This is a file in the archives of the Stanford Encyclopedia of Philosophy. |

- 1. Philosophical and Historical Motivation
- 2. Basic Elements and Assumptions of Game Theory
- 2.1 Utility
- 2.2 Games and Information
- 2.3 Trees and Matrices
- 2.4 The Prisoner's Dilemma as an Example of Strategic-Form vs. Extensive-Form Representation
- 2.5 Solution Concepts and Equilibria
- 2.6 Modular Rationality and Subgame Perfection
- 2.7 On Interpreting Payoffs: Morality and Efficiency in Games
- 2.8 Trembling Hands

- 3. Uncertainty, Risk and Sequential Equilibria
- 4. Repeated Games and Coordination
- 5. Commitment
- 6. Evolutionary Game Theory
- Bibliography
- Other Internet Resources
- Related Entries

Despite the fact that game theory has been rendered mathematically
and logically systematic only recently, however, game-theoretic
insights can be found among philosophers and political commentators
going back to ancient times. For example, Plato, in the
*Republic*, at one point has Socrates worry about the following
situation. Consider a soldier at the front, waiting with his comrades
to repulse an enemy attack. It may occur to him that if the defense is
likely to be successful, then it isn't very probable that his own
personal contribution will be essential. But if he stays, he runs the
risk of being killed or wounded -- apparently for no point. On the
other hand, if the enemy is going to win the battle, then his chances
of death or injury are higher still, and now quite clearly to no
point, since the line will be overwhelmed anyway. Based on this
reasoning, it would appear that the soldier is better off running away
regardless of who is going to win the battle. Of course, if all of the
soldiers reason this way -- as they all apparently *should*,
since they're all in identical situations -- then this will
certainly *bring about* the outcome in which the battle is
lost. Of course, this point, since it has occurred to us as analysts,
can occur to the soldiers too. Does this give them a reason for
staying at their posts? Just the contrary: the greater the
soldiers' fear that the battle will be lost, the greater their
incentive to get themselves out of harm's way. And the greater
the soldiers' belief that the battle will be won, without the need of
any particular individual's contributions, the less reason they have
to stay and fight. If each soldier *anticipates* this sort of
reasoning on the part of the others, all will quickly reason
themselves into a panic, and their horrified commander will have a
rout on his hands before the enemy has even fired a shot!

Long before game theory had come along to show people how to think about this sort of problem systematically, it had occurred to some actual military leaders and influenced their strategies. Thus the Spanish conqueror Cortez, when landing in Mexico with a small force who had good reason to fear their capacity to repel attack from the far more numerous Aztecs, removed the risk that his troops might think their way into a retreat by burning the ships on which they had landed. With retreat having thus been rendered physically impossible, the Spanish soldiers had no better course of action but to stand and fight -- and, furthermore, to fight with as much determination as they could muster. Better still, from Cortez's point of view, his action had a discouraging effect on the motivation of the Aztecs. He took care to burn his ships very visibly, so that the Aztecs would be sure to see what he had done. They then reasoned as follows: Any commander who could be so confident as to willfully destroy his own option to be prudent if the battle went badly for him must have good reasons for such extreme optimism. It cannot be wise to attack an opponent who has a good reason (whatever, exactly, it might be) for being sure that he can't lose. The Aztecs therefore retreated into the surrounding hills, and Cortez had his victory bloodlessly.

This situation, as imagined by Plato and as vividly acted upon by
Cortez, has a deep and interesting logic. Notice that the soldiers are
not motivated to retreat *just*, or even mainly, by their
rational assessment of the dangers of battle and by their
self-interest. Rather, they discover a sound reason to run away by
realizing that what it makes sense for them to do depends on what it
will make sense for others to do, and that all of the others can
notice this too. Even a quite brave soldier may prefer to run rather
than heroically, but pointlessly, die trying to stem the oncoming tide
all by himself. Thus we could imagine, without contradiction, a
circumstance in which an army, all of whose members are brave, flees
at top speed before the enemy makes a move. If the soldiers really
*are* brave, then this surely isn't the outcome any of
them wanted; each would have preferred that all stand and fight. What
we have here, then, is a case in which the *interaction* of
many individually rational decision-making processes -- one process
per soldier -- produces an outcome intended by no one. (Most armies
try to avoid this problem just as Cortez did. Since they can't
usually make retreat *physically* impossible, they make it
*economically* impossible: they shoot deserters. Then standing
and fighting is each soldier's individually rational course of
action after all, because the cost of running is sure to be at least
as high as the cost of staying.)

