newtonizing of Adam Smith, providing mathematical rigor to describe how individual interactions affect a collective economy
...
” It
will become apparent, they asserted, that “this theory of games of
strategy is the proper instrument with which to develop a theory
of economic behavior
...
But the opening sections are remarkably readable, laying
out the authors’ goals and intentions in a kind of extended preamble designed to persuade skeptical economists that their field
needed an overhaul
...
Throughout its early pages, the book
draws on physics as the model for how math can make murky
knowledge precise and practical—in contrast to economics, where
the basic ideas had been expressed so fuzzily that past efforts to
use math had been doomed
...
are often
stated in such vague terms as to make mathematical treatment a
priori appear hopeless because it is quite uncertain what the problems really are,” the authors wrote
...

Von Neumann and Morgenstern were careful to emphasize,
though, that their theory was just a first step
...
”15 But game theory could provide the
foundation for such a theory, by focusing on the simplest of economic interactions as a guide to developing general principles that
would someday be able to solve more complicated problems
...

“The great progress in every science came when, in the study
of problems which were modest as compared with ultimate aims,
methods were developed that could be extended further and further,” von Neumann and Morgenstern declared
...
While economic science as a whole involves the entire complicated system of producing and pricing goods, and earning and spending money, at the
root of it all is the choicemaking of the individuals participating
in the economy
...
Stranded on a desert island, Crusoe was an economy
unto himself
...

Samuel Bowles, an economist at the University of Massachusetts, explained to me that textbooks viewed economics as just the
activities of many individual Robinson Crusoes
...
And that was the standard “neoclassical” view of
economic theory
...

“But there was something odd about it
...
“Game
theory adopts a different framework,” Bowles said
...
”17
And that’s exactly the point that von Neumann and
Morgenstern stressed back in 1944
...
It’s not just the complication of social influences
from other people affecting your choices about the prices of goods
and services
...
“If two or more persons exchange goods
with each other, then the result for each one will depend in general
not merely upon his own actions but on those of the others as
well,” von Neumann and Morgenstern declared
...

Your calculations had to accommodate a mixture of competing

A BEAUTIFUL MATH

goals, maximum utilities for Gilligan, the Skipper too, the millionaire, and his wife, the movie star, the Professor and Mary Ann
...

Indeed, Bentham’s notion of the “greatest possible good of
the greatest possible number” is mathematically meaningless
...
Think about it—you can have zero cost (and no food) or all
the food in the world, at a very high cost
...
In a Gilligan’s
Island economy, it’s not really an issue of wanting the maximum
utility for the maximum number, but rather that all the individuals
want their own personal possible maximum
...
”19 And in
trying to fulfill their desires, every individual’s actions will be influenced by expectations of everyone else’s actions, and vice versa,
the old “I think he thinks I think” problem
...
“And it is this problem which the
theory of ‘games of strategy’ is mainly devised to meet,” von
Neumann and Morgenstern announced
...
It’s one thing to realize
that Gilligan’s Island is more complex than Robinson Crusoe’s; it’s
something else again to figure out how to do the math
...
Then, once you understand how two
people will interact, you can use the same principles to analyze
what will happen when a third person enters the game, and then a
fourth, and so on
...
)
However, you can see how things would rapidly become difficult to keep track of
...
In the
Robinson Crusoe economy, his set of variables encompasses all
those factors that would affect his quest for maximum utility
...
Then Crusoe would need to take all of those
new variables into account, too
...
So the social economy rapidly becomes a mathematical nightmare, it would seem, beyond even the ability of the knowit-all Professor to resolve
...

TAKING SOCIETY’S TEMPERATURE
In drawing analogies between economics and physics, von
Neumann and Morgenstern talked a lot about the theory of heat
(or, as it is more pretentiously known, thermodynamics)
...
In a
similar way, game theory needed to be developed first to give
economists the tools they needed to measure economic variables
properly
...
At the
outset, von Neumann and Morgenstern made it clear that they did
not want to venture into the philosophical quagmire of defining
all the nuances of utility
...
For the
businessman, money (as in profits) is a logical measure of utility;
for consumers, income (minus expenses) is a good measure of utility, or you could think of the utility of an object as the price you
were willing to pay for it
...
So equating utility with money is a
convenient simplifying assumption, allowing the theory to focus

A BEAUTIFUL MATH

on the strategic aspects of how to achieve what you want, without
worrying about the complications involved in defining what you
want
...
Was it even possible, in the first place, to define utility in a numerical way, to make
it susceptible to a mathematical theory? (Bernoulli had proposed a
way to calculate utility, but he had not tried to prove that the
concept could be a basis for making rational choices in a consistent
way
...
And so they had to
show that it was possible to define utility in a mathematically rigorous way
...

As it turned out, utility could be quantified in a way not unlike
the approach physicists used to construct a scientifically rigorous
definition of temperature
...
Utility, or preference, can be thought of as
just a rank ordering
...
But it is not so obvious that you can ascribe a number to how much you prefer A to B, or B to C
...
But nowadays the absolute temperature scale, based on
the laws of thermodynamics, gives temperature an exact quantitative meaning
...

You can get the essence of the method from playing a modified version of Let’s Make a Deal
...
) Suppose Monty
offers you three choices: a BMW convertible, a top-of-the-line big-

VON NEUMANN’S GAMES

screen plasma TV, or a used tricycle
...
So it’s a
simple matter to rank the relative utility of the three products
...
Your choice is to get either the plasma TV,
OR a 50-50 chance of getting the BMW
...
The other door conceals the tricycle
...
If you choose Door Number
1—the plasma TV—you must value it at more than 50 percent as
much as the BMW
...
At some point you will be likely to opt
for the chance to get the BMW, and at that point, you could conclude that the utilities are numerically equal—you value the plasma
TV at, say, 75 percent as much as the BMW (plus 25 percent of the
tricycle, to be technically precise)
...

So far so good
...
And in a smallscale Gilligan’s Island economy, pure strategic choices can be
subverted by things like coalitions among some of the players
...

Temperature is a measure of how fast molecules are moving
...
But you’d have a hard time with Gilligan’s Island, just as it
becomes virtually impossible to keep track of all the speeds of a
relatively few number of interacting molecules
...
(The math behind this is, of course, statistical
mechanics, which will become even more central to the game
theory story in later chapters
...
”21
That was exactly the point made by Asimov’s psychohistorians:
Even though you can’t track each individual molecule, you can
predict the aggregate behavior of vast numbers, precisely what
taking the temperature of a gas is all about
...
Why not do the
same for people? It worked for Hari Seldon
...
“When the number of participants
becomes really great,” von Neumann and Morgenstern wrote,
“some hope emerges that the influence of every particular participant will become negligible
...
The bulk of their book was then devoted to
the issue of finding the best strategy to make the most money
...
A strategy in game theory is a very specific
course of action, not a general approach to the game
...
” A game theory strategy is a defined set
of choices to make for every possible circ*mstance that might arise
...
” And you’d
have other rules for all the other situations
...
In tennis, you might rush the net after every serve (a
pure strategy) or you might rush the net after one out of every
three serves, staying back at the baseline two times out of three (a
mixed strategy)
...

In any event, the question isn’t whether there is always a good
general strategy, but whether there is always an optimum set of

VON NEUMANN’S GAMES

rules for strategic behavior that covers all eventualities
...
You can find the best
strategy using the minimax theorem that von Neumann published
in 1928
...

But its essence can be boiled down into something fairly easy to
remember: When playing poker, sometimes you need to bluff
...
So your strategy should
seek to maximize your winnings, which would have the effect of
minimizing your opponent’s winnings
...

Depending on the game, you may be able to play as well as
possible and still not win anything, of course
...
Still, it is likely that some
strategies will lose more than others, so you would attempt to minimize your opponent’s gains (and your losses)
...
Obviously, then, you would
play that strategy, and if the game is repeated, you would play the
same strategy every time
...
That’s
where game theory gets interesting
...
Say that Bob owes Alice
$10
...
(In the real world, Alice will tell Bob to take a hike and
fork over the $10
...

A BEAUTIFUL MATH

Bob suggests these rules: He and Alice will meet at the library
...
If they both arrive at the same time, he pays $5
...
)
Now, let’s say they live together, or at least live next door to
each other
...
(They are too poor to own
a car, which is why Bob is haggling over the $10
...
So this game is
trivial—both will take the bus, both will arrive at the same time,
and Bob will pay Alice $5
...
” The numbers show
how much the player on the left (Alice) wins
...
If the number is negative, that means the player on top
wins that much (negative numbers signaling a loss for Alice)
...

Obviously, Alice must choose the bus strategy because it always does as well as or better than walking, no matter what Bob

VON NEUMANN’S GAMES

does
...
Walking can do no better and
might be worse
...
So
let’s look at another example, from real-world warfare, a favorite
of game theory textbooks
...

The Allies naturally wanted to bomb the hell out of the convoy
...

Either route would take three days, so in principle the Allies
could get in three days’ worth of bombing time against the convoy
...
Forecasters said the northern
route would be rainy one of the days, limiting the bombing time
to a maximum of two days
...
General Kenney
had to decide whether to send his reconnaissance planes north or
south
...

If the recon planes went north, the bombers would still have time
to get two bombing days in if the convoy went south
...
But
from the Japanese side, you can easily see that going north is the
only move that makes sense
...
By going north, it would get a maximum of two days (and
maybe only one), as good as or better than any of the possibilities
going south
...
(The
Japanese did in fact take the northern route and suffered heavy
losses from the Allied bombers
...
Let’s
revisit Alice and Bob and see what happened after Alice refused to
play Bob’s stupid game
...

In Alice’s version of the game, they go the library every weekday for a month
...
If
they both walk, Bob pays Alice $4
...
If Bob walks and Alice rides
the bus, arriving first, Bob pays $6
...

This game puzzles Bob, too
...

Bob
Bus

Walk

Bus

Walk

Alice

Bob realizes there is no simple strategy for playing this game
...
But Alice, realizing
that, will probably walk, meaning Bob would have to pay her $5
...
But then
Alice might figure that out and ride the bus, so Bob would have to
pay her $6
...

Remember, however, that Alice required the game to be played
repeatedly, say for a total of 20 times
...
(If you did, that
would be a pure strategy—one that never varied
...
She wants to keep Bob guessing
...
So he will
take a mixed strategy approach also
...

In a two-person zero-sum game, you can always find a best
strategy—it’s just that in many cases the best strategy is a mixed
strategy
...
Remember, a mixed strategy is a mix
of pure strategies, each to be chosen a specific percentage of the
time (or in other words, with a specific probability)
...
25 Following the book’s advice, he compares
the payoffs for each choice when Alice walks (the first row of the
matrix) to the values when Alice takes the bus (the second row of
the matrix), subtracting the payoffs in the second row from those
in the first
...
) Those two numbers determine the best ratio for Bob’s two
strategies—2:2, or 50-50
...
It just so happened that in this
case the numbers are equal
...
So she should play the first strategy (bus) three times as often as the second strategy (walk)
...
Bob should ride the bus half the
time and walk half the time
...
Bob could
just flip a coin; Alice might use a random number table, or a game
spinner with three-fourths of the pie allocated to walking and
one-fourth to the bus
...

So you have to keep your opponent guessing
...

If you always raise when dealt a good hand but never when dealt a
poor hand, your opponents will be able to figure out what kind of
a hand you hold
...
But consider a simple two-player version of poker, where Bob
and Alice are each dealt a single card, and black always beats red
...
Alice then plays first, and she may either fold or bet an
additional $3
...
(If both have black or
both have red, they split the pot
...
If he
folds, Alice takes the $13 in the pot; if he calls, they turn over their
cards to see who wins the $16
...
But if she bets, Bob might
think she must have black
...
Bluffing sometimes pays off
...

The question is, how often should Alice bluff, and how often
should Bob call her (possible) bluff ? Maybe von Neumann could
have figured that out in his head, but I think most people would
need game theory
...
Alice can always pass, always bet, pass
with red and bet with black, or bet with red and pass with black
...
If you calculate the
payoffs, you will see that Alice should bet three-fifths of the time
no matter what card she has; the other two-fifths of the time she
should bet only if she has black
...
29 (By the way, another thing game theory can show you
is that this game is stacked in favor of Alice, if she always goes
first
...
)
The notion of a mixed strategy, using some random method to
choose from among the various pure strategies, is the essence of
von Neumann’s proof of the minimax theorem
...

If your opponent doesn’t know game theory, you might do even
better
...
It was about making strategic
decisions—whether in the economy or in any other realm of real
life
...
If you know what outcome you want, game
theory dictates the proper strategy for achieving it
...

In their book, von Neumann and Morgenstern did not speak

A BEAUTIFUL MATH

of a “Code of Nature,” but did allude to game theory as a description of “order of society” or “standard of behavior” in a social
organization
...
“The
mathematical theory of games of strategy,” they wrote, “gains definitely in plausibility by the correspondence which exists between
its concepts and those of social organizations
...
You can find
examples of two-person zero-sum games in real life, but they are
typically either so simple that you don’t need game theory to tell
you what to do, or so complicated that game theory can’t incorporate all the considerations
...
So it’s
no surprise that in applying game theory to situations more complicated than the two-person zero-sum game, von Neumann and
Morgenstern were not entirely successful
...

Nash’s Equilibrium
Game theory’s foundation
Nash’s theory of noncooperative games should now
be recognized as one of the outstanding intellectual
advances of the twentieth century
...

—economist Roger Myerson

As letters of recommendation go, it was not very elaborate, just a
single sentence: “This man is a genius
...
L
...
Within two years,
Duffin’s assessment had been verified
...

Shortly before Nash’s arrival at Princeton, von Neumann and
Morgenstern had opened a whole new continent for mathematical
exploration with the groundbreaking book Theory of Games and
Economic Behavior
...

Nash played the role of Lewis and Clark
...
But before his prolonged departure, Nash successfully steered game theory toward
the mathematical equivalent of manifest destiny
...
By then game theory
had also conquered evolutionary biology and invaded political science, psychology, and sociology
...

There is no doubt that game theory’s wide application throughout
the intellectual world was made possible by Nash’s math
...
“The theory of noncooperative games that
Nash founded has developed into a practical calculus of incentives
that can help us to better understand the problems of conflict
and cooperation in virtually any social, political, or economic
institution
...
But of course it’s not as simple as that
...
Today it is worshiped by some but still ridiculed by others
...
In any event,
game theory has assumed such a prominent role in so many realms
of science that it can no longer intelligently be ignored, as it often
was in its early days
...
But most economists remained dry
...
“The book has accomplished everything except what it started out to do—namely, revolutionize economic theory,” Samuelson wrote
...
In the years
following its publication, Theory of Games and Economic Behavior
was widely reviewed in social science and economics journals
...
”3
“The potentialities of von Neumann’s and Morgenstern’s new approach seem tremendous and may, one hopes, lead to revamping,
and enriching in realism, a good deal of economic theory,”
Hurwicz wrote
...
”4 A more
enthusiastic assessment appeared in a mathematics journal, where a
reviewer wrote that “posterity may regard this book as one of the
major scientific achievements of the first half of the twentieth
century
...
In
1946, the von Neumann–Morgenstern book rated a front page
story in the New York Times; three years later a major piece appeared in Fortune magazine
...
“The techniques applied by the authors in tackling economic problems are of sufficient generality to be valid in political
science, sociology, or even military strategy,” Hurwicz pointed out
in his review
...
“The student of the Theory of Games
...
of the theory into a
fundamental tool of analysis for the social sciences
...
Von Neumann had mastered twoperson zero-sum games, but introducing multiple players led to

A BEAUTIFUL MATH

problems
...

Von Neumann’s approach to multiple-player games was to assume that coalitions would form
...
Many players
might be involved, but if they formed two teams, the teams could
take the place of individual players in the mathematical analysis
...
A key part of two-person zero-sum games was choosing a
strategy that was the best you could do against a smart opponent
...
But if coalitions formed among players in many-person games, as von Neumann believed they would,
that meant your strategy would in fact depend on coordinating it
with at least some of the other players
...
And that’s what
John Nash provided
...
8 But the book reveals a lot
about Nash’s personal troubles
...
He is depicted as immature, selfcentered, arrogant, uncaring, and oblivious
...

Nash was born in West Virginia, in the coal-mining town of
Bluefield, in 1928
...
But by the
time he enrolled at Carnegie Tech (the Carnegie Institute of Tech-

NASH’S EQUILIBRIUM

nology) in Pittsburgh, his choice for major had become chemical
engineering
...
Finding no joy in manipulating laboratory apparatus, Nash
turned to math, where he excelled
...
In
that class Nash conceived the idea for a paper on what came to be
called the “bargaining problem
...
Nevertheless the mathematical principles involved were clearly relevant to more sophisticated economic situations
...
Von Neumann was at the Institute for Advanced Study, just a mile from the university, and
Morgenstern was in the Princeton economics department
...
Nash himself attended a game
theory seminar led by Albert W
...

Shortly after his arrival, Nash realized that his undergraduate
ideas about the “bargaining problem” represented a major new
game theory insight
...

“Bargaining” represents a different form of game theory in
which the players share some common concerns
...
In
this “cooperative” game theory, the goal is for all players to do the
best they can, but not necessarily at the expense of the others
...
A typical real-life bargaining situation would be negotiations between a corporation and a labor
union
...
The problem is to find which way maximizes the benefit
(or utility) for both sides—given that both players are rational (and
know how to quantify their desires), are equally skilled bargainers,
and are equally knowledgeable about each other’s desires
...

(To the athletic minded, a bat might seem more valuable than a
book, while the more intellectually oriented bargainer might rank
the book more valuable than the bat
...
9

SEEKING EQUILIBRIUM
Nash’s bargaining problem paper would in itself have established
him as one of game theory’s leading pioneers
...
It was the paper introducing the
“Nash equilibrium,” eventually to become game theory’s most
prominent pillar
...
Equilibrium means things are in balance,
or stable
...
Biological systems, chemical and
physical systems, even social systems all seek stability
...
If a situation is unstable—as many often are—you can predict
the future course of events by figuring out what needs to happen
to achieve stability
...

NASH’S EQUILIBRIUM

The simplest example is a rock balanced atop a sharply peaked
hill
...
Another common example
of equilibrium shows up when you try to dissolve too much sugar
in a glass of iced tea
...
When the solution reaches equilibrium, molecules will
continue to dissolve out of the pile, but at the same rate as other
sugar molecules drop out of the tea and join the pile
...

It’s the same principle, just a little more complicated, in a
chemical reaction, where stability means achieving a state of
“chemical equilibrium,” in which the amounts of the reacting
chemicals and their products remain constant
...
But it’s often not the case that both original substances
will entirely disappear, leaving only the new one
...
But eventually you may reach a point where the
amount of each substance doesn’t change
...
In
other words, the action continues, but the big picture doesn’t
change
...
Nash had just
this sort of physical equilibrium in mind when he was contemplating stability in game theory
...
10
When equilibrium is reached in a chemical reaction, the quantities of the chemicals no longer change; when equilibrium is
reached in a game, nobody has any incentive to change strategies—so the choice of strategies should remain constant (the game
situation is, in other words, stable)
...
Similarly, in social situations, stability means that everybody is pretty much content with the status quo
...
There’s no impetus for change, so like a rock in
a valley, the situation is at an equilibrium point
...
Whether
using a pure strategy or a mixed strategy, neither player has anything to gain by deviating from the optimum strategy that game
theory prescribes
...
And as you’ll recall, von
Neumann thought the way to analyze large economies (or games)
was by considering coalitions among the players
...
Suppose there are no coalitions, no cooperation among the players
...
Is there always
a set of strategies that makes the game stable, giving each player
the best possible personal payoff (assuming everybody chooses
the best available strategy)? Nash answered yes
...

