Here Prof. Dixit explains game theory and its impact on situations we
encounter every day. "If Nash got a dollar for every time someone wrote or
said 'Nash equilibrium,'" Dixit has said, "he would be a rich
man."
Game theory studies interactive decision-making, where the outcome for each
participant or "player" depends on the actions of all. If you are a
player in such a game, when choosing your course of action or
"strategy" you must take into account the choices of others. But in
thinking about their choices, you must recognize that they are thinking about
yours, and in turn trying to take into account your thinking about their
thinking, and so on.
It would seem that such thinking about thinking must be so complex and
subtle that its successful practice must remain an arcane art. Indeed, some
aspects such as figuring out the true motives of rivals and recognizing
complex patterns do often resist logical analysis. But many aspects of
strategy can be studied and systematized into a science -- game theory.
A Theory is Born
This
science is unusual in the breadth of its potential applications. Unlike
physics or chemistry, which have a clearly defined and narrow scope, the
precepts of game theory are useful in a whole range of activities, from
everyday social interactions and sports to business and economics, politics,
law, diplomacy and war. Biologists have recognized that the Darwinian struggle
for survival involves strategic interactions, and modern evolutionary theory
has close links with game theory.
Game theory got its start with the work of John von Neumann in the 1920s,
which culminated in his book with Oskar Morgenstern. They studied
"zero-sum" games where the interests of two players were strictly
opposed. John Nash treated the more general and realistic case of a mixture of
common interests and rivalry and any number of players. Other theorists, most
notably Reinhard Selten and John Harsanyi who shared the 1994
Nobel Memorial Prize with Nash, studied even more complex games with
sequences of moves, and games where one player has more information than
others.
The Nash Equilibrium
The theory constructs a notion of "equilibrium," to which the
complex chain of thinking about thinking could converge. Then the strategies
of all players would be mutually consistent in the sense that each would be
choosing his or her best response to the choices of the others. For such a
theory to be useful, the equilibrium it posits should exist. Nash used novel
mathematical techniques to prove the existence of equilibrium in a very
general class of games. This paved the way for applications. Biologists have
even used the notion of Nash equilibrium to formulate the idea of evolutionary
stability. Here are a few examples to convey some ideas of game theory and the
breadth of its scope.
The Prisoner's Dilemma
In Joseph Heller's novel Catch-22, allied victory in World War II is a
foregone conclusion, and Yossarian does not want to be among the last ones to
die. His commanding officer points out, "But suppose everyone on our side
felt that way?" Yossarian replies, "Then I'd certainly be a damned
fool to feel any other way, wouldn't I?"
Every general reader has heard of the prisoner's dilemma. The police
interrogate two suspects separately, and suggest to each that he or she should
fink on the other and turn state's evidence. "If the other does not fink,
then you can cut a good deal for yourself by giving evidence against the
other; if the other finks and you hold out, the court will treat you
especially harshly. Thus no matter what the other does, it is better for you
to fink than not to fink -- finking is your uniformly best or 'dominant'
strategy." This is the case whether the two are actually guilty, as in
some episodes of NYPD Blue, or innocent, as in the film LA
Confidential. Of course, when both fink, they both fare worse than they
would have if both had held out; but that outcome, though jointly desirable
for them, collapses in the face of their separate temptations to fink.
Yossarian's dilemma is just a multi-person version of this. His death is
not going to make any significant difference to the prospects of victory, and
he is personally better off alive than dead. So avoiding death is his dominant
strategy.
John Nash played an important role in interpreting the first experimental
study of the prisoner's dilemma, which was conducted at the Rand
Corporation in 1950.
Real-World Dilemmas
Once you recognize the general idea, you will see such dilemmas everywhere.
Competing stores who undercut each other's prices when both would have done
better if both had kept their prices high are victims of the dilemma. (But in
this instance, consumers benefit from the lower prices when the sellers fink
on each other.) The same concept explains why it is difficult to raise
voluntary contributions, or to get people to volunteer enough time, for
worthwhile public causes.
How might such dilemmas be resolved? If the relationship of the players is
repeated over a long time horizon, then the prospect of future cooperation may
keep them from finking; this is the well-known tit-for-tat strategy. A
"large" player who suffers disproportionately more from complete
finking may act cooperatively even when the small fry are finking. Thus Saudi
Arabia acts as a swing producer in OPEC, cutting its output to keep prices
high when others produce more; and the United States bears a disproportionate
share of the costs of its military alliances. Finally, if the group as a whole
will do better in its external relations if it enjoys internal cooperation,
then the process of biological or social selection may generate instincts or
social norms that support cooperation and punish cheating. The innate sense of
fairness and justice that is observed among human subjects in many laboratory
experiments on game theory may have such an origin.
Mixing Moves
In football, when an offense faces a third down with a yard to go, a run up
the middle is the usual or "percentage" play. But an occasional long
pass in such a situation is important to keep the defense honest. Similarly, a
penalty kicker in soccer who kicks exclusively to the goalie's right, or a
server in tennis who goes exclusively to the receiver's forehand, will fare
poorly because the opponent will anticipate and counter the action. In such
situations it is essential to mix one's moves randomly, so that on any one
occasion the action is unpredictable.
Mixing is most important in games where the players' interests are strictly
opposed, and this happens most frequently in sports. Indeed, recent empirical
studies of serving in tennis grand slam finals, and penalty kicks in European
soccer leagues, have found the behavior consistent with the theory.