Another classic source that reproduces exactly this sequence of
reasoning is Shakespeare. In *Henry V*, he has the victor of
Agincourt explain his decision to slaughter French prisoners, in full
view of the enemy, as follows. His own troops observe that the
prisoners have been killed, and observe that the enemy has observed
this. Therefore, they know what fate will await them at the
enemy's hand if they don't win. Metaphorically, but very
effectively, their boats have been burnt. The French prisoners died as
a means by which Henry sent a signal to *his own troops*,
thereby changing their incentives.

These examples might seem to be relevant only for those who find
themselves in sordid situations of cut-throat competition. Perhaps,
one might think, it is important for generals, politicians,
businesspeople and others whose jobs involve manipulation of others,
but the philosopher should only deplore its horrid morality. Such a
conclusion would be highly premature, however. The study of the
*logic* that governs the interrelationships amongst incentives,
strategic interactions and outcomes has been fundamental in modern
political philosophy, since centuries before anyone had an explicit
name for this sort of logic.

Hobbes's *Leviathan* is often regarded as the founding
work in modern political philosophy, the text that began the
continuing round of analyses of the function and justification of the
state and its restrictions on individual liberties. The core of
Hobbes's reasoning can be given quite straightforwardly as
follows. The best situation for all people is one in which each is
free to do as she pleases. Often, such free people will wish to
cooperate with one another in order to carry out projects that would
be impossible for an individual acting alone. But if there are any
immoral or amoral agents around, they will notice that their interests
are best served by getting the benefits from cooperation and not
returning them. Suppose, for example, that you agree to help me build
my house in return for my promise to help you build yours. After my
house is finished, I can make your labour free to me simply by
reneging on my promise. I then realize, however, that if this leaves
you with no house, you will have an incentive to take mine. This will
put me in constant fear of you, and force me to spend valuable time
and resources guarding myself against you. I can best minimize these
costs by striking first and killing you at the first opportunity. Of
course, you can anticipate all of this reasoning by me, and so have
good reason to try to beat me to the punch. Since I can anticipate
*this* reasoning by *you*, my original fear of you was
not paranoid; nor was yours of me. In fact, neither of us actually
needs to be immoral to get this chain of mutual reasoning going; we
need only think that there is some *possibility* that the other
might try to cheat on bargains. Once a small wedge of doubt enters any
one mind, the incentive induced by fear of the consequences of being
*preempted* -- hit before hitting first -- quickly becomes
overwhelming on both sides. If either of us has any resources of our
own that the other might want, this murderous logic will take hold
long before we are so silly as to imagine that we could ever actually
get so far as making deals to help one another build houses in the
first place. Left to their own devices, rational agents will never
derive the benefits of cooperation, and will instead live from the
outset in a state of ‘war of all against all’, in
Hobbes's words. In these circumstances, all human life, as he
vividly and famously put it, will be "solitary, poor, nasty,
brutish and short."

Hobbes's proposed solution to this problem was tyranny. The people can hire an agent -- a government -- whose job is to punish anyone who breaks any promise. So long as the threatened punishment is sufficiently dire -- Hobbes thought decapitation generally appropriate -- then the cost of reneging on promises will exceed the cost of keeping them. The logic here is identical to that used by an army when it threatens to shoot deserters. If all people know that these incentives hold for most others, then cooperation will not only be possible, but will be the expected norm, and the war of all against all becomes a general peace.

Hobbes pushes the logic of this argument to a very strong conclusion,
arguing that it implies not only a government with the right and the
power to enforce cooperation, but an ‘undivided’ government
in which the arbitrary will of a single ruler must impose absolute
obligation on all. Few contemporary political theorists think that the
particular steps by which Hobbes reasons his way to this conclusion
are both sound and valid. Working through these issues here, however,
would carry us away from our topic into complex details of
contractarian political philosophy. What is important in the present
context is that these details, as they are in fact pursued in the
contemporary debates, all involve sophisticated interpretation of the
issues using the resources of modern game theory. Furthermore,
Hobbes's most basic point, that the fundamental justification for
the coercive authority and practices of governments is peoples'
own need to protect themselves from what game theorists call
‘social dilemmas’, is accepted by many, if not most,
political theorists. Notice that Hobbes has *not* argued that
tyranny is a desirable thing in itself. The structure of his argument
is that the logic of strategic interaction leaves only two general
political outcomes possible: tyranny and anarchy. Rational agents then
choose tyranny as the lesser of two evils.