Nash derived his proof in different ways, using either of two
fixed-point theorems—one by Luitzen Brouwer, the other by
Shizuo Kakutani
...
Take two identical sheets of paper,
crumple one up, and place it on top of the other
...
That’s the fixed point
...
(The map represents the crumpled up piece of paper
...
Applying the same
principle to the players in a game, Nash showed that there was
always at least one “stable” point where competing players’ strategies would be at equilibrium
...
D
...

such that each player’s mixed strategy maximizes his payoff if the
strategies of the others are held fixed
...
To put it more colloquially, says economist Robert
Weber, you could say that “the Nash equilibrium tells us what we
might expect to see in a world where no one does anything
wrong
...
”13
Von Neumann was dismissive of Nash’s result, as it did turn
game theory in a different direction
...
“The concept of the Nash
equilibrium is probably the single most fundamental concept in
game theory,” declared Bowles
...
”14

GAME THEORY GROWS UP
Nash published his equilibrium idea quickly
...
Titled “Equilibrium Points in n-Person Games,” the paper established concisely (although not particularly clearly for
nonmathematicians) the existence of “solutions” to many-player
games (a solution being a set of strategies such that no single
player could expect to do any better by unilaterally trying a different strategy)
...
D
...
”

A BEAUTIFUL MATH

Von Neumann and Morgenstern, Nash politely noted in his
paper, had produced a “very fruitful” theory of two-person zerosum games
...
“Our
theory, in contradistinction, is based on the absence of coalitions in
that it is assumed that each participant acts independently, without
collaboration or communication with any of the others
...
When you think about it, that approach pretty much sums
up many social situations
...

“The distinction between non-cooperative and cooperative games
that Nash made is decisive to this day,” wrote game theorist Harold
Kuhn
...
In the natural world, everything
seeks stability, which means seeking a state of minimum energy
...
It’s because of the law of gravity
...
It’s because of the laws of thermodynamics
...
A chemical reaction reaches an equilibrium enforced by the laws of thermodynamics; an economy should reach a Nash equilibrium dictated by
game theory
...
There are usually
complicating factors
...
When people are involved, all sorts of new sources of
unpredictability complicate the game theory playing field
...
18 )
Nevertheless, Nash’s notion of equilibrium captures a critical
feature of the social world
...
So if you want to apply
game theory to real life, you need to devise a game that captures
the essential features of the real-life situation you’re interested in
...

Consequently game theorists have invented more games than
you can buy at Toys R Us
...
There’s the stag hunt
game, the ultimatum game, and the “so long sucker” game
...
But by far the most famous of all such games
is a diabolical scenario known as the Prisoner’s Dilemma
...
In “The Mystery of Marie Roget,” Poe described
a murder believed by Detective Dupin to have been committed by
a gang
...
“Each one of a gang, so
placed, is not so much
...
“He betrays eagerly and early that
he may not himself be betrayed
...

As it happened, the Prisoner’s Dilemma in game theory was
first described by Nash’s Princeton professor, Albert W
...
At that time, Tucker was visiting Stanford and had men-

A BEAUTIFUL MATH

tioned his game theory interests
...
20
You know the story
...
So if both keep mum, both will get a year in prison
...
If only one squeals, the
partner gets five years
...

Alice
Keep Mum

Rat

Keep Mum

1, 1

5, 0

Rat

0, 5

3, 3

Bob

Years in prison for Bob, Alice

If you look at this game matrix, you can easily see where the
Nash equilibrium is
...
Think about it
...
The police shine a light in Bob’s face and spell out the
terms of the game
...
He ponders what
Alice might do
...
But suppose Alice keeps mum
...
No matter which strategy Alice chooses, Bob’s best choice is betrayal,
just as Poe’s detective had intuited
...
The only stable outcome is for both
to agree to testify, ratting out their accomplice
...
But they are
interrogated separately, with no communication between them permitted
...
If they both keep mum (that
is, they cooperate with each other), they spend a total of two years
in prison (one each)
...
But when they rat each other out, they
serve a total of six years—a worse overall outcome than any of the
other pairs of strategies
...
From the standpoint of game theory and Nash’s math, the
choice is clear
...

In real life, of course, you never know what will happen, because the crooks may have additional considerations (such as the
prospect of sleeping with the fishes if they rat out the wrong guy)
...
Sometimes people temper
their choices with considerations of fairness, and sometimes they
act out of spite
...
But that doesn’t detract from the
importance of the Nash equilibrium, as economists Charles Holt
and Alvin Roth point out
...
” So if people cooperate (at least
at first) in a Prisoner’s Dilemma situation, Nash’s math tells us that
such cooperation, “because it is not an equilibrium, is going to be
unstable in ways that can make cooperation difficult to maintain
...
But obviously you cannot easily assess any social situation by calculating the Nash equilibrium
...
While Nash
showed that there is always at least one equilibrium point, it’s another matter to figure out what that point is
...
) Remember, each player’s “strategy” will typically be a mixed
strategy, drawn from maybe dozens or hundreds or thousands (or
more) of pure “specific” strategies
...

THE PUBLIC GOOD
It’s not hopeless, though
...
The idea is that some
members in a community reap the benefits of membership without paying their dues
...
At first glance,
the defector wins this game—getting the benefit of enjoying
Morse and Poirot without paying a price
...
If
everybody defected, there would be no benefit for anybody
...

Similarly, suppose your neighborhood association decided to
collect donations to create a park
...
If
everybody reasons the same way, though, there will be no park
...
You
can imagine a third strategy, called reciprocating
...
Computer simulations of this
kind of game suggest that a mix of these strategies among the
players can reach a Nash equilibrium
...
One study,
reported in 2005, tested college students on a contrived version of
the public goods game
...

The experimenter then doubled the number of tokens in the pot
...

When the game ended (after a random number of rounds), all the
tokens were then evenly divided up among all the players
...
Of course, if nobody put
any in to begin with, nobody reaped the benefit of the experimenter’s largesse, kind of like a local government forgoing federal
matching funds for a highway project
...
But if you want to get
a better payoff than anyone else, you should put in less than the
others
...
On the other hand, everybody in the
group will get more if you put more in the pot to begin with
...
)
When groups of four played this game repeatedly, a pattern of
behavior emerged
...
Since all
the players learned at some point how much had been contributed,
they could adjust their behavior accordingly
...

Over time, the members of each group earned equal amounts
of money, suggesting that something like a Nash equilibrium had
been achieved—they all won as much as they could, given the
strategy of the others
...
“Our results support the view that our
human subject population is in a stable
...
22
Knowing about the Nash equilibrium helps make sense of results
like these
...
There’s more to
game theory than the Nash equilibrium, of course, but it is still at
the heart of current endeavors to apply game theory to society
broadly
...
Models of all of these situations, plus many others, can be built using game theory’s complex
mathematical tools
...
It’s not necessary to know all those
details of game theory history, but it is important to know that
game theory does have a rich and complex history
...

Even today game theory remains very much a work in progress
...
In fact, if you peruse the various accounts of
game theory, you are likely to come away confused
...

Some presentations seem to suggest that game theory should
predict human behavior—what choices people will make in games

NASH’S EQUILIBRIUM

(or in economics or other realms of life)
...
Or some experts will say that game theory
predicts what a “rational” person will do, acknowledging that
there’s no accounting for how irrational some people (even those
playing high-stakes games) can be
...

To me, it seems obvious that basic game theory does not always successfully predict what people will do, since most people
are about as rational as pi
...
There may always be additional considerations in making a
“rational” choice that have not been included in game theory’s
mathematical framework
...
In principle you could use game theory
to analyze lots of ordinary games, like checkers, as well as many
problems in the real world where the concept of game is much
broader
...
The idea is that when faced
with deciding what to do in some strategic interaction, the math
can tell you which move is most likely to be successful
...

The question is, are there ever any such circ*mstances? Early
euphoria about game theory’s potential to illuminate social issues
soon dissipated, as a famous game theory text noted in 1957
...
This has not turned out to be the case
...
There’s
always a lack of patience in the scientific world; many people want
new ideas to pay off quickly, even when more rational observers

A BEAUTIFUL MATH

realize that decades of difficult work may be needed for a theory
to reach maturity
...

In an afterword to the 60-year-anniversary edition of Theory
of Games, Ariel Rubenstein acknowledged that game theory had
successfully entrenched itself in economic science
...
“The distinction between economic theorist and game theorist has virtually disappeared
...
“Game theory is not a box of magic tricks that can help us
play games more successfully
...
25
He scoffed at theorists who believed game theory could actually predict behavior, or even improve performance in real-life strategic interactions
...
“The fact that the academics
have a vested interest in it makes it even less credible
...
“Game theory does not tell us which action is preferable or
predict what other people will do
...
”26
OK—maybe this book should end here
...
I think
Rubenstein has a point, but also that he is taking a very narrow
view
...

Scientists make models
...
Game theory is all about making models
of human interactions
...
No map of Los
Angeles shows every building, every crack in every sidewalk, or

NASH’S EQUILIBRIUM

every pothole—if it showed all that, it wouldn’t be a map of Los
Angeles, it would be Los Angeles
...
A
...

Naturally, game theory introduces simplifications—it is, after
all, a model of real-life situations, not real life itself
...
You don’t have to worry about the chemical composition of
the moon and sun when predicting eclipses, only their masses and
motions
...
The atmosphere is a physical system, but Isaac Newton was no meteorologist
...
But after a few centuries, physics
did get to the point where it could offer reasonably decent weather
forecasts
...

In his book Behavioral Game Theory, Colin Camerer addresses
these issues with exceptional insight and eloquence
...
But it’s clearly a mistake
to think that therefore there is something wrong with game
theory’s math
...
27 Besides, game theory (in its
original form) is based on players’ behaving rationally and selfishly
...
In that case, incorporating better knowledge of human psychology (especially in social situations) into
game theory’s equations can dramatically improve predictions of
human behavior and help explain why that behavior is sometimes
surprising
...
“The goal is not
to ‘disprove’ game theory
...
28

A BEAUTIFUL MATH

As it turns out, game theory is widely used today in scientific
efforts to understand all sorts of things
...

Economist Thomas Schelling, of the University of Maryland, understood in the 1950s that game theory offered a mathematical
language suitable for unifying the social sciences, a vision he articulated in his 1960 book The Strategy of Conflict
...
29
Schelling paid particular attention to game-theoretic analysis
of international relations, specifically (not surprising for the time)
focusing on the risks of armed conflict
...
And he identified various counterintuitive conclusions about
conflict strategy that game theory revealed
...
But the signal sent to the enemy—that the
oncoming army had no way to retreat—would likely diminish the
opposition’s willingness to fight
...

Schelling’s insights also extended to games where all the players desire a common (coordinated) outcome more than any particular outcome—in other words, when it is better for everybody
to be on the same page, regardless of what the page is
...
It doesn’t matter what restaurant (as long as the
food is not too spicy); the goal is for everyone to be together
...
Schelling shed considerable
light on the game-theoretic issues involved in reaching coordinated solutions to such social problems
...

2005’s other economics Nobel winner, Robert Aumann, has
long been a leading force in expanding the scope of game theory
to many disciplines, from biology to mathematics
...
In particular, Aumann
analyzed the Prisoner’s Dilemma game from the perspective of
infinitely repeated play, rather than the one-shot deal in which
both players’ best move is to rat the other out
...

Aumann’s “repeated games” approach had wide application,
both in cases where it led to cooperation and where it didn’t
...
Game theory helps to
show why certain common forms of collective behavior materialize under such circ*mstances
...

While Nobel Prizes shine the media spotlight on specific
achievements of game theory, they tell only a small portion of the
whole story
...
Economics is full of applications, from guiding
negotiations between labor unions and management to auctioning

A BEAUTIFUL MATH

licenses for exploiting the electromagnetic spectrum
...
Game theory is valuable for understanding
terrorist organizations and forecasting terrorist strategies
...
You can call on game theory to explain why the
numbers of male and female births are roughly equal, why people
get stingier as they get older, and why people like to gossip about
other people
...
For it is in biology that game theory has demonstrated its
power most dramatically, in explaining otherwise mysterious outcomes of Darwinian evolution
...

Smith’s Strategies
Evolution, altruism, and cooperation
The stunning variety of life forms that surround us, as
well as the beliefs, practices, techniques, and behavioral forms that constitute human culture, are the
product of evolutionary dynamics
...

—Herbert Gintis, Game Theory Evolving

In the winter of 1979, Cambridge University biologist David
Harper decided it would be fun to feed the ducks
...

Daily foraging is important for ducks, as they must maintain a
minimum weight for low-stress flying
...
They
therefore ought to be good at finding food fast, in order to maintain an eat-as-you-go lifestyle
...
So he cut up some white bread
into precisely weighed pieces and enlisted some friends to toss the
pieces onto the pond
...
But then Harper’s helpers
began tossing the bread onto two separated patches of the pond
...
The second was slower, tossing out the bread balls
just once every 10 seconds
...
When I ask
people what they would do, I inevitably get a mix of answers (and
some keep changing their mind as they think about it longer)
...
But all the other ducks
might have the same idea
...
So the choice of the optimum strategy isn’t immediately obvious, even for people
...

After all, foraging for food is a lot like a game
...
You want to get as much as you
can
...
As these were university ducks, they
were no doubt aware that there is a Nash equilibrium point,
an arrangement that gets every duck the most food possible when
all the other ducks are also pursuing a maximum food-getting
strategy
...
In this case the calculation is pretty simple: The ducks all get their best possible deal if
one-third of them stand in front of the slow tosser and the other
two-thirds stand in front of the fast tosser
...
They split into two groups almost precisely the size that
game theory predicted
...
Even
then, the ducks eventually sorted themselves into the group sizes
that Nash equilibrium required, although it took a little longer
...
Game theory
was designed to describe how “rational” humans would maximize
their utility
...
2 The duck experiment shows, I think, that there’s
more to game theory than meets the eye
...
Game theory captures
something about how the world works
...
And it was in fact the realization
that game theory describes biology that gave it its first major scientific successes
...
Many experts believe that it explains the mystery of human cooperation, how civilization itself could emerge
from individuals observing the laws of the jungle
...

LIFE AND MATH
I learned about evolution and game theory by visiting the Institute
of Advanced Study in Princeton, home of von Neumann during
game theory’s infancy
...
By the late 1990s, though, the institute had decided to
plunge into the 21st century a little early by initiating a program in
theoretical biology
...
Nowak
was an accomplished mathematical biologist who had mixed bio-

A BEAUTIFUL MATH

chemistry with math during his student years at the University of
Vienna, where he earned his doctorate in 1988
...
I visited him in Princeton in the fall of 1998 to
inquire about the institute’s plans for mixing math with the science
of life
...
Underlying much of his work was a common
theme that at the time I really didn’t appreciate: the pervasive relevance of game theory
...
In biology almost everything involves interaction
...
There
are the fierce interactions of immune system cells battling viruses,
or toxic molecules tangling with DNA to cause cancer
...

Evolutionary processes shape the way that such interactions
occur and what their outcomes will be
...
Evolution is about virtually everything in biology—
the physiology of individuals, the diversity of appearances within
groups, the distribution of species in an ecosystem, and the behavior of individuals in response to other individuals or groups interacting with other groups
...
“Game theory has been very successfully used in evolution,” Nowak told me
...
”3
In particular, game theory helps explain the evolution of social
behavior in the animal (including humans) kingdom, solving a perplexing mystery in the original formulation of Darwinism: Why
do animals cooperate? You’d think that the struggle to survive
would put a premium on selfishness
...
Human civilization could never have
developed as it has without such widespread cooperation; finding
the Code of Nature describing human social behavior will not be
possible without understanding how that cooperation evolved
...

GAMES OF LIFE
In the 1960s, even before most economists took game theory seriously, several biologists noticed that it might prove useful in explaining aspects of evolution
...

He was “an approachable man with unruly white hair and
thick glasses,” one of his obituaries noted, “remembered by colleagues and friends as a charismatic speaker but deadly debater, a
lover of nature and an avid gardener, and a man who enjoyed
nothing better than discussing scientific ideas with young researchers over a glass of beer in a pub
...
He died in 2004
...
As a child, he enjoyed collecting beetles and bird-watching, foreshadowing his future biological interests
...
During World War II he did engineering research on airplane stability,
but after the war he returned to biology, studying zoology under
the famed J
...
S
...

In the early 1970s, Maynard Smith received a paper to review
that had been submitted to the journal Nature by an American researcher named George Price
...
Price’s paper was too long for Nature, but the issue remained in the back of Maynard Smith’s mind
...
5
Eventually, Maynard Smith showed that game theory could
illuminate how organisms adopt different strategies to survive the
slings and arrows of ecological fortune and produce offspring to
carry the battle on to future generations
...
All animals participate; so do plants, so do bacteria
...
Is it a better strategy to be a short tree or a tall
tree? To be a super speedy quadruped or a slower but smarter
biped? Animals don’t choose their strategies so much as they are
their strategies
...
If every animal (plant,
bug) is a different strategy, then why are there so many different
forms of life out there, why so many different strategies for surviving? Why isn’t there one best strategy? Why doesn’t one outperform all the others, making it the sole survivor, the winner of the
ultimate fitness sweepstakes? Darwin, of course, had dealt with
that issue, explaining how different kinds of survival advantages
could be exploited by natural selection to diversify life into a smorgasbord of species (like the specialization of workers in Adam
Smith’s pin factory)
...

In doing so, Maynard Smith perceived the need to modify classical game theory in two ways: substituting the evolutionary ideas
of “fitness” for utility and “natural selection” for rationality
...
” In biology, though, “fitness, or
expected number of offspring, may be difficult to measure, but it
is unambiguous
...
”6 And “rationality” as a strategy for human game players exhibits two “snags,” Maynard Smith noted: “It is hard to decide
what is rational, and in any case people do not behave rationally
...
”7
To illustrate his insight, he invented a clever but simple
animal-fighting game
...

Imagine such a world, a “bird planet” populated solely by birds
...
Now suppose these birds all decide that being hawkish is
the best survival strategy
...
But even the winner may
suffer some injuries, incurring a cost that diminishes its benefits
from getting the food
...
well, for the birds
...

Upon encountering some food, he eats only if no other bird is
around
...

The dove might miss a few meals, but at least he’s not losing his
feathers in fights
...
When they meet each other, they share the food
...

Consequently, Maynard Smith noted, an all-hawk population
is not an “evolutionary stable strategy
...
On the other hand, it is equally true
that an all-dove society is not stable, either
...
Only when more hawks begin to appear
will there be any danger of dying in a fight
...
If hawks are rare, a
hawkish strategy is best because most of the opponents will be
doves and will run from a fight
...
So a society should evolve to include a mix of
hawks and doves
...
Maynard Smith showed how game theory described this situation perfectly, with an evolutionary stable strategy
being the biological counterpart of a Nash equilibrium
...
In many sorts of
games there can be more than one Nash equilibrium, and some of
them may not be evolutionary stable strategies
...
Such an ecosystem would not be evolutionarily stable
...
In any case, the
birds have to choose to play hawk or dove just as the ducks had to
decide which bread tosser to favor
...

Exactly what those percentages are depends on the precise costs
of fighting compared to the food you miss by fleeing
...
If Bird 1 is a hawk and Bird 2 a
dove, the dove flies away and gets 0, the hawk gets all the food (2)
...

(Or you could say that one dove defers to the other half the time,
the 1 point each signifying a 50-50 chance of either bird getting
the food
...
9 (Keep in mind that, mathematically, you could have a mix of hawks and doves, or just birds that
play mixed strategies
...
)10
Obviously this is a rather simplified view of biology
...
But you can see the basic idea, and you should also be
able to see how game theory could describe situations with added
complexity
...
In fact, like human boxing or football fans, some birds do
like to watch the gladiators of their group slug it out in a good
fight (as do certain fishes)
...

Spectating may be wired into animal genes by evolutionary history, and maybe game theory has something to do with it
...
But
you don’t have to be a spectator to avoid the danger of a fight
...
So why
watch? The answer emerges naturally from game theory
...

Face it: You can’t always run from a fight
...
On the other hand, looking for a
fight at every opportunity is not so smart, either—the battle may

A BEAUTIFUL MATH

exact a greater cost than the benefit of acquiring the resource
...
The observers (or “eavesdroppers” in
biolingo) could choose to be either a hawk or a dove when it’s
their turn to fight—depending on what they’ve observed about
their adversary
...
In this game, the eavesdropper knows
whether its opponent has won or lost its previous fight
...

“An individual that is victorious in one round is more likely to
win in the next, because its opponent is less likely to mount an
escalated challenge,” Johnstone concluded
...
Alas, the math shows otherwise
...

Why? Because of the presence of spectators! If nobody is
watching, it is not so bad to be a dove
...
With spectators around, acting like a dove guarantees that you’ll face an aggressive opponent in your next fight
...

So the presence of spectators encourages violence, and watching violence today offers an advantage for the spectators who may
be fighters tomorrow
...

But don’t forget that adding spectators is just one of many

SMITH’S STRATEGIES

complications that could be considered in the still very simplified
hawk-dove game
...