Commitments
Greater freedom of action seems obviously desirable. But in games of
bargaining that need not be true, because freedom to act can simply become
freedom to concede to the other's demands. Committing yourself to a firm final
offer leaves the other party the last chance to avoid a mutually disastrous
breakdown, and this can get you a better deal. But a mere verbal declaration
of firmness may not be credible. Devising actions to make one's commitments
credible is one of the finer arts in the realm of strategic games. Members of
a labor union send their leaders into wage bargaining with firm instructions
or mandates that tie their hands, thereby making it credible that they will
not accept a lower offer. The executive branch of the U.S. government engaged
in international negotiations on trade or related matters can credibly take a
firm stance by pointing out that the Congress would not ratify anything less.
And a child is more likely to get the sweet or toy it wants if it is crying
too loudly to hear your reasoned explanations of why it should not have it.
Thomas Schelling pioneered the study of credible commitments, and other
more complex "strategic moves" like threats and promises. This has
found many applications in diplomacy and war, which, as military strategist
Karl von Clausewitz told us long ago, are two sides of the same strategic
coin.
Information and Incentives
Suppose you have just graduated with a major in computer science, and have an
idea for a totally new "killer app" that will integrate PCs, cell
phones, and TV sets to create a new medium. The profit potential is immense.
You go to venture capitalists for finance to develop and market your idea. How
do they know that the potential is as high as you claim it to be? The idea is
too new for them to judge it independently. You have no track record, and
might be a complete charlatan who will use the money to live high for a few
years and then disappear. One way for them to test your own belief in your
idea is to see how much of your own money you are willing to risk in the
project. Anyone can talk a good game; if you are willing to put enough of your
money where your mouth is, that is a credible signal of your own true
valuation of your idea.
This is a game where the players have different information; you know the
true potential of your idea much better than does your prospective financier.
In such games, actions that reveal or conceal information play crucial roles.
The field of "information economics" has clarified many previously
puzzling features of corporate governance and industrial organization, and has
proved equally useful in political science, studies of contract and tort law,
and even biology. The award of the Nobel
Memorial Prize in 2001 to its pioneers, George Akerlof, Michael Spence,
and Joseph Stiglitz, testifies to its importance. What has enabled information
economics to burgeon in the last twenty years is the parallel development of
concepts and techniques in game theory.
Aligning Interests, Avoiding Enrons
A related application in business economics is the design of incentive
schemes. Modern corporations are owned by numerous shareholders, who do not
personally supervise the operations of the companies. How can they make sure
that the workers and managers will make the appropriate efforts to maximize
shareholder value? They can hire supervisors to watch over workers, and
managers to watch over supervisors. But all such monitoring is imperfect: the
time on the job is easily monitored, but the quality of effort is very
difficult to observe and judge. And there remains the problem of who will
watch over the upper-level management. Hence the importance of compensation
schemes that align the interests of the workers and managers with those of the
shareholders. Game theory and information economics have given us valuable
insights into these issues. Of course we do not have perfect solutions; for
example, we are just discovering how top management can manipulate and distort
the performance measures to increase their own compensation while hurting
shareholders and workers alike. This is a game where shareholders and the
government need to find and use better counterstrategies.
From Intuition to Prediction
While reading these examples, you probably thought that many of the lessons of
game theory are obvious. If you have had some experience of playing similar
games, you have probably intuited good strategies for them. What game theory
does is to unify and systematize such intuitions. Then the general principles
extend the intuitions across many related situations, and the calculation of
good strategies for new games is simplified. It is no bad thing if an idea
seems obvious when it is properly formulated and explained; on the contrary, a
science or theory that takes simple ideas and brings out their full power and
scope is all the more valuable for that.
In conclusion, I offer some suggestions for further reading to those whose
appetites are whetted by my sampler of examples. (This site's bibliography
includes some Web sites of interest.)
General interest:
Dixit, Avinash, and Barry Nalebuff. Thinking
Strategically: the Competitive Edge in Business, Politics, and Everyday Life.
New York: W.W. Norton, 1991.
Schelling, Thomas. The Strategy of Conflict.
Revised edition. Cambridge: Harvard University Press, 1980.
Elementary textbook:
Dixit, Avinash, and Susan Skeath. Games of
Strategy. New York: W.W. Norton, 1999.
Advanced textbooks:
Fudenberg, Drew, and Jean Tirole. Game Theory.
Cambridge, Massachusetts: MIT Press, 1991.
Myerson, Roger . Game Theory: Analysis of
Conflict. Cambridge, Massachusetts: Harvard University Press, 1991.
Business applications:
Brandenberger, Adam, and Barry Nalebuff. Co-opetition.
New York: Doubleday, 1996.
McMillan, John. Games, Strategies, and
Managers. Reprint. New York: Oxford University Press, 1996.
Political science applications:
Brams, Steven. Rational Politics: Decisions,
Games, and Strategy. Reprint. Boston: Academic Press, 1989.
Ordeshook, Peter, A Political Theory Primer.
New York: Routledge, 1992.
Applications to law:
Baird, Douglas, Gertner, Robert, and Randal Picker. Game
Theory and the Law. Cambridge, Massachusetts: Harvard University Press,
1994.
Biology:
Maynard Smith, John. Evolution and the Theory of
Games. Cambridge, England: Cambridge University Press, 1982.
Classic books about game theory:
Luce, R. Duncan, and Howard Raiffa. Games and
Decisions: Introduction and Critical Survey. New York: Wiley, 1957.
von Neumann, John, and Oskar Morgenstern. Theory
of Games and Economic Behavior. Second edition. Princeton, New Jersey:
Princeton University Press, 1947.
Avinash
Dixit, John J. F. Sherrerd '52 University Professor of Economics at Princeton
University, is