The reasoning of Cortez, of Henry V and of Hobbes's political
agents has a common logic, one derived from their situations. In each
case, the aspect of the environment that is most important to the
agents' achievement of their preferred outcomes is the set of
expectations and possible reactions to their strategies by other
agents. The distinction between acting *parametrically* on a
passive world and acting *non-parametrically* on a world that
tries to act in anticipation of these actions is fundamental. If you
wish to kick a rock down a hill, you need only concern yourself with
the rock's mass relative to the force of your blow, the extent to
which it is bonded with its supporting surface, the slope of the
ground on the other side of the rock, and the expected impact of the
collision on your foot. The values of all of these variables are
independent of your plans and intentions, since the rock has no
interests of its own and takes no actions to attempt to assist or
thwart you. By contrast, if you wish to kick a person down the hill,
then unless that person is unconscious, bound or otherwise
incapacitated, you will likely not succeed unless you can disguise
your plans until it's too late for him to take either evasive or
forestalling action. The logical issues associated with the second
sort of situation are typically much more complicated, as a simple
hypothetical example will illustrate.

Suppose first that you wish to cross a river that is spanned by three bridges. (Assume that swimming, wading or boating across are impossible.) The first bridge is known to be safe and free of obstacles; if you try to cross there, you will succeed. The second bridge lies beneath a cliff from which large rocks sometimes fall. The third is inhabited by deadly cobras. Now suppose you wish to rank-order the three bridges with respect to their preferability as crossing-points. Your task here is quite straightforward. The first bridge is obviously best, since it is safest. To rank-order the other two bridges, you require information about their relative levels of danger. If you can study the frequency of rock-falls and the movements of the cobras for awhile, you might be able to calculate that the cobra bridge is four times more dangerous than the rocky bridge, and the rocky bridge 20% more dangerous than the unobstructed bridge. Your reasoning here is strictly parametric because neither the rocks nor the cobras are trying to influence your actions, by, for example, concealing their typical patterns of behaviour because they know you are studying them. It is quite obvious what you should do here: cross at the safe bridge. Now let us complicate the situation a bit. Suppose that the bridge with the rocks was immediately before you, while the safe bridge was a day's difficult hike upstream. Your decision-making situation here is slightly more complicated, but it is still strictly parametric. You would have to decide whether the cost of the long hike was worth exchanging for the penalty of a 20% chance of being hit by a rock. However, this is all you must decide, and your probability of a successful crossing is entirely up to you; the environment is not interested in your plans.

However, if we now complicate the situation in the direction of non-parametricity, it becomes much more puzzling. Suppose that you are a fugitive of some sort, and waiting on the other side of the river with a gun is your pursuer. She will catch and shoot you, let us suppose, only if she waits at the bridge you try to cross; otherwise, you will escape. As you reason through your choice of bridge, it occurs to you that she is over there trying to anticipate your reasoning. It will seem that, surely, choosing the safe bridge straight away would be a mistake, since that is just where she will expect you, and your chances of death rise to certainty. So perhaps you should risk the rocks, since these odds are much better. But wait ... if you can reach this conclusion, your pursuer, who is just as rational and well-informed as you are, can anticipate that you will reach it, and will be waiting for you if you evade the rocks. So perhaps you must take your chances with the cobras; that is what she must least expect. But, then, no ... if she expects that you will expect that she will least expect this, then she will most expect it. This dilemma, you realize with dread, is general: you must do what your pursuer least expects; but whatever you most expect her to least expect is automatically what she will most expect. You appear to be trapped in indecision. All that might console you a bit here is that, on the other side of the river, your pursuer is trapped in exactly the same quandary, unable to decide which bridge to wait at because as soon as she imagines committing to one, she will notice that if she can find a best reason to pick a bridge, you can anticipate that same reason and then avoid her.