Size and skill come into play as well
...
If the birds know their own skill levels accurately, overall fighting might be diminished
...
)12
In any case, policy makers who would feel justified in advocating wars based on game theory should pause and realize that real
life is more complicated than biologists’ mathematical games
...
And in
fact, game theory can help show how that civilized state
came about
...
Without game theory, cooperative
human social behavior is hard to understand
...
Even more important, game theory helps to
show how the best strategies might differ as circ*mstances change
...

When evolutionists talk about circ*mstances changing, typically they’ll be referring to something like the climate, or the
trauma of a recent asteroid impact
...
And that’s why
game theory is essential for understanding evolution
...
In
other words, the best survival strategy depends on who else is
around and how they are behaving
...

A BEAUTIFUL MATH

Using the language of evolution, success in the survival game
equates to “fitness
...
Obviously
some individuals score better in this game than others
...
Using this metaphor, you can
think of fitness—or the goal of a game—as getting a good vantage point, living on the peak of a mountain with a good view of
your surroundings
...
Some latitude–longitude positions will put you on high
ground; some will leave you in a chasm
...
It’s just another way of saying
that some combinations of features and behaviors improve your
chance to survive and reproduce
...

In a fitness landscape (just like a real landscape) there can, of
course, be more than one peak—more than one combination of
properties with a high likelihood for having viable offspring
...
) In a landscape with many fitness peaks, some
would be “higher” than others (meaning your odds of reproducing
are more favorable), but still many peaks would be good enough
for a species to survive
...
A natural disaster—a hurricane like Katrina,
say, or an earthquake and tsunami—can literally reshape the landscape, and a latitude and longitude that previously gave you a great
view may now be a muddy rut
...
Something like this seems to be what happened to the
dinosaurs
...
Simply suppose that some new species
moves into the neighborhood
...
So as evolution

SMITH’S STRATEGIES

proceeds, the fitness landscape changes
...
No species is a Robinson Crusoe alone on an island
...

Recognizing this ever-shifting evolution landscape is the key
to explaining how cooperative behavior comes about
...

KIN AND COOPERATION
It’s not that nonhuman animals never cooperate
...
But such social insect societies can easily be explained by
evolution’s basis in genetic inheritance
...
By cooperating they enhance the prospect
that their shared genes will be passed along to future colonies
...
As Maynard Smith’s teacher J
...
S
...
(On average, you
share one-half of a sibling’s genes, one-eighth of a cousin’s
...
Somehow, humans evolved to cooperate with strangers
...
The other was language
...
“One is that they have a language which allows us
to talk about everything
...
Animals can talk about a lot
of things and signal about a lot of things to each other, but it
seems that they are limited to a certain finite number of things that
they can actually tell each other
...
“There must have
been a transition in evolution,” Nowak said, that allowed humans
to develop this “infinite” communication system
...
“Humans are the only species
that have solved the problem of large-scale cooperation between
nonrelated individuals,” Nowak pointed out
...
”13
Charles Darwin himself noted this “altruism” problem
...
But humans (many of them, at least) possess a compelling instinct to be helpful
...
(He was the baseball manager of
the mid-20th century who was famous for saying “Nice guys finish last
...
If you help out your
neighbor, maybe someday your neighbor will return the favor
...
”) But that explanation
doesn’t take you very far
...
Yet people often help
others whom they will probably never see again
...
Suppose you help out a stranger whom you never
see again, but that stranger—overwhelmed by your kindness—
becomes a traveling Good Samaritan, rendering aid to all sorts of
disadvantaged souls
...

Such “indirect reciprocity,” Nowak told me, had been mentioned long ago by the biologist Richard Alexander but was generally dismissed by evolutionary biologists
...
Nowak, though, had explored the idea
of indirect reciprocity in detail with the mathematician Karl
Sigmund in Vienna
...
The secret to altruism, Nowak suggested, is the
power of reputation
...
”
The importance of reputation explains why human language
became important—so people could gossip
...

“It’s interesting how much time humans spend talking about other
people, as though they were constantly evaluating the reputations
of other people,” Nowak said
...
A cooperative population makes language more important
...
Language is essential for
this
...
In the
Prisoner’s Dilemma game, for instance, both players come out
ahead if they cooperate
...
In a one-shot game against
an unknown opponent, the smart play is to defect
...
In
situations where the game is played repeatedly, cooperation offers
the added benefit of enhancing your reputation
...
Working out the math to prove that indirect
reciprocity can infuse a large society with altruistic behavior turned

A BEAUTIFUL MATH

up some problems
...
When I
next encountered Nowak, in 2004 at a complexity conference in
Boston, his story had grown more elaborate
...
The essential
backdrop was a famous game theory tournament held in 1980,
organized by the political scientist Robert Axelrod at the University of Michigan
...
He invited game theory experts to submit a strategy for
playing Prisoner’s Dilemma (in the form of a computer program)
and then let the programs battle it out in a round-robin competition
...

Of the 14 strategies submitted, the winner was the simplest—
an imitative approach called tit for tat, submitted by the game theorist Anatol Rapoport
...
After that, the player
does whatever its opponent did in the preceding round
...
Whenever
the opponent defects, though, the tit-for-tat player defects on the
next play and continues to defect until the opponent cooperates
again
...
But in a large number of rounds versus
many different opposition strategies, tit for tat outperforms the
others on average
...

Once tit for tat emerged as the winner, it seemed possible that
even better strategies might be developed
...
Of the contestants
in the second tournament, only one entered tit for tat
...

You can see how playing tit for tat enhances opportunities for

SMITH’S STRATEGIES

cooperation in a society
...
And if they don’t, you won’t
...
Just because tit for
tat won Axelrod’s tournament, that doesn’t mean it’s the best strategy in the real world
...

In his talk at the conference, Nowak explored some of the
nuances of the tit-for-tat strategy in a broader context
...
The
mathematics of evolutionary game theory, based on analyzing an
infinitely large population, seems to confirm that expectation
...

But if you keep calculating what would happen if the game
continues, it gets still more complicated
...
If you work out what would happen in such a
game, the tit-for-tat strategy becomes less successful than a modified strategy called “generous tit for tat
...

“Generous tit for tat is a strategy that starts with cooperation,
and I cooperate whenever you cooperate, but sometimes I will cooperate even when you defect,” Nowak explained
...
”16
As the games go on, the situation gets even more surprising,
Nowak said
...
” Oh Happy Days
...
As soon
as everybody cooperates, an always-defect strategy can invade, just
like a hawk among the doves, and clean up
...
“And this,” said Nowak, “is the theory of war and
peace in human history
...
If indirect reciprocity isn’t responsible for that cooperation, what is? Lately, one popular view
seems to be that cooperation thrives because it is enforced by the
threat of punishment
...

Among the advocates of this view are the economists Samuel
Bowles and Herbert Gintis and the anthropologist Robert Boyd
...
” A strong reciprocator rewards cooperators but punishes defectors
...
Rather than playing the
Prisoner’s Dilemma game—a series of one-on-one encounters—
strong reciprocity researchers conduct experiments with various
versions of public goods games
...
In a typical public goods game, players are given “points” at the outset (redeemable for real money later)
...
Then each player receives a fraction of the community fund
...
Altruistic players will share some of their points to increase
the payoff to the whole group
...
As we’ve seen, humankind comprises all three sorts

SMITH’S STRATEGIES

of players
...

In one such test of a public goods game,18 most players began
by giving up an average of half their points
...
In one test, nearly three-fourths
of the players donated nothing by round 10
...
That is to say, more of the players
became reciprocators
...
When low-amount donors were
ridiculed, the cheapskates coughed up more generous contributions in later rounds
...
Shame, apparently, induced improved behavior
...
So it may have been in the evolutionary past that
groups containing punishers—and thus more incentive for cooperation—outsurvived groups that did not practice punishment
...
(“Ingrained” might not be just in the genes,
though—many experts believe that culture transmits the punishment attitude down through the generations
...
Bowles
and Gintis have suggested that the punishment might have
consisted of ostracism, making the cost to the punisher relatively
low but still inflicting a significant cost on the noncooperator
...
The
human race plays a mixed strategy
...
I came across one paper
showing that, in fact, altruism could evolve solely through benefits
to the altruistic individual, not necessarily to the group, based on
simulations of yet another popular game
...
Behavioral game theorists believe that getting to
the roots of human social behavior—understanding the Code of
Nature—ultimately requires knowing what makes individuals tick
...
And the
popular way of doing that has spawned a hybrid discipline uniting
game theory, economics, psychology, and neuroscience in a controversial new discipline called neuroeconomics
...

—Sigmund Freud, Project for a Scientific Psychology, 1895

Sigmund Freud really wanted to understand the brain
...
He planned
to decipher the code linking the brain’s physical processes to the
mysteries of the mind
...
But Freud found the brain science of
the late 19th century too immature to link cranial chemistry to
thought and behavior
...

Others never even dreamed of achieving the “brain physics”
that Freud envisioned
...
These “behaviorists” decreed that psychology should stick to
observing behavior, studying stimulus and response
...
The black box concealing the brain turned translucent as molecular medicine revealed some of its inner workings
...

And so the infant neuroscience that Freud abandoned over a century ago has now matured, nearly to the point of fulfilling his
original intention
...

And even though they regarded game theory as a window into
human behavior, game theory’s originators themselves did not
imagine that their math would someday advance the cause of brain
science
...
1 But in the late 1990s, game theory turned out to be
just the right math for bringing neuroscience and economics together, in a new hybrid field known as neuroeconomics
...
To win a game, or survive in the
jungle, or succeed in business, you need to know how to play your
cards
...
You have to know when
to hold ’em and know when to fold ’em
...
Winners excel at making smart snap judgments
...

Animals know this
...
“Do I chase this new prey or do I continue nibbling on
my last kill?” Berns and Montague wrote in Neuron
...
Hesitation is bad for their health
...
Yet somehow animal brains add up all the
factors and compute a course of action that enhances the odds of
survival
...

Brains have evolved a way to compare and choose among behaviors, apparently using some “common currency” for valuing one
choice over others
...
Just as money replaced the barter system—
providing a common currency for comparing various goods and
services—nerve cell circuitry evolved to translate diverse behavioral choices into the common currency of brain chemistry
...
But neuroscientists began to figure it all out only when they joined forces with
economists inspired by game theory
...
Von
Neumann and Morgenstern showed how utility could be rigorously defined and derived logically from simple axioms, but still
thought of utility in terms of money
...

Putting game theory into experimental action, though, showed
that people don’t always do that
...
And people turned out not to be utterly
rational, but pretty darn emotional
...

A BEAUTIFUL MATH

GAMES AND EMOTIONS
You might think (and some people do) that game theory therefore
becomes irrelevant to the real world of human social interaction,
because people are not rational seekers of maximum utility, as game
theory allegedly assumes
...
Game theory actually
only tells you what people would do if they were “rationally” maximizing their utility
...

There is, however, another interpretation of what’s going on
...
And
maybe “emotional” and “rational” are not mutually exclusive descriptions of human behavior
...

Actually, most economists have long recognized that people
are emotional
...
“One of the things
mainstream economists have said is, well, rationality is mathematically precise,” he said
...
But there
are a lot of ways to not be rational
...
”
And if anything can happen, there’s no hope of finding a mathematical handle on the situation
...
”
This argument seems very much like the strategy of looking
for lost keys only under the lamp post, because you couldn’t see
them if they were anywhere else
...
But Camerer and other

FREUD’S DREAM

behavioral economists would rather first figure out what behavior
is actually like
...
and ask them
for some help,” Camerer said
...
’”3
Of course, there was a time—as in Freud’s day—when psychologists couldn’t have provided very reliable answers to the questions about brain processes underlying human behavior
...
Human emotions, for instance, are no longer as much of a mystery as
they used to be
...
Not to mention getting high on
drugs
...
Consequently human behavior, economic and otherwise, can now be
analyzed in terms other than the economist’s “rational” and monetary notion of utility
...
And that’s
just one of the insights that the new discipline of neuroeconomics
is providing into human economic behavior
...
His “Human Neuro-imaging Laboratory” is a cutting-edge
model of advanced technology in the service of science, with 100
or so computers, walls lined with plasma screen monitors, and stateof-the-art brain scanning machines
...

“We’re quantifying the mind and human experience,” he said
...
”4

A BEAUTIFUL MATH

Montague began his scientific life in mathematics and biophysics, but foresight warned him that physics was not the wave of the
future
...
Why not put math to use in comprehending cognition as well as the cosmos? He began to work on
computational modeling of brain processes, and then proceeded
to peer deep into real brains, exploiting a technology provided by
physics to revolutionize psychology
...
The behaviorist psychology of the
early 20th century, proselytized by B
...
Skinner, had left its imprint on general beliefs about brain and behavior
...
It turned
out to be a misguided notion of both science and the brain
...
Radioactive atoms could be attached to critical molecules, allowing their activity
to be observed in living brains, providing clues to what brains were
doing while animals were behaving
...
Ultimately this method,
known as magnetic resonance imaging, or MRI, became widely
used in medicine to “see” beneath the skin
...
5
“It can make a movie of the dynamic blood flow changes in
every region of your brain,” Montague said
...
You can watch
how patterns of activity change in different parts of the brain as
its owner performs various behaviors
...
“There’s a kind of sea change of belief in what you can and
can’t explain,” he said
...
The experiments are working
beautifully
...
”6
A new scientific discipline to exploit these technological abilities seems to have emerged almost out of nowhere
...
7 Before
that, people like Montague had been referring to their studies as
“neural economics
...
Glimcher and Platt had measured
nerve-cell activity in the brains of monkeys performing a decisionmaking task
...

Monkeys, of course, are not obsessed with money, but they do
really enjoy getting squirts of fruit juice and can be fairly easily
trained to perform all sorts of tasks for a juice-squirt reward
...

Looking at a light earned a squirt of juice
...
It didn’t take the monkey long to figure
that out
...
) If the
experimenters changed the high-reward squirt to the other light,
the monkey caught on right away and preferred the new highreward light
...
But in this case, Platt and Glimcher also recorded the activity of a nerve cell in a region of the monkey brain
that processes visual input and is involved in directing eye movement
...
)

100

A BEAUTIFUL MATH

Now here’s the tricky part of the experiment
...
When
the accessible light appeared, that nerve cell fired electrical impulses, as nerve cells do when stimulated
...
No surprises there
...
To an oldschool neurophysiologist, that would be surprising
...
Somehow the neuron linked to
that visual stimulus “knew” which light was the Big Gulp of juice
dispensers
...
8
Of course, that experiment was just a start, but it opened a lot
of scientists’ eyes to the possibility of understanding economic
decision making by looking inside the brain
...
Montague recalls the skepticism expressed by
one of the economists attending, who saw no reason to believe
that brain chemicals had anything to do with economics
...
“If your brain
doesn’t generate economic behavior, what kind of ghost horses do
you believe in?” Even worse, the economist didn’t even think his
remarks were particularly provocative
...
“I might still be stunned by that
...
A special issue of Neuron, published in October
2002, included a passel of papers on human decision making,
many of them exploring the new insights offered by neural economic studies
...
The paper noted various lines
of evidence supporting the idea that a circuit of activity linking
two parts of the brain—one at the front, behind the forehead, and
another deep in the brain’s middle—helps govern choice making
by producing more or less dopamine
...

Dopamine had long been known as the brain’s chief pleasure
molecule, linked to behavior that produces pleasant feelings
...
Actually,
the brain’s dopamine currency seems tuned to the expectation of
pleasure (or reward of some sort)
...

If a choice produces precisely the predicted reward, the dopamine
cells maintain a constant level of activity
...
If the reward
disappoints, dopamine production is curtailed
...
When the anticipated rewards aren’t realized, the dopamine monitoring system tells the brain to change its
behavior
...

A critical point, noted by Montague and Berns, is that all brains
are not alike
...
Some people make a risky choice only when expecting a huge reward; others gamble for the fun of it
...

In one experiment described by Montague and Berns, people
chose either A or B on a computer screen and then watched a bar
on the screen to see whether their choice earned a reward
...
) As
the game went on, the computer adjusted the rewards, based on
the player’s choices
...
When A’s payoffs
dropped, some players noticed right away and quickly switched to
choosing B more often
...
It appeared that some
brains are more inclined to take risks than others—some players
play conservatively; others are risk-takers
...
” “I call them conservative and risky because
you can make good jokes about that,” he said
...
” But the labels don’t really matter
...
Sure enough, patterns of brain activity differed in the two
groups, particularly in a small clump of brain cells called the
nucleus accumbens
...

The neatest thing, though, is that you can tell who the risk
takers and play-it-safers are from their brain scans just after the
very beginning of the game, even while their behaviors are still identical
...
Early in the game, two players can behave identically,
making exactly the same choices
...

“The people that ended up on average being risky are different
from these people right away—nobody even jumps categories,”
Montague told me
...

So neuroeconomics thus offers economists a tool they had not
possessed before, giving hope that by getting inside people’s heads,
science might really be on the road to finding the Code of Nature
that governs human behavior
...
In a study by Alan Sanfey and colleagues, participants
in an experiment played the ultimatum game, one of the favorites
of behavioral game theorists
...
Suppose you get $100
...
Then you have to give all the
money back, and nobody wins anything
...
Therefore, a
game theorist might conclude, you should offer a low amount—
$10, say, or even $1—so that you will then walk away with the
most money possible
...
If you offer $10, for instance, you’re much more likely to
walk away with zero than $90, as the stranger will probably reject
your offer just to punish you, even at personal expense
...

So this is another case where naive game theory, in assuming
that everybody will maximize their money, makes an incorrect prediction, as many economic experiments had already established
...
In this case, the prize was only $10—
science doesn’t have budgets like Who Wants to Be a Millionaire?—
but the principle was the same
...
But not always
...

Stronger brain activity in the front part of a brain region
known as the insula (an area known to be associated with negative

104

A BEAUTIFUL MATH

emotions, such as anger and disgust) was common in players who
were more likely to reject low offers
...
That region is known to be involved in
monitoring conflict—in this case, the conflict between the choice
of punishing a cheapskate or turning away money
...
can lead people to sacrifice sometimes considerable financial gain in order to punish their partner for the slight,” Sanfey and
his collaborators reported in Science
...
But behavioral game
theory, Camerer noted, can relax these assumptions and still learn a
lot about human behavior
...

Montague’s subjects at Baylor, for instance, play similar behavioral games that reveal the quirks of human economic behavior
...
If Player 1 keeps $10 and donates $10, the sum in the pot becomes $30
...

“If you split it 15-15, then in a sense you’ve repaid the trust,”
said Montague
...
At any point in
the game, one player or the other could decide to keep all the
money, so the logical move is to take it all as soon as possible,
before the other player does
...

Traditional economists were not surprised at the results of such
games
...
What’s new in neuroeconomics is eavesdropping on the players’ brains via the MRI scanners while the games
are in progress
...
The scientists watch
as computers record the brain activity of players deciding how
to move or how to react to another player’s move
...
You can back up and look at their
intent to act badly or their intent to invest more,” Montague
said
...
I think it’s cool
...
”11
Neuroeconomics does not always require scanning, though
...
He can relate variant economic behaviors to levels of certain hormones
...
One player, given
$10, offers some of it to another player, who is paid triple the
amount offered
...

Player 2 then can take it all, or give part of it back to Player 1
...
There’s no incentive to earn trust so as to get more money
the next time around
...
But Player
1, anticipating that move, should therefore offer none of the money
to begin with
...

About half of the first-movers offer some money (suggesting that
they are trusting souls), while three in four of the responders give
some back (suggesting that they are trustworthy)
...
It turns
out that among the trustworthy players, blood tests revealed higher

106

A BEAUTIFUL MATH

levels of oxytocin, a hormone linked to pleasure and happiness
...
“It tells us that people
are very much responsive to their environment,” Zak told me when
I visited him at Claremont
...
”12
Zak believes that the relationship between trust and oxytocin
is central to understanding many of the world’s economic
ills
...
Trust levels, in turn, are
a good indicator of a country’s economic well-being
...

hom*o NEUROECONOMICUS
For all of its intriguing findings, neuroeconomics doesn’t excite
everybody, like the economist who perplexed Montague by not
caring about the brain
...
To
them, it only matters what people do; it doesn’t matter which part
of the brain is busy when they do it
...
They want the Code of Nature, the
scientific understanding of humanity sought by 18th-century
thinkers such as David Hume and Adam Smith
...
”13
Rustichini, of the University of Minnesota, points out that
Adam Smith’s great works—Theory of Moral Sentiments and Wealth
of Nations—were part of a grand plan to codify the nature of

FREUD’S DREAM

107

human civilization, to explain how selfish individuals manage to
cooperate sufficiently well to establish elaborate functioning societies
...
Modern neuroscience has begun to show how sympathy works, by identifying “mirror neurons,” nerve cells in the brain that fire their
signals both in performing an action and when viewing someone
else performing that same action
...
Scientists scanning the brains of players participating in a repeated Prisoner’s Dilemma game, for
instance, have identified regions in the brain that are active in players who prefer cooperating rather than the “purely rational” choice
to defect
...
In this
game, players who feel cheated may assess a fine on the defector
(even though they must pay the price of reducing their own earnings by half the amount of the fine they impose)
...
That suggests that some
people derive pleasure from punishing wrongdoers—the payoff is
in personal satisfaction, not in money
...
(Since this punishment
is costly to the individual but beneficial to the group as a whole, it
is known as “altruistic punishment
...
Some players prefer to cooperate
while others choose to defect, and some players show a stronger
desire than others to inflict punishment
...
The human race plays a mixed strategy in the game of life
...
People just differ, dancing to their
own personal drummer
...
It’s understanding those differences,
Camerer says, that will make such a break with old schools of
economic thought
...
In an economy with millions
of people, everybody is clearly not going to be completely alike in
behavior
...
A real mix
...
“It’s much easier to say that there’s one kind of person and
there’s a million of them
...
”
So for the sake of computational simplicity, economists would operate as though the world was populated by millions of one generic type of person, using assumptions about how that generic
person would behave
...
“It
was, well, let’s just stick to one type of person
...
”
And in a way, that is a very natural thing for economists to
want to do
...
“And so
loosely speaking, the more individual difference there is, the better
that might be for the economy—as long as you get people in the
right jobs
...
”16

FREUD’S DREAM

109

Zak, who has also performed studies to localize the brain’s
computing of utility, notes that such work revolutionizes the kinds
of questions that economists can study
...
“Now we can ask
all kinds of questions about that
...
”17
Yet while neuroeconomics may provide the foundation for
understanding individual behavior and differences, it cannot alone
provide the Code of Nature, or a science of human behavior
like Asimov’s psychohistory
...
It’s in understanding
human culture that science must seek a Code of Nature, and game
theory provides the best tool for that task
...

—La Rochefoucauld

You don’t need to know about game theory to understand the
ultimatum game
...

Decades before economists invented the ultimatum game,1
something very much like it appeared in the 1941 movie The Maltese Falcon
...

Spade (played by Humphrey Bogart) has just made a deal with the
criminal Kasper Gutman (Sydney Greenstreet)
...

“I’d like to give you a word of advice,” Gutman whispers to
Spade
...
” Gutman knew that people react negatively
to the perception of being treated unfairly
...

So why bother with game theory? If you can figure out human nature just by observing how people behave, whether in the
real world or the lab, perhaps game theory is nothing more than
110

SELDON’S SOLUTION

111

superfluous mathematics
...

Actually, though, game theory provides a more sophisticated
and quantitative tool for describing human nature than the intuition of criminals
...
Fairness, trust, and
other social conditions do affect how people play games and make
economic choices
...

Game theory’s math doesn’t really tell you what people want, but
rather how people should behave in order to achieve what they
want
...
“It has many times been
claimed that certain game-theoretic solutions—such as Nash equilibrium
...
“While it may well be true that human subjects do
not behave according to these solutions in many situations, few
experiments actually provide evidence for this
...
Such tests do not disprove game theory, though; instead, they suggest that something is
wrong with the experimenter’s assumptions
...
Such factors as altruism and
spite, Weibull notes, affect the outcome that players prefer to reach,
and they make their choices accordingly
...
”3 In some cases, social
context (say, the norms of a person’s peer group) dictates choices
that appear inconsistent with both personal self-interest and con-

112

A BEAUTIFUL MATH

cern for the welfare of others
...
4

THE NATURE OF HUMAN NATURE
By getting a grip on the nuances of social preferences, game theory
enhances its prospects for forging a science of human behavior, a
Code of Nature for predicting social phenomena
...
It presumes that there is such a thing as
“human nature” to begin with for game theory to describe
...

After all, when economists play the ultimatum game with college
students, the results come out pretty much the same, whether in
Los Angeles, Pittsburgh, or even Tokyo
...
They are devotees of a discipline known as evolutionary psychology, a widely publicized field
contending that human behavior today reflects the genetic selection imposed on the species during the early days of human evolution
...

A typical advocate of this view is Harvard psychologist Steven
Pinker, who argued his beliefs with considerable passion in a book
called The Blank Slate
...
General features of human nature have been programmed by evolution and
stored on a genetic hard drive that guides the brain’s development
...
“The study of humans from an evolutionary perspective has shown that many psychological faculties (such as our
hunger for fatty food, for social status, and for risky sexual liaisons)
are better adapted to the evolutionary demands of our ancestral

SELDON’S SOLUTION

113

environment than to the actual demands of the current environment,” Pinker wrote
...

On the surface, it might seem that it would be a good thing for
game theory—and the rest of the human sciences—if this idea is
right
...
After all, the concept that a Code of Nature exists might be
interpreted to mean that there is some universal behavioral program to which all members of the human species conform
...
And it turns out that rather than bolstering evolutionary
psychology, game theory helps to show why it breaks down
...

COMPARING CULTURES
In Prelude to Foundation, the first prequel to Asimov’s Foundation
Trilogy, a young Hari Seldon delivers a talk at a mathematics conference on the planet Trantor, capital world of the Galactic Empire
...
Naturally the emperor receives word of this talk (in the
galactic future, politicians pay more attention to science than they
do today) and invited Seldon to an audience
...
to predict the future,
not in full detail, of course, but in broad sweeps; not with certainty, but with calculable probabilities
...
Seldon, in fact, was skeptical that he would ever
succeed
...
“To take into account the various
attitudes and impulses of mind adds so much complexity that there
lacks time to take care of all of it
...
“However theoretically
possible a psychohistorical analysis may be, it is not likely that it
can be done in any practical sense,” he admitted
...

By the end of the book Seldon realized that Trantor was a microcosm of the galaxy, home to hundreds of societies each with their
own mores and customs
...

Toward the end of the 20th century, Earth-bound anthropologists independently arrived at a similar scheme for analyzing human social behavior
...
College students in postindustrial society, it turns out, are not perfectly
representative of the entire human race
...
The rules were the same as with college students: One player

SELDON’S SOLUTION

115

is given a sum of money and must offer a share of it to the second
player
...

By the time Henrich tried the game in Peru, it had been widely
played with college students, who usually make offers averaging
more than 40 percent of the pot
...
Sometimes lower amounts would be offered, but they
would usually be rejected
...

“We both expected the Machiguenga to do the same as everybody else,” UCLA anthropologist Robert Boyd told me
...
”9
Could it be that the Machiguenga actually understood the
rational-choice rules of game theory, while everybody else in the
world let emotions diminish their payoffs? Or would other isolated cultures behave in the same way? Soon Henrich, Boyd, and
others acquired funding from the MacArthur Foundation, and later
the National Science Foundation, to repeat the games in 15 smallscale societies on four continents
...

From Fiji to Kenya, Mongolia to New Guinea, people played the
ultimatum game not just the way college students did, or the way
economic theory dictated, but any way they darn well pleased
...
But in other cultures, low offers were
frequently made but typically rejected
...
But in
some societies such generous offers were likely to be refused
...
10
“It really makes you rethink the nature of human sociality,”
Henrich, now at Emory University in Atlanta, told me
...
Whatever your theory is about
human behavior, you have to account for that variation
...
And it appears that
the differences in behavior are indeed rooted in culture-specific
aspects of the group’s daily life
...
Such choices apparently depend not so much on individual idiosyncrasies as on the sorts of economic activity a society
engages in
...
More experience participating in markets, the research suggested, produces not cutthroat
competition, but a greater sense of fairness
...
So their market-based economic activity is very limited, and their behavior is selfish
...

Orma average offers are similar to those found with American
college students
...
College students find their low offers are usually
rejected, but in some societies any offer is accepted, no matter how
low
...
Even so, Torguud offers averaged between 30 and 40 percent—despite the fact that the offerer
would surely get more by offering less
...
At the same time,
inflicting punishment (by rejecting an offer) is not highly regarded
there, either
...
Among the Aché of Paraguay, for example, hunters often leave

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the day’s game on the outskirts of their village
...
When playing
the ultimatum game, the Aché typically make high offers, often
more than half
...

In other societies, though, the cultural influences play out differently
...

Nonsharers, though, risk ostracism, social scorn, and negative gossip
...

On the other hand, high offers do not always signify a culture
imbued with altruism
...
The reason, it seems, is that among the Au and
Gnau accepting a gift implies an obligation to reciprocate in the
future
...

Colin Camerer, one of the economists collaborating with the
anthropologists in the cross-cultural games, observes that this result is just another twist in the cultural influence on economic behavior
...
“So the money is turned down because
they don’t want to be insulted, and they don’t want to be in
debt
...
Rather than purely testing
economic behavior, the games actually tapped into patterns of cultural practice
...

For instance, the Orma quickly recognized a similarity between
real life and a variant of the ultimatum experiment, the public
goods game (which we encountered in Chapters 3 and 4)
...
Ensminger would then double
the pot and divide it equally among the four players
...

“That really changed our thinking a lot about what was going
on when people are in an experiment,” Camerer told me in one of
our conversations at Caltech
...
” In other words, the initial belief was that “when you present the game, it’s like a smart kid
sitting down to play Monopoly or poker
...
But
these subjects treat it as like analogical reasoning—what is this like
in my life?”13
So what the game theory experiments have shown is that life
differs in different cultures, and economic behavior reflects those
differences in cultural life
...

Human culture is not monolithic—it’s like a mixed strategy in game
theory
...
When I visited Boyd
in his office—on the third floor of Haines Hall on the UCLA
campus—our discussion turned to that problem in pursuing the
general notion of human nature and the basic principles of human
behavior
...

“We have this weird, I think untenable, situation in the social
sciences,” he said
...
And the students come over here to
sociology, one floor down, and they get told no, that’s all wrong,
this is right
...
And then they go to the

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psychology department and they get told something different
again
...
It’s not acceptable that the economists are
happy with their world and the sociologists are happy with their
world, and this persists in an institution which is supposed to be
about getting at the truth
...
In particular, merging the
abstract math of game theory with the real-world immersion of
anthropologists and other social scientists has begun to show how
disparate views of human nature may be drawn closer to how life
really works
...
”

GAMES, GENES, AND HUMAN NATURE
The fairness displayed in many societies and the variety of behaviors among them are hard to reconcile with the view that human
psychology is universally programmed by the evolutionary past
...
The game experiment project argues
otherwise, posing a conundrum for evolutionary psychologists
...
ruthlessly selfish, they would have said, ‘See, I told
you so,’” said Boyd
...
that’s
not a comfortable fact for them
...
” He pointed out, though, that evolution
remains important to human psychology
...
“The question is, how did it work?”
And as Camerer pointed out, evolutionary psychologists can
always retreat to the fallback position that the ancestral environment programmed people to be different
...
“I

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think the hard story about cultural universality, you can reject,”
Camerer said
...
These are evidence-driven conclusions about evolutionary psychology’s limitations
...

One of the more interesting critiques comes from philosopher
David Buller, of Northern Illinois University in Dekalb, who critically assessed the methodological rigor underlying several of evolutionary psychology’s claimed “successes” and found that the
evidence for them was actually ambiguous
...
”
“Evolutionary Psychologists argue that our psychological adaptations are ‘modules,’ or special-purpose ‘minicomputers,’ each
of which evolved during the Pleistocene to solve a problem of
survival or reproduction faced by our hunter-gatherer ancestors,”
Buller wrote
...
Evolutionary
psychologists say their work explains sex differences in jealousy,
an innate ability to detect “cheating” (as when someone fails to
perform an obligation incurred in return for receiving some benefit), and a tendency of parents to abuse stepchildren more than
their own genetic offspring
...
In some

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cases the data on which the claims are based may be biased or
incomplete, and sometimes the research methods are not rigorous
enough to exclude alternative explanations for the findings
...
“Although
the Evolutionary Psychology paradigm is a bold and innovative
explanatory framework, I believe it has failed to provide an accurate understanding of human psychology from an evolutionary
perspective,” he wrote
...
The
Evolutionary Psychology view ascribes enormous power to the role
of genetic endowment in directing human behavior; many scientists, philosophers, and scholars of other stripes find the belief in
the dictatorial determinism of genetic power to be particularly
distasteful
...
Without genes, of course,
there is no behavior—because there would be no brain, and no
body, to begin with
...
In recent
years, the most thoughtful investigators of this issue have tended
to agree that genes do matter, to some degree or another
...
And modern neuroscience does even provide some
evidence for modularity in many brain functions, as Evolutionary
Psychologists argue
...
A brain hardwired for certain behaviors
ought to be, in fact, hardwired
...

“One of the surprises of the last few years is the fact that we’re
learning that the brain is hardwired for change,” says Ira Black, of
the Robert Wood Johnson Medical School in New Jersey
...
”18
Heredity does wire some predispositions into the brain, to be
sure, but it’s a mistake to believe that experience must somehow
defy the brain’s genetic hardwiring
...

“You are flexible because of your genes, not in spite of them,”
declare neuroscientists Terrence Sejnowski and Steven Quartz in
their book Liars, Lovers, and Heroes
...
”19
So most experts would agree that genes are important, and
genetic variation can influence propensities toward different kinds
of behavior
...
Even animals,
often portrayed as mere “gene machines” responding to stimuli with
programmed responses, actually exhibit a lot of variability in their
behavior that cannot be ascribed to genetic variations
...
For years, scientists have annoyed mice by dipping their tails into a cup of hot water (typically
about 120 degrees Fahrenheit)
...
Sure enough, the mice do not like having their tails
dipped into hot water; as soon as you put the tail in, the mouse will
jerk it out
...
Experimenters have found
that some mice react, on average, in a second or less; others might

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take three or four
...
Since the environmental conditions are apparently just
the same, it is tempting to conclude that differences in this simple
behavior reflect some difference in the mice’s genes
...

As it turns out, Jeffrey Mogil of McGill University in Montreal
and collaborators at the University of Illinois had been dipping
mouse tails in hot water for more than a decade and had accumulated plenty of data with which to perform such an analysis
...
Keep
the environmental conditions constant (the water temperature
should be precisely 49 degrees Celsius, for example) and some
genetic strains, on average, do flip their tails out of the water faster
than others
...
After reviewing the scores of more than 8,000 irritated mice, Mogil’s team
found that all sorts of things influence reaction speed
...
20 In other words, genes aren’t even as important as which
researcher is handling the mouse
...
Environmental influences were responsible for 42 percent of the performance differences, with 19
percent attributed to interactions between environment and genes
...
)
Mogil and collaborators concluded that the laboratory environment plays an important role in the way mice behave, either
masking or exaggerating the effects under genetic control
...
More complicated behaviors would probably be
even more susceptible to environmental effects, the researchers
observed
...
Genes, environment, and culture interact to produce a multiplicity of behaviors in
mice, and in people
...

It shouldn’t be surprising that cultures differ around the world as
well, that the planet is populated by a “mixed strategy” of cultures,
rooted in a mixture of influences on how behavior evolves
...
It
is the mixed human nature that, on reflection, should be obvious in
a world ruled by game theory
...
As we’ve seen,
evolutionary game theory does not predict that a single behavioral
strategy will win the game
...
Game theory’s rules induce instead a multiplicity of
strategies, leading to a diverse menagerie of species practicing different sorts of behaviors to survive and reproduce
...
Game theory
guarantees that evolution will produce a diversity of species, a
mixture of behaviors, and in the case of the human race, a multiplicity of cultures
...
Nash’s original game theory math was construed and interpreted a little too narrowly
...
But that was because the math originated and
operated in an abstract realm of assumptions and calculations
...

“My goal is to get the mathematicians to loosen their grip on
game theory and get away from thinking about a game
...
Instead, he said,
playing games can be thought of as something “like an X-ray about
a thing that’s happening in the world
...

It becomes a tool for grappling with the complexity of human
behavior and understanding the innumerable interactions that drive
human history
...

Of course, Asimov’s character had many real-life predecessors
who sought a similar science of society
...

Quetelet’s Statistics and
Maxwell’s Molecules
Statistics and society, statistics and physics
The mob has many heads but no brains
...
are in reality never inconsistent, but however capricious they may appear only
form part of one vast system of universal order
...
He simply said you could describe
masses of people in the same way you describe masses of molecules
...
And so he reasoned that a
sufficiently advanced science could do the same thing with people
...
1 “It was the science of mobs; mobs in their billions
...
” So while any one
person could do his or her own thing, society might collectively
126

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127

exhibit patterns of behavior that equations could capture
...
As one of Asimov’s characters explained, “The laws of
history are as absolute as the laws of physics, and if the probabilities of error are greater, it is only because history does not deal
with as many humans as physics does atoms, so that individual
variations count for more
...
On the other hand, in the mid19th century math seemed similarly useless for physicists pondering the complexities of molecular motion in gases
...
How
could anyone grasp the inner workings of a mass of molecules
too numerous to count and too small to be seen? Yet the Scottish
physicist James Clerk Maxwell found a way, by using statistics—
mathematical descriptions of the average behavior of large groups
of molecules
...

Although you couldn’t say exactly what any one molecule was up
to, you could predict precisely what a sufficiently large group of
molecules would do in certain circ*mstances
...
Similar methods were
developed to deal with matter in all sorts of situations
...
You can use the
statistical approach to describe a substance’s magnetic or electric
properties, or whether it will snap or stretch when under tension
...

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While Asimov’s vision remains a science fiction dream, it is
now closer to reality than probably even he would have thought
possible
...
Physicists have
applied the statistical approach to analyzing the economy, voting
behavior, traffic flow, the spread of disease, the transmission of
opinions, and the paths people take when fleeing in panic after
somebody shouts Fire! in a crowded theater
...
This isn’t a new idea, and physicists didn’t
have it first
...
As the science journalist Philip Ball has
observed, “by seeking to uncover the rules of collective human
activities, statistical physicists are aiming to return to their roots
...
And that history contains
hints of ideas that can, in retrospect, be seen as similar to key
aspects of game theory—foreshadowing an eventual convergence
of all these fields in the quest for a Code of Nature
...
In a
sense it goes back to ancient times, of course, resembling at least
partially the old notion of a “natural law” of human behavior or a
Code of Nature
...
Even
before Newton, though, the rise of mechanistic physical science
inspired several philosophers to consider a similarly rigorous approach to society
...
Descartes, Galileo, and other pioneers of modern science
advocated a mechanical, cause-and-effect view of the cosmos that
ultimately led to Newton’s definitive system of physics, published
in his Principia in 1687
...
One was Thomas Hobbes, whose famous
work Leviathan described the state of society that (Hobbes believed) maximized the well-being of all its members
...
Otherwise, he argued, a dog-eat-dog mentality of
unrestrained human nature would guarantee life to be “nasty, brutish, and short
...
The Hobbes approach was to assess the interacting preferences of various individuals and figure out how best to achieve the best deal for
everybody
...
As such, Hobbes’s Leviathan could
be seen as an early effort to understand society mathematically,
with the prescient indication that something like game theory
would be a good mathematical instrument for the task
...
The scientist and politician Sir William Petty,
a student of Hobbes, advocated the scientific study of society in a
quantitative way
...
Graunt and
others began to keep track of births and deaths and analyzed the
data much like the way that baseball fans pore over batting averages today
...
“The idea that
there were laws that stood in relation to society as Newton’s mechanics stood in relation to the motion of the planets was shared
by many,” writes Ball
...
Physics, as
Newton had sculpted it, was the science of certainty, his dictatorial
laws of motion determining how things happened
...
Much about human behavior seemed to depend on chance—
the luck of the draw (as in games!)
...

Early studies of probability theory predated Newton, starting
with the mid-17th-century work of Blaise Pascal and Pierre
Fermat—their idea being to figure out how to win at dice or card
games
...