We know from experience that, in situations such as this, people do
not usually stand and dither in circles forever. As we'll see
later, there *is* a rational solution -- that is, a best
rational action -- available to both players. However, until the 1940s
neither philosophers nor economists knew how to find it
mathematically. As a result, economists were forced to treat
non-parametric influences as if they were complications on parametric
ones. This is likely to strike the reader as odd, since, as our
example of the bridge-crossing problem was meant to show,
non-parametric features are often fundamental features of
decision-making problems. Part of the explanation for game
theory's relatively late entry into the field lies in the
problems with which economists had historically been concerned.
Classical economists, such as Adam Smith and David Ricardo, were
mainly interested in the question of how agents in very large markets
-- whole nations -- could interact so as to bring about maximum
monetary wealth for themselves. Smith's basic insight, that
efficiency is best maximized by agents freely seeking mutually
advantageous bargains, was mathematically verified in the twentieth
century. However, the demonstration of this fact applies only in
conditions of ‘perfect competition,’ that is, when firms
face no costs of entry or exit into markets, when there are no
economies of scale, and when no agents' actions have unintended
side-effects on other agents' well-being. Economists always
recognized that this set of assumptions is purely an idealization for
purposes of analysis, not a possible state of affairs anyone could try
(or should want to try) to attain. But until the mathematics of game
theory matured near the end of the 1970s, economists had to hope that
the more closely a market *approximates* perfect competition,
the more efficient it will be. No such hope, however, can be
mathematically or logically justified in general; indeed, as a strict
generalization the assumption can be shown to be false.

This article is not about the foundations of economics, but it is important for understanding the origins and scope of game theory to know that perfectly competitive markets have built into them a feature that renders them susceptible to parametric analysis. Because agents face no entry costs to markets, they will open shop in any given market until competition drives all profits to zero. This implies that if costs and demand are fixed, then agents have no options about how much to produce if they are trying to maximize the differences between their costs and their revenues. These production levels can be determined separately for each agent, so none need pay attention to what the others are doing; each agent treats her counterparts as passive features of the environment. The other kind of situation to which classical economic analysis can be applied without recourse to game theory is that of monopoly. Here, quite obviously, non-parametric considerations drop out, since there is only one agent under study. However, both perfect and monopolistic competition are very special and unusual market arrangements. Prior to the advent of game theory, therefore, economists were severely limited in the class of circumstances to which they could neatly apply their models.

Philosophers share with economists a professional interest in the
conditions and techniques for the maximization of human welfare. In
addition, philosophers have a special concern with the logical
justification of actions, and often actions must be justified by
reference to their expected outcomes. Without game theory, both of
these problems resist analysis wherever non-parametric aspects are
relevant. We will demonstrate this shortly by reference to the most
famous (though not the most typical) game, the so-called
*Prisoner's Dilemma*, and to other, more typical,
games. In doing this, we will need to introduce, define and illustrate
the basic elements and techniques of game theory. To this job we
therefore now turn.

Though we might no longer be moved by scruples derived from
*psychological* behaviorism, many theorists continue to follow
Samuelson's way of understanding utility because they think it
important that game theory apply to *any* kind of agent -- a
person, a bear, a bee, a firm or a country -- and not just to agents
with human minds. When such theorists say that agents act so as to
maximize their utility, they want this to be part of the
*definition* of what it is to be an agent, not an empirical
claim about possible inner states and motivations. Samuelson's
conception of utility, defined by way of *Revealed Preference
Theory* (RPT) introduced in his classic paper
(Samuelson (1938))
satisfies this demand.

Some other theorists understand the point of game theory
differently. They view game theory as providing an explanatory account
of strategic reasoning. For this idea to be applicable, we must
suppose that agents at least sometimes do what they do in
non-parametric settings *because* game-theoretic logic recommends
certain actions as the rational ones. Still other theorists interpret
game theory *normatively*, as advising agents on what to do in
strategic contexts in order to maximize their utility. Fortunately for
our purposes, all of these ways of thinking about the possible uses of
game theory are compatible with the tautological interpretation of
utility maximization. The philosophical differences are not idle from
the perspective of the working game theorist, however. As we will see
in a later section, those who hope to use game theory to explain
strategic *reasoning*, as opposed to merely strategic
*behavior*, face some special philosophical and practical
problems.