Probability became more useful to physics (and the rest of
science) with the development of the theory of measurement errors during the 18th century, particularly in astronomy
...
For a being with intelligence capable of analyzing the circ*mstances of all the bodies in
the universe, and the forces operating on them, all movements great
and small could be foreseen by applying Newton’s laws, Laplace
declared
...
”5
Laplace recognized full well, though, that no human intelligence possessed such grand ability
...
Laplace wrote extensively on the issue of probability

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131

and uncertainties, focusing especially on the inevitable errors that
occurred whenever measurements were made
...
No matter how good
your instruments, uncontrollable factors will prevent you from measuring positions with arbitrary precision
...
But such random errors do
not render your measurements hopelessly inaccurate
...
If the measurements are careful enough, for example, small errors will be
more common than somewhat larger errors, and huge errors would
be even rarer
...
Another
was Carl Friedrich Gauss, the German mathematician whose name
was given to the now familiar bell-shaped curve that depicts how
random measurement errors are distributed around the average
value (the “Gaussian distribution”)
...
The math describing the curve tells you how to estimate the likelihood that the true
value differs from the mean by any given amount
...
Like others of his era, Laplace recognized the relevance of
statistics to human affairs, and applied the error curve to such issues as the ratio of male to female births
...

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SOCIAL PHYSICS
Quetelet, who was born in Ghent in 1796, made a major mathematical contribution to society that most Americans today are uncomfortably familiar with, although few people know to blame
Quetelet
...
But
he had much greater vision for applying science to society than
merely telling people how to know when they were overweight
...
He got a job teaching math in
Brussels, where he was soon elected to the Belgian academy of
sciences
...

Quetelet later wrote some widely read popularizations of astronomy and physics for the general reader
...

Quetelet was highly regarded as a teacher and as a person by those
who knew him—he was described as amiable and considerate, tactful and modest, but still a rigorous thinker who expressed his views
strongly
...
He also learned probability theory from Laplace and met
his colleagues Poisson and Fourier, who also had an interest in the
statistics of society
...

Quetelet began to publish papers on the statistical description
of society, and in 1835 authored a detailed treatise on what he
called social physics8 (or social mechanics), introducing the idea of
an “average man” for analyzing social issues
...
“In giving to my
work the title of Social Physics, I have had no other aim than to

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133

collect, in a uniform order, the phenomena affecting man, nearly as
physical science brings together the phenomena appertaining to
the material world,” Quetelet commented
...
“In a given state
of society, resting under the influence of certain causes, regular
effects are produced, which oscillate, as it were, around a fixed
mean point, without undergoing any sensible alterations,” he
wrote
...

Understanding the “average man,” Quetelet contended, was essential for sound government based on an intelligent understanding of human nature
...
Yet certain tendencies would show
up in society more often than others, so statistical methods could
be used to construct an abstract “average” representation of the
typical mix of human characteristics
...
After many shots by many archers, the arrows lodged in the
target form a distinct pattern, some close to the bull’s-eye, some
farther away
...
Even if no arrow had actually hit it,
you could infer the bull’s-eye’s location from the pattern made by
the arrows
...
11
Quetelet collected all the data he could find on such social
variables as birth and death rates, analyzing how such rates differed by location, season, and even time of day
...
He was struck by the constancy of
crime reports in various sorts of categories from year to year
...

“The actions which society stamps as crimes,” Quetelet wrote,
“are reproduced every year, in almost exactly the same numbers;
examined more closely, they are found to divide themselves into
almost exactly the same categories; and if their numbers were sufficiently large, we might carry farther our distinctions and subdivisions, and should always find there the same regularity
...
“Crime
pursues its path with even more constancy than death,” Quetelet
observed
...
Another researcher,
for instance, had shown that property crime in France was higher
in provinces where more children were sent to schools, and
concluded that education caused crime
...
Quetelet correctly chastised such
stupidity
...
The insurance company’s mortality
tables cannot forecast the time of any one person’s death, for instance
...

Quetelet’s exposition of social statistics attracted a great deal
of attention among scientists and philosophers
...
Quetelet responded
not by denying free will, but by observing that it had its limits, and
that human choice was always influenced by conditions and circ*mstances, including laws and moral strictures
...
This “empire
of causes” typically overwhelms free will, which is why, with

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135

knowledge of all the factors affecting someone’s choice, it is usually possible to predict what it will be
...
Fortunately for physics, some of the commentaries on it
reached the hands of James Clerk Maxwell
...
He saw deeply into almost every corner of physics, forever
alert to the hidden principles governing the complexities of physical phenomena
...
And in fact, Maxwell detected an essential
shortcoming in Newton’s laws of motion, too
...
But what about
the submicroscopic molecules from which such objects were made?
Presumably, Newton’s laws would still apply
...
And if you couldn’t describe the motion
of an object’s parts, how could you expect to predict the behavior
of the object?
For a cannonball dropped from the leaning tower of Pisa, the
internal motion of the metal’s atoms made no difference to the rate
at which it fell
...
Suppose you wanted to know how changes in pressure
affected the temperature of the steam in a steam engine, for instance
...

Physicists were not helpless in the face of this question—they
had devised some pretty good formulas for describing how gases
behaved
...
If he could show

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A BEAUTIFUL MATH

how molecular motions produced the observed gross behavior, he
would have achieved both a deeper understanding of the phenomena and have provided strong new evidence for the very existence of atoms and molecules, which was at the time—in the
mid-19th century—a contentious conviction in some circles
...
It was known as the
kinetic theory of gases, originally articulated in 1738 by our old
friend Daniel Bernoulli, who explained the gas laws with a crude
picture of molecules modeled as billiard balls
...
”14 Bernoulli’s idea was based on the (correct) notion that heat is merely the motion of molecules, but in his
day most physicists believed heat to be some sort of fluid
substance (called caloric)
...

One of the major pioneers of thermodynamics was the German physicist Rudolf Clausius
...
He described how the pressure of a gas was related to the
motion of molecules as they impinged on the walls of their container
...
In his
approach, Clausius emphasized the importance of the average velocity of the molecules, and in an 1858 paper introduced the important notion of the average distance that molecules traveled
between collisions (a distance labeled the “mean free path”)
...
In his approach, Maxwell applied the sort of
statistical thinking that Quetelet had promoted
...
(Herschel, of course, was familiar with Quetelet as a fellow astronomer
...
Buckle, himself clearly influenced by Quetelet, believed
that science could discover the “laws of the human mind” and that
human actions are part of “one vast system of universal order
...
)
Buckle was another of the 19th century’s most curious characters
...

When he was 18 his father, a maritime merchant, died, leaving the
son sufficient funds to tour Europe and pursue his hobbies of history and chess
...
He also became a prolific bibliophile,
amassing a library exceeding 20,000 books
...
Originally planned to focus on the Middle Ages, the work eventually took on broader aims
and became the History of Civilization in England (by which Buckle
actually meant the history of civilization, period)
...
He criticized the “metaphysical” (or philosopher’s) approach to the issue, advocating instead the “historical” method (by
which he basically meant the scientific method)
...
is, in its origin, always the same,
and consists in each observer studying the operations of his own
mind,” Buckle wrote
...
”16 Buckle could not resist remarking that the
metaphysical method “is one by which no discovery has ever been
made in any branch of knowledge
...
“Every thing we at
present know,” Buckle asserted, “has been ascertained by studying

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A BEAUTIFUL MATH

phenomena, from which all causal disturbances having been
removed, the law remains as a conspicuous residue
...
”17
Much of Buckle’s philosophy echoes Quetelet, including similar slams against the idea of unfettered free will
...
“If, however, I were capable of correct reasoning, and if, at the same time, I had a complete knowledge both of his disposition and of all the events by
which he was surrounded, I should be able to foresee the line of
conduct which, in consequence of those events, he would adopt,”
Buckle pointed out
...
Game
theory is, in fact, all about understanding what choice would (or
should) be made if all the relevant information influencing the
outcome of the decision is known
...

Since sorting out the nuances of all the influences exceeds science’s
powers, the nature of human behavior must be described instead
by the mathematics of statistics
...

“The most comprehensive inferences respecting the actions of men
are derived from this or from analogous sources: they rest on statistical evidence, and are expressed in mathematical language
...
Though Maxwell found Buckle’s book
“bumptious,” he recognized it as a source of original ideas, and the
statistical reasoning that Buckle applied to society seemed just the
thing that Maxwell needed to deal with molecular motion
...
“We cannot, therefore, ascertain the actual motion of any one of these
molecules; so that we are obliged to
...
”20 That statistical
method, he showed, could indeed reveal “uniformities” in molecular behavior
...
21
The essential feature of Maxwell’s work was showing that the
properties of gases made sense not if gas molecules all flew around
at a similar “average” velocity, as Clausius had surmised, but only if
they moved at all sorts of speeds, most near the average, but some
substantially faster or slower, and a few very fast or slow
...
In subsequent collisions, a fast molecule might be
either slowed down or speeded up
...

Just as Quetelet’s average man was fictitious, and key insights
into society came from analyzing the spread of features around the
average, understanding gases meant figuring out the range and
distribution of molecular velocities around the average
...

As Maxwell refined his ideas during the 1860s, he showed
that when the velocities reached the bell-shaped distribution, no
further net change was likely
...
) Any specific molecule might speed up or slow down, but the
odds were strong that other molecules would change in speed to
compensate
...
When a gas reached that state—in which

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A BEAUTIFUL MATH

further collisions would cause no net change in its overall condition—the gas was at equilibrium
...
And it’s an analogy that has
more than merely lexical significance
...
And just as the Nash equilibrium is typically a mixed set of
strategies, a gas seeks an equilibrium state with a mixed distribution of molecular velocities
...
It’s such an important concept for game theory (and for science generally) that it’s worth a brief interlude to mercilessly
pound the idea into your brain (possibly with a silver hammer)
...
How do the molecules in a gas share
the total amount of energy that the gas possesses? One possibility
is that all the molecules move at something close to the average, as
Clausius suspected
...
Clearly, there are lots of possible combinations
...
It’s just that some combinations of velocities are more likely
than others
...
It’s
easy to calculate the probability distribution for pennies, because
you know that the odds of heads versus tails are 50-50
...
5, or onehalf
...
) In the long run, therefore, you’ll find that the
average number of heads per trial is something close to 5 (if you’re

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using a fair coin)
...
Half the trials could turn up 10
heads, for instance, while the other half turned up zero every time
...

What actually happens is that the number of trials with different numbers of heads is distributed all across the board, but with
differing probabilities—about 25 percent of the time you’ll get 5
heads, 20 percent of the time 4 (same for 6), 12 percent of the
time 3 (also for 7)
...
1 percent
of the time, or once in a thousand)
...
Maxwell’s insight was that the same kind of
probability distribution governs the possible allocations of energy
among a mess of molecules
...

Imagine you are repeatedly playing a simple game like matching pennies, in which you guess whether your opponent’s penny
shows heads or tails
...
Your choices need to be made randomly, so that they will reflect the proper probability distribution
for equally likely alternatives
...
If you are
choosing with true randomness, 1 percent of the time you’ll choose
heads 9 times out of 10, for instance
...
To keep your opponent guessing, you should
serve one way or the other at random
...
Professionals, on the
other hand, do approach the ideal distribution more nearly, suggesting that game theory does indeed capture something about
optimal behavior, and that humans do have the capability of learning how to play games with game-theoretic rationality
...
In many situations, over time, people do learn
how to play games in a way so that the results coincide with Nash
equilibrium
...

STATISTICS RETURNS TO SOCIETY
Of course, real-life situations, the rise of civilization, and the evolution of culture and society are much more complicated than flipping coins and playing tennis
...
In most realms of physics and chemistry, the phenomena in
need of explanation are rarely split between two equally likely
outcomes, so computing probability distributions is much more
complicated than the simple 50-50 version you can use with
pennies
...
Willard Gibbs consequently expended enormous
intellectual effort in devising the more elaborate formulas that
today are known as statistical mechanics, or sometimes simply
statistical physics
...

The success of statistical mechanics in physics has driven the
belief among many physicists that it could be applied with similar
success to society
...
Everything from the flow of funds in the stock market to the flow of
traffic on interstate highways has been the subject of statisticalphysics study
...
But the closing years of the 20th century
saw an explosion of new research in that arena, and as the 21st
century opened, that trend turned into a tidal wave
...
The use of statistical physics to
describe such networks has propelled an obscure branch of math
called “graph theory” into the forefront of social physics research
...

Bacon’s Links
Networks, society, and games
Unlike the physics of subatomic particles or the largescale structure of the universe, the science of networks is the science of the real world—the world of
people, friendships, rumors, disease, fads, firms, and
financial crises
...

If you had said so four centuries ago, you would have meant
Francis Bacon, the English philosopher who stressed the importance of the experimental method for investigating nature
...

Nowadays, though, when you mention Bacon and science in
the same breath you’re probably talking not about Francis, but
Kevin, the Hollywood actor
...

After all, everybody has heard by now that Kevin Bacon is the
most connected actor in the movie business
...
John Belushi and Demi
Moore, for instance, are linked via Bacon through his roles in Animal House (with Belushi) and A Few Good Men (with Moore)
...

By mid-2005, Bacon had appeared in films with nearly 2,000
other actors, and he could be linked in six steps or fewer to more
than 99
...
Bacon’s notoriety in this regard has become legendary,
even earning him a starring role in a TV commercial shown during
the Super Bowl
...
Bacon’s role in the network of actors motivated
mathematicians to discover new properties about all sorts of networks that could be described with the tools of statistical physics
...

In fact, the new network math has begun to resemble a blueprint for a science of human social interaction, a Code of Nature
...
Many
researchers believe, however, that there is—or will be—a connection
...
What started out as a game
about Kevin Bacon’s network may end up as a convergence of the
science of networks and game theory
...
They devised a party game in which players tried to find the shortest path
of movies linking Bacon to some other actor
...
They soon

146

A BEAUTIFUL MATH

launched a research project that spawned a Web page providing
instant calculations of how closely Bacon was linked to any other
actor
...
org
...
Another 169,274 can be linked to
Bacon through one intermediary, giving them a Bacon number of
2
...
On average, Bacon can be linked to the 770,269 linkable actors in the
movie database1 in about 2
...
And out of those 770,269 in
the database, 770,187 (almost 99
...

So studies of the Kevin Bacon game seemed to verify an old
sociological finding from the 1960s, when social psychologist
Stanley Milgram conducted a famous mail experiment
...
On average, it took a little more than five mailings to
reach the stockbroker, suggesting the notion that any two people
could be connected, via acquaintances, by less than “six degrees of
separation
...

From a scientific standpoint, the Bacon game and Guare’s play
came along at a propitious time for the study of networks
...

NETWORKS ARE US
When I was growing up, “network” meant NBC, ABC, or CBS
...
As the world’s cultural focus shifted from TV

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to computers, though, the notion of network expanded far beyond its origins
...
Networks permeate government, the
environment, and the economy
...

Businesses engage networks of buyers and sellers, producers and
consumers, and even networks of insider traders
...

Atlases depict networks of rivers and roads
...
Bodies contain
networks of organs, blood vessels, muscles, and nerves
...

Of all these networks, though, one stands out from the
crowd—the Internet and the World Wide Web
...
) During the early 1990s, awareness of
the Internet and Web spread rapidly through the population, bringing nearly everybody in contact with a real live example of a network in action
...
True, the word “network” already had its informal uses, for such things as groups of friends or
business associates
...

Throughout the scientific world, networks inspired a new
viewpoint for assessing some of society’s most perplexing problems
...
Discovering the laws governing networks could provide critical clues for how to protect—
or attack—everything from power grids and ecosystems to Web
sites and terrorist organizations
...

Using math to study networks is not entirely new, though
...
In the mid-20th century, Paul Erdös and Alfréd
Rényi developed the math to describe networks in their most abstract representation—essentially dots on paper connected by lines
...

Such drawings of dots and lines are technically known as graphs,
so traditional network math is known as graph theory
...
The nodes may be any of various objects or entities, such as
people, or companies, or computers, or nations; the links may be
wires connecting machines, friendships connecting people, common film appearances connecting movie actors, or any other
common property or experience
...
There are networks of people who share common investments, political views,
or sexual partners
...
Its dots and lines resemble real
networks about as much as a scorecard resembles a baseball field
...
Same with graphs
...
In a random network, every node is an equal among many,
and few nodes get much more or less than a fair share of links
...
(In a network of sexual partners, for ex-

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149

ample, some people have many more “links” than average—an important issue in understanding the spread of HIV
...

Erdös and Rényi knew full well that their dots and lines did
not capture the complexities of real-world networks
...
Describing random connections was a mathematically feasible thing to do; describing all the complexities of real-world
networks was not
...

But then a paper appearing in the British journal Nature began
to change all that
...
”3

NETWORK MANIA
A few years later, when I met Strogatz at a complexity conference,
I asked him why networks had become one of math’s hottest topics in the late 1990s
...
“If you
ask me when did this really start, I think it started in 1998 when
our paper appeared in Nature on what we called small-world networks
...
It really was a
case of culture preparing the conditions for the advance of
science
...
“So it was in the air
...
The Bacon game became famous just about the time that
the public became aware of the Internet, thanks to the arrival of
the World Wide Web
...
Not only was the Web a high-profile example of a vast, elaborate network, it made many other networks accessible for research
...
Data on
the metabolic reactions in nematode worms or the gene interactions in fruit flies could similarly be collected and transmitted
...
“People started to think about
things as networks
...
“We didn’t think
of them so much as networks,” Strogatz said
...
”
With the Web it was different
...
You had to browse, link by link
...
“In many different branches of science,”
Strogatz observed, “the kind of thinking that we call network
thinking started to take hold
...
They showed how to
make a model of a “small-world” network, in which it takes only a
few steps on average to get from any one node of the network to
any other
...
But Strogatz
thinks some of those surprises have been misrepresented as being
responsible for network math’s revival
...
Others have suggested that “clustering” of links
(small groups of nodes connected more than randomness would
suggest) was the key discovery
...
“The reason it
caught on, I think, is because we were the first to compare networks from different fields and find that there were similar properties across fields
...

Such commonalities gave people hope that network math could be
more than a tedious chore of sorting out links in one kind of
network and then moving on to the next
...

SMALL WORLDS
One of the key common features of different networks is that
many of them do in fact exhibit the small-world property
...

So discovering the rules governing small-world networks may be
the key to forecasting the social future
...
In a regular network (what is often called a
regular lattice), the nodes are
connected only to their nearby
neighbors
...
The
dots representing the nodes are
connected to their immediate
neighbors on either side by the
line representing the circle
...
Each node
would then be connected to four others—two neighbors on each
side
...

Some nodes would be linked only to other nodes nearby; some
would be connected to nodes on the other side of the circle;
some would be connected both to neighbors and to distant nodes
...
That’s what it means to be random
...
Regular
networks, though, are not so
easy to navigate
...

But what happens, Watts
Random network
and Strogatz wondered, with an
“in-between” network—neither completely regular nor completely
random? In other words, suppose you started with a regular network and then added just a few links at random between other
nodes
...
But that intermediate
network retains an important feature of the regular network—its
nearby nodes are still more highly connected than average (that is,
they are “clustered”), unlike random networks where clustering is
mostly absent
...
But the fact that you
needed very few shortcuts to
make the network world small
implied that small-world networks might be common in nature
...
elegans
...

“Thus,” Watts and Strogatz concluded, “the small-world phenomenon is not merely a curiosity of social networks nor an artifact of an idealized model—it is probably generic for many large,
sparse networks found in nature
...

In just the way that statistical physics made it possible to tame the
complexities of a jumble of gas molecules, mathematicians could
use similar math to compute a network’s defining properties
...
“Everybody pointed out, isn’t this remarkable
that these totally different networks have these properties in common—how would you have ever thought that?” Strogatz said
...
The average number of steps
to get from any one node to any other—the “path length”—is one
such parameter
...
A relatively high clustering rate is one of the counterintuitive
features of small-world networks
...
On
the other hand, the high clustering coefficient is completely unlike
that in random networks, but is more similar to that in regular
networks
...
Since my
sister Sue, for instance, has a friend Debby and a friend Janet, it is
more likely than average that Debby and Janet also know each
other
...
) “There is a tendency to form triangles, and you
wouldn’t see that in random networks,” Strogatz pointed out
...
(The “degree” of a node
is the number of other nodes it is linked to
...
Being well connected, after all, is
what makes the average path length between Bacon and other actors so short
...
Taking the average
number of steps to link to another actor as the gauge, he doesn’t
even rank in the top 1,000!
It turns out, in fact, that Bacon’s true importance for networks
had nothing to do with how special he is, but rather how typical
he is
...
And the existence of
such hubs turns out to be a critical common feature of many realworld networks
...
679 steps
...
95,

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155

ranked 1,049th
...
) In second place
was Christopher Lee (2
...
698)
...
Among women, the most connected was Karen Black, in 21st place
...
Choose any two
actors at random
...
And it would be unusual to need more than four
...

(And remember, I said you need to choose two at random
...
These two actors are from entirely different eras, and
Lohan is young and has been in relatively few films
...
An aging Rathbone appeared in
Queen of Blood (1966) with the pre–Easy Rider Dennis Hopper
...
The
short path between Rathbone and Lohan was made possible by the
hub provided by Hopper, who is, in fact, much more connected
then Bacon
...
)
Hubs like Hopper make the actor network a small world
...

Such large hubs generally do not exist, though, in either random or regular networks
...
In a random network,
shortcuts exist, but they might be very hard to find because prominent hubs would be very rare
...
Only a few
would have a lot more links than average, or a lot less
...
But in
many small-world networks, there is no such “typical scale” of the
number of links
...
” In scale-free networks, many lonely nodes will have
hardly any connections at all, some nodes will be moderately well
connected, and a few will be superconnected hubs
...
”
In a groundbreaking paper published in Science in 1999, Réka
Albert and Albert-László Barabási of Notre Dame University noted
the scale-free nature of many kinds of networks, and consequently
the usefulness of power laws for describing them
...
(They “salivate over power laws,”
Strogatz says—apparently because power law discoveries in other
realms of physics have won some Nobel Prizes
...
Cities, for example
...
S
...
Same with earthquakes
...

In their Science paper, Barabási and Albert showed how the
probability that a node in a scale-free network is linked to a given
number of other nodes diminishes as the number of links increases
...
Nodes with few links are common, like small earthquakes;

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nodes with huge numbers of links are relatively rare, like huge
earthquakes
...
In other words, a good
theory of networks should explain not only how Kevin Bacon (or
Dennis Hopper) can be so connected, but also why networks are
analogous to earthquakes
...
As networks grow by adding new nodes, Barabási
and Albert hypothesized, new links do not form at random
...
In other words, the rich get richer, and the result of such a
growth process is a scale-free network with very rich hubs
...
8
While their “preferential attachment” scheme did indeed predict the formation of hubs, it did not explain many other aspects
of real-world networks, including clustering
...
Barabási and
Albert’s original work, for instance, suggested that the networks
explored by Watts and Strogatz were scale-free as well as being
small worlds
...
The power grid is a small world but
isn’t scale-free, and neither is the neural network of C
...
Still, there are many examples of networks that are
both small-world and scale-free, with the World Wide Web being
one spectacular example
...

Following Barabási and Albert’s pioneering efforts to quantify
network evolution, many other groups have joined the hunt to
identify all the important qualities of networks and devise math-

158

A BEAUTIFUL MATH

ematical models to explain them
...
So their top scientists are busy
investigating network math themselves
...
When I visited the
Microsoft research labs outside Seattle, they outlined to me their
efforts to identify the features that network math needs in order to
capture the essence of the Web’s structure
...
“No one really planned the
Internet, and certainly no one planned the structure of the World
Wide Web
...
A further important feature is the preferential attachment
identified by Barabási that conditions how a network grows, or
ages
...
But it’s
not always true that the oldest pages are the most connected
...
AltaVista, for example, once was the Cadillac of Web search engines
...
So different sites must
earn links not only by virtue of age, but also beauty—or “fitness
...
“All
other things equal, the older sites will on average have more links,
but if one site is more fit than another, that compensates for age
...
If I’m twice as fit and I’m half as old, I should tend to have
about the same number of connections
...
” Unlike the Internet,
where wires run both ways, Web page hyperlinks go in only one

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direction
...
(Although, of course, it should
...
A few sites have a huge number of links, more have a
medium number, and most have very few links at all
...
In the strongly
connected component, or SCC, you can move from any page to
another by following hyperlinks one page at a time
...
“I can follow a series of hyperlinks and get to that person’s
page, and that person can follow a series of hyperlinks and get
back to my page
...
Some pages get linked to
from a page in the SCC but don’t link back to an SCC page
...

Building mathematical models that reproduce all these features
of the Web is still a work in progress
...
“I think there is going to be a mathematics of networks,” said Chayes
...
”

BACK TO THE GAMES
Since game theory also claims dominion over describing human
behavior, I asked Chayes whether it had any role to play in the
new math of social networks
...
“We are
trying to explain why these network structures have evolved the
way they have evolved, and that’s really a game theory problem,”

160

A BEAUTIFUL MATH

Chayes said
...
”
In fact, Chayes, Borgs, and collaborators have shown how math
that is similar at least in spirit to a game theory approach can
explain the emergence of preferential attachment in an evolving
network (rather than merely assuming it, as Barabási and Albert
did)
...
(It’s kind of like buying a car—you
can get a cheap one that will cost you more to keep running, or
shell out more up front for high performance with low maintenance
...

More explicit uses of game theory have been called on to explain the evolution of other kinds of networks
...
The interplay of thousands of proteins ends up determining how cells behave, which is
often a matter of life and death
...

Biologists would, of course, naturally assume that cellular metabolism should evolve to reach some “optimal” condition for fueling cell activity most efficiently
...
“Thus, by evolving towards
optimal properties, organisms change their environment, which in
turn alters the optimum,” note computational biologist Thomas
Pfeiffer and biophysicist Stefan Schuster
...
For example, a key molecule in the network of cellular chemistry is ATP, which provides the energy
needed to drive important metabolic processes
...
To stay alive, a cell needs a

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constant source of ATP, so the reaction “assembly” line has to operate 24/7
...
An important question in cellular biology is whether cells should prefer to produce ATP as rapidly as
possible, or as efficiently as possible (that is, with pathways that
produce greater quantities of ATP from the same amount of raw
material, getting more ATP bang for the buck)
...

The best strategy, a game-theoretic analysis shows, depends on
the various other organisms in the vicinity competing for resources
...
After all, if a population of microbial cells are competing for food, it would seem
best for the group for each microbe to make the most efficient use
of the available food supply, so there will be enough to go around
...
What’s best for the individuals
doesn’t compute to be the overall best deal for the group
...
“In the framework of
evolutionary game theory, slow and efficient ATP production can
be seen as altruistic cooperative behavior, whereas fast and inefficient ATP production can be seen as selfish behavior
...
Game theory math suggests
that in scenarios where a microbe’s neighbors eat a different kind
of food (so there is no competition for a single resource), more
efficient production of ATP at the expense of speed would be a
better survival strategy
...
In multicellular organisms, though, cells
behave more cooperatively with their neighbors, evolving reaction
pathways that produce ATP more efficiently
...
Game theory hasn’t exactly cured cancer yet,
but insights into such properties of cancer cells may contribute to
progress in fighting it
...
Evolutionary game theory’s assault on the cooperation problem—how altruistic behavior can
evolve in societies of seemingly selfish individuals—has relied
mainly on playing the Prisoner’s Dilemma game under a variety of
circ*mstances
...
In one version, though, the agents
face such decisions only in interactions with their immediate neighbors (the game, in other words, is “spatially structured”)
...

But perhaps the Prisoner’s Dilemma does not always capture
the essence of real life very accurately
...
One candidate is the
“snowdrift” game, in which the best strategic choice differs from
the classic Prisoner’s Dilemma
...
In the snowdrift game, your best move is to defect
only if your opponent cooperates
...
12 As it turns out, spatial constraints also
influence the evolution of cooperation in the snowdrift game, but
in a different way—inhibiting cooperation rather than enhancing
it
...

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However, as physicists Francisco Santos and Jorge Pacheco have
pointed out, the “spatial constraint” of agents interacting only with
their neighbors is not realistic, either
...
Merging the math of scale-free networks with game
theory, the physicists found that cooperation ought to emerge with
either the Prisoner’s Dilemma or snowdrift games
...
13
Numerous other papers have explored links between game
theory and network math
...
Networks are, after
all, complex systems that have grown and evolved over time
...
(One
paper specifically models a version of the Prisoner’s Dilemma game
showing how repeated play can lead to a complex network in a
state that the authors refer to as a “network Nash equilibrium
...

In fact, physicists building their version of a Code of Nature
with the tools of statistical mechanics (as did Asimov’s Hari Seldon)
have turned increasingly to using those tools on a network-based
foundation
...

Asimov’s Vision
Psychohistory, or sociophysics?
“Humans are not numbers
...

—Dietrich Stauffer

In 1951—the same year that John Nash published his famous paper on equilibrium in game theory—Isaac Asimov published the
novel Foundation
...
Asimov’s books eventually
became the most famous science fiction trilogy to appear between
Lord of the Rings and Star Wars
...
1
Mixing psychology with math, psychohistory hijacked the
methods of physics to forecast—and influence—the future course
of social and political events
...

As a result, Asimov’s vision is no longer wholly fiction
...
At various schools and institutes around the world, collaborators from diverse departments are creating new hybrid disciplines,
with names like econophysics, socionomics, evolutionary economics, social cognitive neuroscience, and experimental economic anthropology
...
The
National Science Foundation has identified “human and social dynamics” as a special funding initiative
...
Some examine voting patterns in diverse populations, how crowds behave when fleeing in panic, or
why societies rise and fall
...
Still
others analyze how rumors, fads, or new technologies spread
...
Put them all together, and Asimov’s
idea for a predictive science of human history no longer seems
unthinkable
...

Among the newest of these enterprises—and closest to the
spirit of Asimov’s psychohistory—is a field called sociophysics
...
Like Asimov’s
psychohistory, sociophysics is rooted in statistical mechanics, the
math used by physicists to describe systems too complex to expose
the intimate interactions of their smallest pieces
...

Taking society’s temperature isn’t quite as straightforward as it
is with, say, gas molecules in a room
...
To use statistical physics to take
society’s temperature, physicists first have to figure out where to
stick the thermometer
...
While molecules collide, people connect, in
various sorts of social networks
...

Social networks have now provided physicists with the perfect
playground for trying out their statistical math
...
After all, von Neumann and
Morgenstern themselves pointed out that statistical physics provided a model giving hope that game theory could describe large
social groups
...
And game theory provides the
proper mathematical framework for describing how competitive
interactions produce complex networks to begin with
...

SOCIOCONDEMNATION
Network math offers many obvious social uses
...
And because ideas can
spread like epidemics, similar math may govern the spread of opinions and social trends, or even voting behavior
...
Early
attempts to apply statistical physics to such problems met with
severe resistance, though, as Serge Galam has testified
...
Galam pursued his education in statistical physics but with a

ASIMOV’S VISION

167

concern—its methods were so powerful that all the important
problems of inert matter might soon be solved! So he began to
advocate the use of statistical physics outside physics, especially
for analyzing human phenomena, and published several papers
along those lines
...
The response from other physicists was not
enthusiastic
...
To
suggest humans could behave like atoms was looked upon as a
blasphemy to both hard science and human complexity, a total
non-sense, something to be condemned
...
There are some enthusiasts, though, and international
conferences have been devoted to sociophysics and related topics
...

Part of this acceptance probably stems from the growing popularity of an analogous discipline known as econophysics, a much
more developed field of study
...
Many young physicists
have taken their skills in this field to Wall Street, where they can
make money without the constant fear of government budget cuts
...
It should ultimately encompass econophysics within it, along with everything else in the
realm of human interactions
...
But
whatever anybody thinks of this research, there is certainly now a
lot of it
...

Now working in France, he has studied such social topics as the
spread of terrorism, for instance, trying to identify what drives the
growth of terrorism networks
...
S
...
”4
Other researchers have produced opinion-spreading papers that
try to explain whether an extreme minority view can eventually
split a society into two polarized opposite camps, or even eventually become an overwhelming majority
...
There is no point in trying to be
completely realistic—no amount of math could capture all the
nuances in the process by which even a single individual formed
his or her opinions, let alone an entire population
...
If the math
then reproduces something recognizable about human behavior, it
can be further refined in an attempt to inch closer to reality
...
Human beings are not particles—they bear not the
slightest similarity to atoms or molecules
...
In his paper introducing statistical considerations to the study of gases, Maxwell applied his math to a system
containing “small, hard and perfectly elastic spheres acting on one
another only during impact
...
But he believed that insights into the
behavior of real molecules might emerge by analyzing a simplified
system
...
”5 Today, physicists hope

ASIMOV’S VISION

169

to find a similar analogy between particles and people that will
lead to an improved knowledge of the functioning of society
...

She was interested in how opinions form and change among members of a society
...
6 “The question is if the laws on the microscopic scale
of a social system can explain phenomena on the macroscopic
scale, phenomena that sociologists deal with,” she wrote
...
“Indeed, we are individuals,” she wrote, “but
in many situations we behave like particles
...
Sometimes what one person
does or thinks depends on what someone else is doing, just as one
particle’s behavior can be affected by other particles in its vicinity
...
Then, the next morning, four people stare
skyward, and soon others stop as well, all looking up for no reason
other than to join in the behavior of the crowd
...

Another sort of phase transition, of the type that attracted her
attention, is the sudden appearance of magnetism in some materials cooled below a certain temperature
...
A material like iron can be magnetic because its atoms possess magnetic properties, thanks to the
arrangement of their electrons, the electrically charged fuzzballs
that shield each atom’s nucleus
...
(You can view the spins as around an
axis either pointing up or pointing down, corresponding to
whether the electron spin is clockwise or counterclockwise
...

If enough atoms align themselves in one particular direction,
though, others will follow—kind of like the way if enough people
look up to the sky, everybody else will, too
...
It’s as though each atom
checks to see how its neighbor’s electrons are spinning
...
Consequently the spin of one iron electron can
influence the spin of its neighbor, inducing it to take on the same
orientation
...
But iron and a few other materials possess
some properly positioned electrons without partners
...
)
As scientists began to understand this aspect of magnetism,
they wondered if such local interactions between neighbors could
explain the global phase transition from the nonmagnetic to magnetic state
...
The problem was not in the basic
idea, though—it was that Ising analyzed only a one-dimensional
system, like a string of spinning beads on a necklace
...

Magnetism could thus be understood as a collective phenomenon stemming from the interactions of individuals—sort of like
pack journalism
...
J
...
Similarly, rapid large-scale
changes mimicking phase transitions occur in biology or the
economy, such as mass extinctions or stock market crashes
...

Sznajd-Weron set out to devise an Ising-like model of social
opinions, trying out a very simplified approach that would be easy
to handle mathematically
...
If you
start out with opinions at random, how would the system change
over time? Sznajd-Weron proposed a model based on the idea
of “social validation
...

Sznajd-Weron’s model of society was pretty simple—
something like one long street with houses on only one side
...

To start out, the houses all have opinions at random
...
Based on neighboring opinions, each house may (or may not) modify its own
...
Each of that pair has another

172

A BEAUTIFUL MATH

neighbor (House 9 and House 12)
...
If houses 10 and 11 disagree, though, House 9 will adjust its
opinion to agree with House 11, and House 12 will change to
agree with House 10
...
):
If Si = Si+1 then Si–1 = Si and Si+2 = Si
If Si = -Si+1 then Si–1 = Si+1 and Si+2 = Si
In other words, when the two neighbors (10 and 11) agree, the
two outside neighbors will share that opinion
...
Why
should that be? No reason, it’s just a model
...
No matter how the opinions started out, the neighborhood eventually reached one of three
stable situations—either all the houses voting yes, all no, or a 5050 split
...
”)
Since not all societies are dictatorships or stalemates, the model
does not reflect the true complexity of the real world
...
And you don’t
need to know what all those other factors are to improve the
model—you just need to know that they exist
...
With a sufficiently
high social temperature, the system can stay in some disordered
state, more like a democracy, rather than becoming a stalemate or
dictatorship
...
So in the years since her proposal, she and others have
worked on extensions of the model
...
With the people
aligned on a grid, everybody has four neighbors, a pair has six
neighbors, and a block of four has eight neighbors
...
Or
two neighbors paired with the same spin can change the spins of
their six neighbors
...

SOCIONETWORKS
Clearly, though, the way to get more social realism is to apply such
rules not to simple strings or grids but to the complex social networks discovered in the real world
...
One approach examines the
general idea of “contagion”—the spread of anything through a
population, whether infectious disease or ideas, fads, technological
innovations, or social unrest
...

In some cases, a small starting “seed” (a literal virus, perhaps, or
just a new idea) can eventually grow into an epidemic
...

Peter Dodds and Duncan Watts (of the Watts-Strogatz network
paper) of Columbia University have shown that what happens can
depend on how much more likely a second exposure is to infect an
individual than a first exposure
...
Such results imply that the best way to hamper or advance contagion would be strategies that increase or reduce the
odds of infection
...

“Our results suggest that relatively minor manipulations
...
8 It sounds like just the sort of thing that Hari Seldon
incorporated into psychohistory, so that his followers could subtly
alter the course of future political events
...
So
some experts question how useful the statistical mechanics approach to society will ultimately be
...

“It really is the right language for discussing enormous systems of
whatever it is, whether it’s people or neurons or spins in a magnet
...
But I worry that a lot of these physicist-style models of social
dynamics are based on a real dopey view of human psychology
...
Game theory has given economists and other social scientists
the tool for quantifying human psychology in ways that Freud
could only dream of
...
And once you have a better picture of human
psychology—in particular, a picture that depicts the psychological

ASIMOV’S VISION

175

variations among individuals—you need game theory to tell you
what happens when those individuals interact
...
It is yet again similar to the situation
with molecules in a gas
...
But atoms and molecules can interact in more complicated ways
...

Similarly, the behavior of people depends on how they are
affected by what other people are doing, and that’s what game
theory is supposed to be able to describe
...
”10 Numerous efforts have
been made to apply game theory in just that way
...

Keep in mind that in game theory, a player’s choices should
depend on what the other players are choosing
...
In simple sociophysics models based on neighbors
interacting, the global collective behavior results from purely local
influences
...
It may be, for instance, that the average choices of all the
other players is the most important influence on any one
individual’s choice (in physics terms, that would correspond to a
“mean-field theory” version of statistical mechanics)
...
But sometimes (actually, almost all the time) those conditions are not satisfied
...
There are situations where the game is too
complex and too many people are involved to choose a foolproof
decision using game theory
...
This
problem was made famous by Brian Arthur, an economist at the
Santa Fe Institute, in the early 1990s
...
(It was reminiscent of baseball player Yogi Berra’s famous comment about the New York City
restaurant Toots Shor’s
...
”)
Arthur saw in the El Farol situation a problem of decision
making with limited information
...

Above some level of attendance, it’s no fun
...

In 1997, Damien Challet and Yi-Cheng Zhang developed the
mathematics of the El Farol problem in detail, in the form of what
they called the minority game
...
11
In the basic version of the game, each would-be bar patron (in
mathematical models, such customers are called “agents”) possesses
a memory of how his or her last few bar-going decisions have
turned out
...
) Suppose that Friday
is your regular drinking night, and you can remember what hap-

ASIMOV’S VISION

177

pened three weeks back
...
They
were therefore the losers, as the minority of players avoided the
crowd by staying at home
...

On the other hand, your strategy might be to go based only on the
results of the past week, regardless of what happened the two
weeks before
...
Over time, the agents will learn to use the strategies that work the best most often
...

You don’t have to be a drinker to appreciate the usefulness of
the minority game for describing social situations
...
You can imagine
many such scenarios in economics, for instance, such as when it’s
better to be a buyer or a seller
...

Further work on the minority game has shown that in some
circ*mstances it is possible to predict which choice is likely to be
in the minority on next Friday night
...
As the number of
players goes down (or their memory capacity goes up), at some
point the outcome is no longer random and can be predicted with
some degree of statistical confidence
...
And it certainly is a far cry from Asimov’s psychohistory
...
Today’s nonfictional anthropologists have used game theory to demonstrate such cultural
diversity, but it’s something else again to ask game theory to explain it
...

At first glance, the prospects for game theory encompassing
the totality of cultural diversity seem rather bleak
...
People are not totally rational beings acting
purely out of self-interest as traditional game theory presumes, for
example
...
And societies develop radically different cultural patterns of collective behavior
...