Since game theory involves formal reasoning, we must have a device for
thinking of utility maximization in mathematical terms. Such a device
is called a *utility function*. The utility-map for an agent is
called a ‘function’ because it maps *ordered
preferences* onto the real numbers. Suppose that agent *x*
prefers bundle *a* to bundle *b* and bundle *b*
to bundle *c*. We then map these onto a list of numbers, where
the function maps the highest-ranked bundle onto the largest number in
the list, the second-highest-ranked bundle onto the next-largest
number in the list, and so on, thus:

bundleThe only property mapped by this function isa3bundle

b2bundle

c1

bundlea7,326bundle

b12.6bundle

c-1,000,000

The numbers featuring in an ordinal utility function are thus not
measuring any *quantity* of anything. A utility-function in
which magnitudes *do* matter is called ‘cardinal’.
Whenever someone refers to a utility function without specifying which
kind is meant, you should assume that it's ordinal. These are the
sorts we'll need for the first set of games we'll
examine. Later, when we come to seeing how to solve games that involve
*randomization* -- our river-crossing game from Part 1 above,
for example -- we'll need to build cardinal utility functions.
The technique for doing this was given by
von Neumann & Morgenstern (1947),
and was an essential aspect of their invention of game theory. For
the moment, however, we will need only ordinal functions.

We assume that players are economically rational. That is, a player can (i) assess outcomes; (ii) calculate paths to outcomes; and (iii) choose actions that yield their most-preferred outcomes, given the actions of the other players. This rationality might in some cases be internally computed by the agent. In other cases, it might simply be embodied in behavioral dispositions built by natural, cultural or economic selection.

Each player in a game faces a choice among two or more possible
*strategies*. A strategy is a predetermined ‘programme of
play’ that tells her what actions to take in response to
*every possible strategy other players might use*. The
significance of the italicized phrase here will become clear when we
take up some sample games below.

A crucial aspect of the specification of a game involves the
information that players have when they choose strategies. The
simplest games (from the perspective of logical structure) are those
in which agents have *perfect information*, meaning that at every
point where each agent's strategy tells her to take an action, she
knows everything that has happened in the game up to that point. A
board-game of sequential moves in which in which both players watch
all the action (and know the rules in common), such as chess, is an
instance of such a game. By contrast, the example of the
bridge-crossing game from Section 1 above illustrates a game of
*imperfect information*, since the fugitive must choose a bridge
to cross without knowing the bridge at which the pursuer has chosen to
wait, and the pursuer similarly makes her decision in ignorance of the
actions of her quarry. Since game theory is about rational action
given the strategically significant actions of others, it should not
surprise you to be told that what agents in games know, or fail to
know, about each others' actions makes a considerable difference to
the logic of our analyses, as we will see.

It was said above that the distinction between sequential-move and
simultaneous-move games is not identical to the distinction between
perfect-information and imperfect-information games. Explaining why
this is so is a good way of establishing full understanding of both
sets of concepts. As simultaneous-move games were characterized in the
previous paragraph, it must be true that all simultaneous-move games
are games of imperfect information. However, some games may contain
mixes of sequential and simultaneous moves. For example, two firms
might commit to their marketing strategies independently and in
secrecy from one another, but thereafter engage in pricing competition
in full view of one another. If the optimal marketing strategies were
partially or wholly dependent on what was expected to happen in the
subsequent pricing game, then the two stages would need to be analyzed
as a single game, in which a stage of sequential play followed a stage
of simultaneous play. Whole games that involve mixed stages of this
sort are games of imperfect information, however temporally staged
they might be. Games of perfect information (as the name implies)
denote cases where *no* moves are simultaneous (and where no
player ever forgets what has gone before).

It was said above that games of perfect information are the
(logically) simplest sorts of games. This is so because in such games
(as long as the games are finite, that is, terminate after a known
number of actions) players and analysts can use a straightforward
procedure for predicting outcomes. A rational player in such a game
chooses her first action by considering each series of responses and
counter-responses that will result from each action open to her. She
then asks herself which of the available final outcomes brings her the
highest utility, and chooses the action that starts the chain leading
to this outcome. This process is called *backward induction*
(because the reasoning works backwards from eventual outcomes to
present decision problems).