As Jenna Bednar and Scott Page of the University of Michigan
have described it, game theory would seem hopeless as a way to
account for the defining hallmarks of cultural behavior
...
”12 But human cultures aren’t like that
...
But behavior differs dramatically from one
culture to the next
...
When incentives change, behavior often remains stubbornly stuck to cultural
norms
...

“Cultural differences—the rich fabric of religions, languages,
art, law, morals, customs, and beliefs that diversifies societies—and
their impact would seem to be at odds with the traditional game
theoretic assumption of optimizing behavior,” say Bednar and
Page
...
”13
But game theory has a remarkable resilience against charges of
irrelevance
...
“Surprisingly,” Bednar and Page declare, “game theory is
up to the task
...
But the effort to figure out optimal behaviors in a complicated situation is often considerable
...
With limited brain power (and everybody’s is), you
can’t always afford the cost of calculating the most profitable
response
...
You are in fact engaging in an ensemble of many different games simultaneously, imposing an even greater drain on your
brain power
...
”
So Alice and Bob (remember them?) may be participating in a
whole bunch of other games, requiring more complicated calculations than they needed back in Chapter 2
...
Even if they face identical situations in the
one game they play together, their choices might differ, depending
on the difficulty of all the other games they are playing at the
same time
...
”
In other words, with limited brain power, and many games to
play, the “rational” thing to do is not to calculate pure, ideal game
theory predictions for your choices, but to adopt a system of general guidelines for behavior, like the Pirate’s Code in the Johnny

180

A BEAUTIFUL MATH

Depp movie Pirates of the Caribbean
...
Cultural patterns of behavior emerge as individuals tailor a toolkit of strategies to apply in various situations,
without the need to calculate payoffs in detail
...

Thus agents in different environments may play the same game
differently
...
In
the various games, incentives for the self-interested agents differed,
to simulate different environmental conditions
...

This approach doesn’t explain everything about culture, of course,
but it shows how playing games can illuminate aspects of society
that at first glance seem utterly beyond game theory’s scope
...

In any event, recent developments in the use of statistical physics in describing networks and society—and game theory’s intimate relationship with both—instill a suspicion that game theory
and physics are somehow related in more than a superficial way
...
In fact, that’s exactly what has
already happened in economics
...
”
“The substantial contribution of physics to economics is still in
an early stage, and we think it fanciful to predict what will ultimately be accomplished,” wrote the authors, Doyne Farner and
Eric Smith of the Santa Fe Institute and Yale economist Martin

ASIMOV’S VISION

181

Shubik
...
”16
Yet there are areas of overlap, they note, and “striking empirical regularities suggest that at least some social order
...
” The role of markets in setting
prices, allocating resources, and creating social institutions involves
“concepts of efficiency or optimality in satisfying human desires
...
In physics, analogous concepts correspond to physical systems treated with statistical mechanical math
...

One possible weakness in the analogy between physics and
game theory, though, is that physics is more than just statistical
mechanics
...
If the physics–game theory
connection runs deep, there should be a quantum connection as
well
...

Meyer’s Penny
Quantum fun and games
Do games have anything deeper to say about physics,
or vice versa? Maybe
...

—Chiu Fan Lee and Neil F
...

Captain Jean-Luc Picard places a penny heads up in a box, so
that it can be touched but not seen
...
Without knowing what Q has done, Picard then must decide
to flip, or not flip, the coin as well
...
He
either flips the penny or leaves it alone
...

They play the game 10 times, and Q wins them all
...

The penny-flipping game is an old game theory favorite
...
(Whether
you flip the coin or leave it alone corresponds to veering out of the
path of the oncoming car or continuing straight on
...
Ten wins in a row for one player defies any
reasonable definition of luck
...
But the wiser
Picard would have pondered the situation a little longer and eventually would have realized that Q’s name must be short for quantum
...

As it turns out, Earth’s physicists did not need an alien to teach
them about quantum games
...

It was an unexpected twist in the story of game theory, as quantum
games disrupted the understanding of traditional “classical” games
in much the way that quantum mechanics disturbed the complacency of classical physics
...
And it may even be (though
perhaps not until the 24th century) that quantum games will cement the merger of game theory and physics
...

Now, if you’ve been reading carefully all this time, it might
seem a little unfair that, after coming to grips with the complexities of game theory, network math, and statistical mechanics, you
must now face the bewildering weirdness of quantum physics on
top of it
...
Besides, you don’t
need to know everything there is to know about quantum physics
to see how quantum game theory works
...

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A BEAUTIFUL MATH

QUANTUM TV
I’ve described this quantum confusion before (in my book The Bit
and the Pendulum) by relating it to television
...
(Nowadays they usually arrive via cable
...
The realm of atoms, molecules, and particles even smaller
works in a similar way
...
In particular,
you cannot say that a particle occupies any specific location
...

An observation will find it located in one of the many possible
positions that the quantum equations allow
...
”
In recent years, it has become generally agreed that humans are not
necessary to perform an observation or measurement on a particle
...

That is to say, an atom, on its own, cannot be said to occupy a
specific location
...
This phenomenon is known as decoherence
...

This feature of quantum physics has been an endless source of
controversy and consternation for physicists and nonphysicists
alike
...
In the subatomic world, reality is fuzzy, encompassing a
multiplicity of possibilities
...
It’s not just that you don’t know where an atom is—
it occupies no definite location, but rather occupies many locations
simultaneously
...

Personally, I find this to be an uncanny analogy
...
In the math of quantum physics, the location of a particle cannot be definitively determined,
but only described probabilistically—perhaps you will find it in
Region A 70 percent of the time, and in Region B 30 percent of
the time
...
There’s no reason to believe that the math of
molecules would be relevant to making choices in economic games
...

To be sure, some experts have doubted that quantum games
offer any real benefits that could not be obtained in other ways
...
And new technologies have begun to make
experimental demonstrations of quantum games possible
...
In a way, the surprise is that nobody did it
sooner
...
And the
initial impetus for quantum games stemmed from the fact that von
Neumann was also a pioneer in developing digital computers
...
“I was giving a talk to the whole

186

A BEAUTIFUL MATH

research division and I wanted to come up with something new to
talk about,” Meyer told me when I visited him at his office on the
UCSD campus in La Jolla
...
“And of course von Neumann is also responsible
for the architecture of modern computers to a large extent, so
that’s relevant to Microsoft also,” Meyer noted
...
So I thought, OK, how can I put
these things together?” It seemed obvious that the thing to do was
explore the possibility of a quantum version of game theory
...
Von Neumann had shown that in two-player
zero-sum games each player could always have a “best” strategy,
but that it was not always a pure strategy—making the same play
(for given circ*mstances) every time
...

“Now the fact that it’s called a mixed strategy versus a pure
strategy is not an accident,” Meyer pointed out
...
You have
pure states and you have mixed states—mixed states are probability distributions over pure states
...
”
So Meyer’s talk at Microsoft explored a way of bringing quantum theory’s multiple “mixed state” realities to game theory
...
It’s a simple game where the idea is simply to outguess your opponent, since there is no particular logic involved in
deciding whether to flip the penny or not
...

In a nonquantum or “classical” version of the game, Picard’s
best strategy would be to flip the coin half the time (in other
words, he should flip a coin to decide whether to flip the coin),
thereby making sure there will be no pattern to detect
...
If
both players observe those strategies, they should each win half
the time
...

In Meyer’s quantum scenario, Picard still must play classically
...

In the lingo of quantum information physics, such a dualvalued head-tail combination is known as a qubit—short for a
“quantum bit” of information
...
A classical coin must fall either
heads or tails, but a quantum coin is permitted multiple possibilities, a mix of heads and tails at the same time
...
)
In actual quantum information experiments, the “coin” is usually a particle of light—a photon—and heads or tails might correspond to how the photon is spinning (the direction in which its
axis of spin is pointing)
...
Filters (like the polarized lenses of
sunglasses) can block or transmit polarized light depending on its

188

A BEAUTIFUL MATH

orientation, usually designated as horizontal or vertical
...
(Of course, you could also transmit a photon
oriented in between vertical and horizontal—that is, tilted
...
)
For Meyer’s penny, turning it to heads or tails corresponds to
setting the orientation of a polarizing filter—showing the head,
say, but hiding the tail side of the coin
...
Since Q plays first, he can use his quantum magic to flip the
coin into a 50-50 mix of heads and tails
...
) Therefore it doesn’t matter what Picard does on
his next move
...
Q can then perform a reverse
quantum move that will return the coin to its original condition—
heads up
...
If you define heads as a spin
pointing north on the z axis (in the “+z” direction), then tails would
be pointing the opposite way (south, or the -z direction)
...
Q , however, can perform a
quantum twist to the spin, pointing it “east” (along the +x axis)
...
2 Picard’s strategy of
flipping the coin half the time—guaranteed by classical game
theory to be the best strategy he can play—turns out to be a worthless strategy against a quantum player
...
Game

MEYER’S PENNY

189

theory supposedly tells you the best strategy you can play in a
game
...

The footnote reads, in bold type, that game theory tells you the
best strategy only if you are able to ignore the multiple possibilities
of quantum physics
...

QUANTUM DILEMMAS
Meyer’s paper reporting the substance of his Microsoft talk was
published in Physical Review Letters in 1999
...
And
in the next few years, dozens of other papers began to explore a
whole gamut of quantum games
...
Some papers applied quantum game principles to economics, suggesting, for instance, that
the multiple possibilities of quantum physics might be applied to
selecting the best mix of stocks in a portfolio, or in making decisions about when and whether to buy or sell
...

In some cases, merely allowing a “referee” to mediate between players, without any quantum magic, can achieve the same effects
...
After thinking this through, however,
Meyer concluded that there are still ways to make games truly
quantum in character
...
“But once you start
thinking about adding classical communication
...
” In other words, if the mediators or players are
allowed to use quantum communication systems, quantum benefits
may indeed be achieved
...
“So it’s not implausible that you could
...
”4
If so, he said, various real-life problems might be addressed
with quantum game theory
...
Combinatorial auctions, such as the bidding
by many companies for various licenses to be issued by the government, could perhaps be managed more efficiently by using quantum information to coordinate the bids
...
“There’s a huge realm to play
in here
...
and it might
even be practical at some point
...
Qubits can be used to
transmit secret codes with uncrackable quantum protection from
eavesdroppers, guaranteeing that the code cannot be intercepted
without detection
...
Quantum-coded signaling to military satellites is well
within the realm of technological possibility within the future
timeframe of Pentagon budget planning
...
In fact, one of
the neatest things about quantum games is that they could give
quantum computers something to do
...
If they can be
scaled up to a useful size, quantum computers could exploit the
multiple quantum realities to do many calculations simultaneously,
thereby drastically shortening the time it takes to solve some very
hard problems
...
Massive database searching could be faster with a quantum computer, for example, and breaking secret codes is certainly
something you wouldn’t want to try without one
...
For small numbers,
that’s easy to do: 15, for instance, is obviously the product of the
two prime numbers 5 and 3; 35 is the product of the primes 5 and
7
...
The way secret
coding systems are set up, you can encode a message if you know
the long number, but decode it only if you know its two prime
factors
...
But in
1994, the mathematician Peter Shor showed that the primes could
be discovered quickly—if you had a quantum computer at your
disposal
...
Designing and building a quantum com-

192

A BEAUTIFUL MATH

puter is much simpler on paper than in practice, though, and it will
doubtless be decades before you’ll be able to buy one at Best Buy
...
In fact, factoring 15 has been achieved with a quantum computer based on the same technology used in MRI medical imaging
...
In the following year, Chinese physicists Lan
Zhou and Le-Man Kuang outlined how to set up a quantum game
communication system using lasers and mirrors and other optical
devices in a paper published in Physics Letters A
...
When two particles of
light (photons) are emitted simultaneously from the same atom, for
example, they retain an ethereal connection—measuring one seems
to affect the other even if it is meters, miles, or light-years away
...

When two photons are entangled, they share quantum information in a peculiar way
...
And then the other one stops spinning,
too! So suppose I possess two pennies, spinning within opaque
boxes, entangled in such a way that if one is observed to be heads,
the other will turn up tails
...
She can’t resist opening the package right away and discovers the penny on the bottom of the box, showing heads
...
Even if I don’t look in my box, I know
damn well that the penny shows tails (once my sister has called to

MEYER’S PENNY

193

tell me that hers is heads)
...
The same thing happens for real with actual photons
of light, when you measure not heads or tails but how the photon
is spinning or the orientation of its polarization
...
In
a quantum game, entangled particles can carry information about a
player’s choice in such a way that one choice will influence another
...
In the classical game, the
players typically choose to defect because they cannot be sure that
their partner will cooperate
...
But each individual prisoner’s best strategy is to rat out the
other (to avoid the risk of a much longer sentence)
...
“We have a dilemma,” write quantum game theorists
Adrian Flitney and Derek Abbott, “some form of which is responsible for much of the misery and conflict throughout the world
...
That’s what playing with entangled photons offers
...
You can shoot your photon into the maze in different
ways—so that it would end up in the “defect” detector, for instance, or in the “cooperate” detector
...
But the maze can be set up so that the photons from the two
players become entangled, with the result that both will end up
cooperating
...

This work shows that quantum game theory, at least in principle, could be used to alter in a deep and profound way the choices

194

A BEAUTIFUL MATH

people make given the choices of others
...
The idea is that a neighborhood group proposes building
a project of public benefit, such as a park, to be paid for by voluntary contributions
...
But standard game theory
suggests that many people who want the park will contribute little
or no money, reasoning that others will fork over enough to pay
for it
...

In 2003, scientists from HP Labs in Palo Alto, California,
posted a paper on the Internet showing how a quantum public
goods game provides strategies that reduce the temptation to freeload
...
So if you send
your pledge via a quantum information channel, its message can
depend on the messages from the other contributors
...
Using quantum-entangled photons to communicate their intentions could allow a coordination of commitments that otherwise couldn’t be
guaranteed
...
8

QUANTUM VOTING
The same principle could be applied to other sorts of community
communication issues, including voting, especially in elections with
multiple candidates
...

MEYER’S PENNY

195

Here, I think, is a real potential for coping with some of the
mathematical problems inherent in today’s democratic system of
voting
...
Here’s how it can work:
In a primary election, Candidate A gets 37 percent of the votes,
Candidate B gets 33 percent, and Candidate C gets 30 percent
...
But for most of candidate B’s
voters, C was the second choice
...
So if C were running against
A alone, C would win
...
But in the primary, C finished third so the ultimate winner
will be A or B
...
A
quantum voting scheme could, by incorporating multiple possibilities in the voting, reach a more “democratic” election result
...
And it may even be
possible that quantum game theory underlies much deeper aspects
of nature and of life
...
In particular,
Azhar Iqbal of the University of Hull in England argues that quantum entanglement could influence the interactions of molecules
leading to a more stable mix of ingredients than would otherwise
occur (in analogy to an evolutionary stable strategy for organisms
in an ecosystem)
...
9 If there’s anything
to this—and it would seem to be far too early to say—then you
could imagine something like quantum game theory playing a role
in the origin of stable sets of self-replicating molecules—in other
words, life itself
...
)

196

A BEAUTIFUL MATH

In any case, quantum game theory offers a new perspective on
both games and physics, with implications awaiting further exploration
...

Life and physics, it seems, are all mixed up
...

Pascal’s Wager
Games, probability, information, and ignorance
Although this may seem a paradox, all exact science is
dominated by the idea of approximation
...
He wrote a genius-caliber treatise
on geometry at age 16 and invented a rudimentary computer to
assist the calculations of his tax-collector father
...
He died at 39, leaving a legacy, in the words of the
mathematician E
...
Bell, as “perhaps the greatest might-have-been
in history
...
What Pascal offered
was not religious counseling on the evils of gambling, but mathematical advice on how to win
...
What’s more, out of Pascal’s religious ruminations came an
idea about probability that was to emerge centuries later as a critical concept in mathematics, with particular implications for game
theory
...
You need to know
what’s at stake
...
Or you
might consider playing it safe by betting on a sure thing even if
the payoff was small
...

Pascal framed this issue in his religious writings, specifically in
the context of making a wager about the existence of God
...
If you believe
in God, and that belief turns out to be wrong, you haven’t lost
much
...
Even if God is a low-probability deity, the
payoff is so great (basically, infinite) that He’s a good bet anyway
...
“Let us estimate these two chances
...
Wager, then, without hesitation that
He is
...
3 It illustrated the kind
of reasoning that goes into calculating the “mathematical expectation” of an economic decision—you multiply the probability of
an outcome by the value of that outcome
...

Pascal’s wager is often cited as the earliest example of a mathbased approach to decision theory
...
And when your best decision depends on what other people are deciding, simple decision
theory no longer applies—making the best bets becomes a problem in game theory
...
) Still, probabilities and expected payoffs remain intertwined with game theory in a profound and complicated
way
...
And probability arises not only in making measurements and testing hypotheses, but also in the very description of physical phenomena, particularly in the realm of
statistical physics
...
So game theory’s intimate relationship with
probabilities, I’d wager, is one of the reasons why it finds such
widespread applicability in so many different scientific contexts
...

So far, attempts to devise a sociophysics for describing society
have mostly been based not on game theory, but on statistical mechanics (as was Asimov’s fictional psychohistory)
...
In
fact, the mixed strategies used by game players to achieve a Nash
equilibrium are probability distributions, precisely like the distributions of molecules in a gas that statistical physics quantifies
...

That is to say, they can be expressed using the same mathematical
language
...
It’s just that few people have noticed it
...
Among
them is David Wolpert, a physicist-mathematician at NASA’s Ames
Research Center in California
...
He pursues his own
intuitions and interests along the amorphous edges separating (or
connecting) physics, math, computer science, and complexity
theory
...

In early 2004, Wolpert’s name caught my eye when I noticed a
paper he posted on the World Wide Web’s physics preprint page
...
In fact, as
Wolpert showed in the paper that attracted my attention to this
issue in the first place, a particular approach to statistical mechanics turns out to use math that is equivalent to the math for noncooperative games
...
The mixed strategies used by players to achieve a Nash
equilibrium are probability distributions, just like the distribution
of energy among particles described by statistical physics
...
I asked
what had motivated him to forge a link between game theory and
statistical physics
...

Wolpert had been working on collective machine learning systems, situations in which individual computers, or robots, or other
autonomous devices with their own individual goals could be coordinated to achieve an objective for the entire system
...
He noticed similarities in his work to a paper published in
Physical Review Letters about nanosized computers
...

PASCAL’S WAGER

201

“The editor actually came back and said ‘Well,
...
“And I was annoyed
...
After all, a
bunch of agents with their own agendas, yet pursuing a common
goal, is entirely analogous to players in a game seeking a Nash
equilibrium
...
5
Games deal with players; physics deals with molecules
...
The mix of all the players’
strategies would then be like the combined set of motion states of
all the molecules, as ordinarily described by statistical physics
...
You could
then do exactly the same sort of calculation for the combined
strategies of all the players in a game
...

“Those topics are fundamentally one and the same,” he wrote
in his paper
...
”6
Wolpert’s mathematical machinations were rooted in the idea
of “maximum entropy,” a principle relating standard statistical
physics to information theory, the math designed to quantify the
sending and receiving of messages
...
Wolpert, for one, calls Jaynes’s
work “gloriously beautiful” and thinks that it’s just what scientists
need in order “to bring game theory into the 21st century
...

It seems essentially simple but nevertheless poses tricky complications
...
In any event, its explanation requires a

202

A BEAUTIFUL MATH

brief excursion into the nature of probability theory and information theory, the essential threads tying game theory and statistical
physics together
...
Even today there exist separate schools of
probabilistic thought, generally referred to by the shorthand labels
of “objective” and “subjective
...

In a way, that’s a bit surprising, since probability theory really
lies at the very foundation of science, playing the central role in
the process of analyzing experimental data and testing theories
...
You’d think they’d have it all
worked out by now
...
There are different philosophies
and approaches to science
...

Science is like grammar
...
A true grammarian does not tell people
how they should speak, but codifies the way that people actually
do speak
...

That’s why science is not all experiment, and not all theory, but a
complex interplay of both
...
And in
most realms of science, you need math to test the mesh
...
(Different ideas about
how to perform the test, then, lead to different conceptions of
probability
...
Laplace and
others showed the way to estimate how far off your measurement
was likely to be from the true value for a particular degree of
confidence
...
He concluded that there was only one chance
in 11,000 that the true mass of Saturn would deviate from the
then-current measurement by more than 1 percent
...
6 percent
...