We will have much more to say about backward induction and its
properties in a later section (when we come to discuss equilibrium and
equilibrium selection). For now, we have described it just in order
to use it to introduce one of the two types of mathematical objects
used to represent games: *game-trees*. A game-tree is an
example of what mathematicians call a *directed graph*. That
is, it is a set of connected nodes in which the overall graph has a
direction. We can draw trees from the top of the page to the bottom,
or from left to right. In the first case, nodes at the top of the page
are interpreted as coming earlier in the sequence of actions. In the
case of a tree drawn from left to right, leftward nodes are prior in
the sequence to rightward ones. An unlabelled tree has a
structure of the following sort:

The point of representing games using trees can best be grasped by visualizing the use of them in supporting backward-induction reasoning. Just imagine the player (or analyst) beginning at the end of the tree, where outcomes are displayed, and then working backwards from these, looking for sets of strategies that describe paths leading to them. Since a player's utility function indicates which outcomes she prefers to which, we also know which paths she will prefer. Of course, not all paths will be possible because the other player has a role in selecting paths too, and won't take actions that lead to less preferred outcomes for him. We will present some examples of this interactive path-selection, and detailed techniques for reasoning through them, after we have described a situation we can use a tree to depict.

Figure 1

Trees are used to represent *sequential* games, because they
show the order in which actions are taken by the players. However,
games are sometimes represented on *matrices* rather than
trees. This is the second type of mathematical object used to
represent games. Matrices, unlike trees, simply show the outcomes,
represented in terms of the players' utility functions, for every
possible combination of strategies the players might use. For example,
it makes sense to display the river-crossing game from
Section 1 on a matrix, since in that game
both the fugitive and the hunter have just one move each, and each
chooses their move in ignorance of what the other has decided to
do. Here, then, is *part of* the matrix:

Figure 2

The fugitive's three possible strategies -- cross at the safe
bridge, risk the rocks and risk the cobras -- form the rows of the
matrix. Similarly, the hunter's three possible strategies --
waiting at the safe bridge, waiting at the rocky bridge and waiting at
the cobra bridge -- form the columns of the matrix. Each cell of the
matrix shows -- or, rather *would* show if our matrix was
complete -- an *outcome* defined in terms of the players'
*payoffs*. A player's payoff is simply the number
assigned by her ordinal utility function to the state of affairs
corresponding to the outcome in question. For each outcome, Row's
payoff is always listed first, followed by Column's. Thus, for
example, the upper right-hand corner above shows that when the
fugitive crosses at the safe bridge and the hunter is waiting there,
the fugitive gets a payoff of 0 and the hunter gets a payoff of 1. We
interpret these by reference to their utility functions, which in this
game are very simple. If the fugitive gets safely across the river he
receives a payoff of 1; if he doesn't he gets 0. If the fugitive
doesn't make it, either because he's shot by the hunter or
hit by a rock or struck by a cobra, then the hunter gets a payoff of 1
and the fugitive gets a payoff of 0.

We'll briefly explain the parts of the matrix that have been
filled in, and then say why we can't yet complete the
rest. Whenever the hunter waits at the bridge chosen by the fugitive,
the fugitive is shot. These outcomes all deliver the payoff vector (0,
1). You can find them descending diagonally across the matrix above
from the upper right-hand corner. Whenever the fugitive chooses the
safe bridge but the hunter waits at another, the fugitive gets safely
across, yielding the payoff vector (1, 0). These two outcomes are
shown in the second two cells of the top row. All of the other cells
are marked, *for now*, with question marks. Why? The problem
here is that if the fugitive crosses at either the rocky bridge or the
cobra bridge, he introduces parametric factors into the game. In these
cases, he takes on some risk of getting killed, and so producing the
payoff vector (0, 1), that is independent of anything the hunter
does. We don't yet have enough concepts introduced to be able to
show how to represent these outcomes in terms of utility functions --
but by the time we're finished we will, and this will provide the
key to solving our puzzle from
Section 1.

Matrix games are referred to as ‘normal-form’ or
‘strategic-form’ games, and games as trees are referred to
as ‘extensive-form’ games. The two sorts of games are not
equivalent, because extensive-form games contain information -- about
sequences of play and players' levels of information about the
game structure -- that strategic-form games do not. In general, a
strategic-form game could represent any one of several extensive-form
games, so a strategic-form game is best thought of as being a
*set* of extensive-form games. When order of play is irrelevant
to a game's outcome, then you should study its strategic form,
since it's the whole set you want to know about. Where order of
play *is* relevant, the extensive form *must* be
specified or your conclusions will be unreliable.