But what does probability itself really mean? If you ask people
who ought to know, you’ll get different answers
...
You observe in what fraction of all cases that
event happens and thereby measure its objective probability
...
Measuring how
often something happens gives you a frequency, not a probability,
the subjectivists maintain
...
Dozens of books have been devoted to that controversy, which is largely irrelevant to game theory
...

Subjective statistics often goes under the label of Bayesian,
after Thomas Bayes, the English clergyman who discussed an approach of that nature in a paper published in 1763 (two years after
his death)
...
In any case,
the Bayesian viewpoint today comes in a variety of flavors, and
there is much disagreement about how it should be interpreted and
applied (perhaps because it is, after all, subjective)
...
It’s just that in some
cases it seems more convenient, or more appropriate, to use one
rather than another, as Jaynes pointed out half a century ago
...
He noted that both views, subjectivist and objectivist, were needed in physics, but that for some types of problems only the subjective approach would do
...
If you are given a box full of particles but know nothing
about them—not their mass, not their composition, not their internal structure—there’s not much you can say about their behavior
...
In other words, your ignorance about the behavior of the particles is at a maximum
...
Well, that helps in doing the
calculations, perhaps, but is there any real basis for assuming the
probabilities are equal? Except for certain cases where an obvious
symmetry is at play (say, a perfectly balanced two-sided coin),
Jaynes said, many other assumptions might be equally well justified (or the way he phrased it, any other assumption would be
equally arbitrary)
...
Shannon was interested in quantifying communication, the sending of messages, in a way that would
help engineers find ways to communicate more efficiently (he
worked for the telephone company, after all)
...
Before communication begins,
all messages are possible, so uncertainty is high; as messages are
actually received, that uncertainty is reduced
...
But suppose, for example, that
all you wanted to do was send someone a single English word
(from among all the words in a standard unabridged dictionary,
about half a million)
...
In other words,
you have reduced the uncertainty by half (which so happens to
correspond to one bit of information)
...
He found a formula for a quantity that measures that
uncertainty precisely—the greater the uncertainty, the greater the
quantity
...

Physicists’ entropy is a measure of the disorder in a physical
system
...
Then you remove the partition between the
compartments
...
But something else has happened—you no longer know where the
molecules are
...
Shannon showed that his formula for
entropy in communication—as a measure of ignorance or uncertainty—is precisely the same equation that is used in statistical
mechanics to describe the increasing entropy of a collection of
particles
...
En-

206

A BEAUTIFUL MATH

tropy is synonymous with uncertainty
...

So here’s a clue about what to do when you know nothing
about the probabilities in the system you want to study
...
Assuming maximum entropy/
ignorance, then, is not just an assumption; it’s a factual statement
about your situation
...
In his view, statistical mechanics itself just
became a system of statistical inference about a system
...

In particular, you now can justify the notion that all the possibilities are equally possible
...
Everything not explicitly excluded by the information you’ve got has to be viewed as
having some probability of occurring
...
) And if you know nothing,
you cannot say that any one possibility is more likely than any
other—that would be knowledge
...
But if you know
nothing at all, there’s only one probability distribution that you
can identify for making your bets: the one that has the maximum entropy, the maximum uncertainty, the maximum ignorance
...

This is the magic that makes it possible to make a prediction

PASCAL’S WAGER

207

even when knowing nothing about the particles or people you’re
making the prediction about
...
But it’s still the best possible prediction
you can make, the likeliest answer you can identify, when you know
nothing to begin with
...
“Whether
or not the results agree with experiment, they still represent the
best estimates that could have been made on the basis of the information available
...
Here’s a simple example
...
All you know
are the rules (that is, the laws of nature)—everybody gets a grade,
and the grade has to be A, B, C, D, or F (no incompletes allowed)
...
What is your best prediction of the average grade
for the kids in the class? In other words, how do you find a probability distribution for the grades that tells you which grade average is the most probable?
Applying the maxent or maximum ignorance principle, you
simply assume that the grades can be distributed in all possible
ways—all possible combinations equally likely
...
Another would
be all F’s
...
You could
have 50 C’s, 20 B’s and 20 D’s, 5 A’s and 5 F’s
...

In statistical physics, this sort of thing is called the “canonical
ensemble”—the set of possible states for the molecules in a system
...
Many different possible

208

A BEAUTIFUL MATH

microstates (distributions of grades) can correspond to the same
average (the macrostate)
...
(You’re talking something close to 10 to the
70th power
...
Of all the possible microstate
combinations, many more work out to a C average than to any
other grade
...
But you can get a C average in many different ways—100 C’s, 50 A’s and 50 F’s, 20 students getting each of the five grades, and so on
...
In 100 trials, many combinations give an average of 2, but
only a few will give an average of 0 or 4
...

BACK TO THE GAME
In game theory, a player’s mixed strategy is also a probability distribution, much like grades or penny flips
...
In a multiplayer
game, there is at least one mix of all players’ mixed strategies for
which no one player could do any better by changing strategies—
the Nash equilibrium, game theory’s most important foundational
principle
...

While it’s true that, as Nash showed, all games (with certain qualifications) have at least one Nash equilibrium, many games can have
more than one
...
And even if there is only one Nash equilibrium in a
complicated game, it is typically beyond the capability of a com-

PASCAL’S WAGER

209

mittee of supercomputers to calculate what all the players’ mixed
strategies would have to be
...
In a world where most people
can’t calculate the sales tax on a cheeseburger, that’s a tall order
...
So game theory appears to assume that each player can do what supercomputers can’t
...
Modern approaches to game theory often assume,
therefore, that rationality is limited or “bounded
...
An enormous amount of research, of the highest caliber, has modified and elaborated game
theory’s original formulations into a system that corrects many of
these initial “flaws
...
Nevertheless, many game
theorists often cling to the idea that “solving a game” means finding an equilibrium—an outcome where all players achieve their
maximum utility
...

When I visited Wolpert at NASA Ames, a year after our conversation in Boston, he pointed out that the search for equilibrium
amounts to viewing a game from the inside, from the viewpoint of
one of the participants, instead of from the vantage point of an
external scientist assessing the whole system
...
If you look at it that way, you know you can never
be sure how a game will end up
...
“It’s going to be
the case that whenever you are given partial information about a
system, what must pop out at the other end is a distribution over
possibilities, not a single answer
...
If
you think about it, you realize that no physicist computing the
thermodynamic properties of a gas worries about what an individual molecule is doing
...
You can’t know what each
molecule is up to, but you can calculate, statistically, the macroscopic behavior of all the molecules combined
...
Statistical physicists
studying gases don’t know what individual molecules are doing,
and game theorists don’t know what individual players are thinking
...
Similarly, game theorists ought to be
able to make statistical predictions about what will happen in a
game
...
Scientists have limited information about the systems
they are studying and try to make the best prediction possible
given the information they have
...

All sciences face this sort of problem—knowing something
about a system and then, based on that limited knowledge, trying
to predict what’s going to happen, Wolpert pointed out
...
”12
From this point of view, another sort of mixed strategy enters

PASCAL’S WAGER

211

game theory
...

The scientist describing the game also has a kind of “mixed strategy” of possible predictions about how the game will turn out
...
“If I
give you a game of real human beings, no, you’re not going to
always have the same outcome
...
It’s not going to be the case they are
always going to come up with the exact same set of mixed strategies
...
”
This is clearly taking game theory to another level
...
And how do
you find those probability distributions of mixed strategies? By
maximizing your ignorance, of course
...
Using this approach, you don’t need to assume that the players in a
game have limits on their rationality; such limits naturally appear
in the formulas that information theory provides
...

“When you need a prediction, a probability distribution won’t
do,” said Wolpert
...
” The underlying axioms for the mathematical basis for making such a decision were worked out in the 1950s
by Leonard Savage13 in some detail, but they boil down to something like Pascal’s Wager
...

“If you predict X, but the truth is Y, how much are you hurt?

212

A BEAUTIFUL MATH

Or conversely, how much do you benefit?” Wolpert explained
...
But in other instances
...
”
Decision theory dictates that you should make the prediction
that minimizes your expected loss (“expected” signifying that the
relative probabilities of the choices are taken into account—you
average the magnitudes of loss over all the possibilities)
...

“In other words, for the exact same game, your decision as the
external person making the prediction is going to vary depending
on your loss function,” he says
...
” And so sometimes the likeliest outcome of a game will not be a
Nash equilibrium
...
But when game theory is cast in the informationtheoretic equations of maximum entropy, the answer becomes clear
...
In other words, a player attempting to achieve a maximum
payoff must factor in the cost of computing what it takes to get
that payoff
...

What’s more, individual differences can influence the calculations
...
Temperature relates ignorance
(or uncertainty) to the cost of computing a strategy—more uncertainty about what to do means a higher cost of figuring out what
to do
...

“So what that means,” Wolpert explained, “is that it is literally
true that somebody who is purely rational, who always does the
best possible thing, is cold—they are frozen
...
That just falls
out of the math
...
”14
Temperature, in other words, represents a quantification of irrationality
...
With game players, higher temperature means a
greater chance that they won’t be maximizing their payoff
...
“It’s the exact same thing
...
” You are still likely to play strategies that would increase your payoff, but just how much more
likely depends on your temperature
...
It arises naturally from adopting the viewpoint of somebody looking from outside the game
instead of being inside the game
...
“Game theory has always
had probability theory inside of it, because people play mixed
strategies, but game theory has never actually applied probability
theory to the game as a whole
...
”
Ultimately, the idea of a player’s temperature should allow better predictions of how real players will play real games
...
But if you know
something about the students—maybe all are honors students
who’ve never scored below a B—you can adjust the probability
distribution by adding that information into the equations
...
With
collaborators at Berkeley and Purdue, Wolpert is beginning to test
that idea on real people—or at least, college students
...
Do they actually get
more rational or less rational? What are the correlations between
different individuals’ temperatures? Do I get more rational as you
get less rational?”
If, for instance, one player is always playing the exact same
move, that makes it easier for opponents to learn what to expect
...
“So in these experiments our intention is
to actually look for those kinds of effects
...
It sounded like Wolpert
was saying that all this knowledge could be fed into the probability distribution formulas to improve game theory’s predictive
power
...

“Let’s say that you know something from psychology, and
you’ve gotten some results from experiments,” he said
...
You also
know something about their degree of being risk averse, and this,
that, or the other
...
”
Adding such knowledge about real people into the equations
reduces the ignorance that went into the original probability distribution
...
“It’s
a way of actually integrating game theory with psychology, formally,” Wolpert said
...
quantification of individual human beings’ behavior integrated with an actual
mathematical structure that deals with incentives and utility functions and payoffs
...
“This actually is a way of trying to
get a mathematics of psychohistory in Isaac Asimov’s sense,”
Wolpert said
...
”16
Just as I had suspected
...
It’s the math
that merges game theory with statistical physics
...
It should be called Game Physics
...
But it captures the idea of
psychohistory or sociophysics pretty well
...

Epilogue
Let the physical basis of a social economy be given—
or, to take a broader view of the matter, of a society
...
This consists of not setting up
one rigid system of apportionment
...
This system

...
’
—Von Neumann and Morgenstern,
Theory of Games and Economic Behavior

Despite its title, the science fiction cult classic Ender’s Game isn’t
really about game theory, at least not explicitly
...

It’s all about choosing strategies to achieve goals—about adults
plotting methods for manipulating young Ender Wiggin, Ender
choosing among maneuvers to win on a simulated battlefield, and
Ender’s siblings’ devising tactics for influencing public opinion
...
Ender’s brother Peter, for instance, epitomizes the selfish
rational agent of game theory’s original naive formulation:
217

218

A BEAUTIFUL MATH
Peter could delay any desire as long as he needed to; he could conceal
any emotion
...
He would only do it if the advantages outweighed the
risks
...
1

Ender himself represents the social actor who plays games with
a combination of calculation and intuition, more in line with
the notion of game theory embraced by today’s behavioral game
theorists:
“Every time, I’ve won because I could understand the way my enemy
thought
...
I could tell what they thought I was
doing, how they wanted the battle to take shape
...
I’m very good at that
...
”2

That is, after all, what the modern science of game theory is all
about—understanding how other people think
...
It is also what
Isaac Asimov’s fictional psychohistory was all about, and what
the centuries-long quest by social scientists has been all about—
discerning the drumbeat to which society dances
...

The modern search for a Code of Nature began in the century
following Newton’s Principia, which established the laws of motion and gravity as the rational underpinning of physical reality
...
That dream persisted through the
19th century into the 20th, from Adolphe Quetelet’s desire to
describe society with numbers to Sigmund Freud’s quest for a deterministic physics of the brain
...
By the end of the 20th century, the quest for a Code of Nature was taken up by physicists
who wanted to use those statistics to bring the sciences of society

EPILOGUE

219

and the natural world back together
...

PHYSICS AND EVERYTHING
Historically, the physicist’s notion of everything has been a bit
limited, though
...
In the century just gone by, Einstein added
cosmic time and space to the mix
...
Through
the 20th-century physicist’s eyes, then, “everything” comprised
mass-energy and space-time
...
Awakened by the metaphorical power of the digital computer, astute observers realized
that information was the glue connecting the outside world to its
scientific description
...

Information opened physicists’ eyes to the rest of reality
...
Biology included people
...
So physicists began applying their favorite all-purpose tool—
statistical mechanics—to everything from the stock market to flu
epidemics
...
By the dawn of the
21st century, real-life physicists were trying to do almost exactly
the same thing that Seldon had done, using statistical mechanics to
build mathematical models of society for the purpose of making
predictions
...
Von Neumann and Morgenstern focused on economics, but
clearly viewed economics as simply one (albeit a major) example
of social science in general
...

A few years later, John Nash took a second major step toward a
mathematics of society by introducing the Nash equilibrium into
game theory’s arsenal of ideas
...
The existence of a Nash equilibrium in any
game implied that societies could be stable—nobody having incentive to change their behavior, as any deviation would lower
their payoff if everybody else continued to play the same way
...
Only rarely is one single “pure” strategy consistently your best
bet
...

This idea of a mixed strategy is a recurring theme in game
theory and its applications to various aspects of life and society
...
The human race plays
a mixed strategy, comprising cooperators, competitors, and punishers
...
Even in the physical
realm, quantum physics shows that reality itself is a mixed strategy
at the subatomic level, a feature that game theorists may be able to
exploit to solve their thornie*st dilemmas
...
And probability distributions, it just so happens, are what statistical physics deals

EPILOGUE

221

with as well
...

Like matter and energy, or space and time, game theory and physics are different sides of a coin
...
It’s a neat, tight fit, and it’s a mystery why it took
so long for game theory and physics to mutually realize this underlying relationship
...
Von Neumann referred to the usefulness of statistics in describing large numbers of interacting
agents in an economy
...

Nash, after all, studied chemical engineering and chemistry at
Carnegie Tech before becoming a math major, and his dissertation
at Princeton drew on the chemical concept of “mass action” in
explaining the Nash equilibrium
...
Borrowing the physical concept of equilibrium in chemical systems of molecules, Nash derived an analogous
concept of equilibrium in social systems composed of people
...
The seed of the physics-society link resided
within Nash’s beautiful mind
...
Game theory provides the

222

A BEAUTIFUL MATH

common mathematical language for unifying these sciences, sciences that represent the puzzle pieces that fit together to generate
life, mind, and culture—the totality of collective human behavior
...

After all, both physical and living systems seek stability, or
equilibrium
...
Game
theory shows why reaching an equilibrium point requires mixed
strategies—and how this need for mixed strategies drives the creation of complexity
...
Game theory describes the evolutionary process that produces mixtures of different
species, mixtures of different types of people, mixtures of different strategies that people employ, mixtures of different cultures
that arise in the mixture of environments found around the planet
...
The brains that choose from a mix of strategies
are networks of nerve cells; the societies that exhibit a mixture of
cultures are networks of brains
...

Game theory is not, however, the same as the popular “Theory
of Everything” that theoretical physicists have long sought
...
Once
you know how the pieces of atoms are put together, this view
holds, you don’t need to worry about everything else
...
It’s about the realm of
life that builds itself upon the universe’s physical foundation
...

DANGER
There has always been a danger in seeking a Code of Nature—a
risk that it would be regarded as a dogmatic deterministic view of
human behavior, denying the freedom of the human spirit
...
The idea that a
Code of Nature is inscribed into human genes, advanced in the
1970s under the label sociobiology, evoked a vitriolic response
demonstrating how invective often overwhelms intellect
...

Game theory, on the other hand, offers a possible rapprochement between the advocates of genetic power and the defenders
of human freedom
...
It acknowledges the power of evolution—in fact, it helps to explain evolution’s ability to generate
life’s complexities
...
Game theory poses no universal genecontrolled determinant of human social behavior, but rather requires, as Nash’s math showed, a mixed strategy
...

Game theory’s potential scientific power is so great, I think,
because it is so intellectually commodious—not narrow and confining, but capable of accommodating many seeming contradictions
...
Game theory encompasses the

224

A BEAUTIFUL MATH

coexistence of selfishness and sympathy, competition and cooperation, war and peace
...
Game theory connects simplicity to complexity by reconciling the tension between
evolutionary change and stability
...
Game theory bridges the sciences of mind and mindless
matter
...
It offers a mathematical recipe for making sense of what seems to be a hopelessly
messy world, providing a tangible sign that the Code of Nature is
not a meaningless or impossible goal for scientists to pursue
...

“We want to understand human nature,” says Joshua Greene, a
neuroscientist and philosopher at Princeton
...
”3
Success may still be a long way off
...
From people to cities,
corporations to governments, all of the elements of society must
ultimately mesh
...

“The idea is really to have, in the end, a seamless understanding of the universe, from the most basic physical elements, the
chemistry, the biochemistry, the neurobiology, to individual human
behavior, to macroeconomic behavior—the whole gamut
seamlessly integrated,” says Greene
...
”

Appendix

Calculating a Nash Equilibrium
Consider the simple game discussed in Chapter 2, where Alice and
Bob compete to see how much of a debt to Alice that Bob will
have to pay back
...
The payoffs in the game matrix are the
amounts Bob pays to Alice, so Bob’s “payoff ” in each case is the
negative value of the number indicated
...

In this example, Alice chooses Bus with probability p, and
Walk with probability 1 – p (since the probabilities must add up to
1)
...

Alice can calculate her “expected payoff ” for choosing Bus or
Walk as follows
...

Similarly for Bob
...
In other words, the expected payoff for each choice (Bus or Walk) must be equal
...
)
For Bob, his strategy should not change if

APPENDIX

227

–3p + –5(1 – p) = –6p + –4(1 – p)
Applying some elementary algebra skills, that equation can be recast as:
–3p –5 +5p = –6p –4 + 4p
or
2p = 1 – 2p
so
4p = 1
Which, solving for p, shows that Alice’s optimal probability for
playing Bus is
p = 1/4
So Alice should choose Bus one time out of 4, and Walk 3 times
out of 4
...

A BEAUTIFUL MATH

228

Now let’s say Alice and Bob decide to play the hawk-dove
game, in which the payoff structure is a little more complicated
because what one player wins does not necessarily equal what the
other player loses
...

Bob
Hawk

Dove

Hawk

–2, –2

2, 0

Dove

0, 2

1, 1

Alice

Alice plays hawk with probability p and dove with probability
1 – p; Bob plays hawk with probability q and dove with probability 1 – q
...
Her expected payoff from dove is 0q + 1(1 – q)
...

Bob will not want to change strategies when
–2p + 2(1 – p) = 0p + 1(1 – p)
2 = 1 + 3p
3p = 1
p = 1/3
So p, Alice’s probability of playing hawk, is 1/3
...
Consequently
the Nash equilibrium in this payoff structure is to play hawk onethird of the time and dove two-thirds of the time
...
Those that I found most useful and illuminating:
Camerer, Colin
...
Princeton, N
...
: Princeton
University Press, 2003
...
Game Theory Evolving
...
J
...

Kuhn, Harold W
...
The Essential John Nash
...
J
...

Luce, R
...
Games and Decisions
...

Williams, J
...
The Compleat Strategyst: Being a Primer on the Theory of
Games of Strategy
...

Von Neumann, John and Oskar Morgenstern
...
Sixtieth-anniversary Edition
...
J
...

Two other readable books were very helpful:
Davis, Morton D
...
Mineola,
NY: Dover, 1997 (1983)
...
Prisoner’s Dilemma
...

230

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