The name of the Prisoner's Dilemma game is derived from the following
situation typically used to exemplify it. Suppose that the police have
arrested two people whom they know have committed an armed robbery
together. Unfortunately, they lack enough admissible evidence to get a
jury to convict. They *do*, however, have enough evidence to
send each prisoner away for two years for theft of the getaway
car. The chief inspector now makes the following offer to each
prisoner: If you will confess to the robbery, implicating your
partner, and she does not also confess, then you'll go free and
she'll get ten years. If you both confess, you'll each get 5
years. If neither of you confess, then you'll each get two years
for the auto theft.

Our first step in modeling your situation as a game is to represent it in terms of utility functions. Both you and your partner's utility functions are identical:

Go free 4The numbers in the function above are now used to express your and your partner's2 years 3

5 years 2

10 years 0

Each cell of the matrix gives the payoffs to both players for each combination of actions. Player I's payoff appears as the first number of each pair, Player II's as the second. So, if both of you confess you each get a payoff of 2 (5 years in prison each). This appears in the upper-left cell. If neither of you confess, you each get a payoff of 3 (2 years in prison each). This appears as the lower-right cell. If you confess and your partner doesn't you get a payoff of 4 (going free) and she gets a payoff of 0 (ten years in prison). This appears in the upper-right cell. The reverse situation, in which she confesses and you refuse, appears in the lower-left cell.

Figure 3

You evaluate your two possible actions here by comparing your payoffs
in each column, since this shows you which of your actions is
preferable for each possible action by your partner. So, observe: If
your partner confesses than you get a payoff of 2 by confessing and a
payoff of 0 by refusing. If your partner refuses, you get a payoff of
4 by confessing and a payoff of 3 by refusing. Therefore, you're
better off confessing regardless of what she does. Your partner,
meanwhile, evaluates her actions by comparing her payoffs down each
row, and she comes to exactly the same conclusion that you
do. Wherever one action for a player is superior to her other actions
for each possible action by the opponent, we say that the first action
*strictly dominates* the second one. In the PD, then,
confessing strictly dominates refusing for both players. Both players
know this about each other, thus entirely eliminating any temptation
to depart from the strictly dominated path. Thus both players will
confess, and both will go to prison for 5 years.

The players, and analysts, can predict this outcome using a
mechanical procedure, known as iterated elimination of strictly
dominated strategies. You, as Player 1, can see by examining the
matrix that your payoffs in each cell of the top row are higher than
your payoffs in each corresponding cell of the bottom row. Therefore,
it can never be rational for you to play your bottom-row strategy,
viz., refusing to confess, *regardless of what your opponent
does*. Since your bottom-row strategy will never be played, we can
simply *delete* the bottom row from the matrix. Now it is obvious
that Player II will not refuse to confess, since his payoff from
confessing in the two cells that remain is higher than his payoff from
refusing. So, once again, we can delete the one-cell column on the
left from the game. We now have only one cell remaining, that
corresponding to the outcome brought about by mutual confession. Since
the reasoning that led us to delete all other possible outcomes
depended at each step only on the premise that both players are
economically rational - that is, prefer higher payoffs to lower ones -
there is very strong grounds for viewing joint confession as the
*solution* to the game, the outcome on which its play *must
*converge. You should note that the order in which strictly
dominated rows and columns are deleted doesn't matter. Had we begun by
deleting the right-hand column and then deleted the bottom row, we
would have arrived at the same solution.

It's been said a couple of times that the PD is not a typical game in many respects. One of these respects is that all its rows and columns are either strictly dominated or strictly dominant. In any strategic-form game where this is true, iterated elimination of strictly dominated strategies is guaranteed to yield a unique solution. Later, however, we will see that for many games this condition does not apply, and then our analytic task is less straightforward.

You will probably have noticed something disturbing about the outcome
of the PD. Had you both refused to confess, you'd have arrived at
the lower-right outcome in which you each go to prison for only 2
years, thereby *both* earning higher utility than you receive
when you confess. This is the most important fact about the PD, and
its significance for game theory is quite general. We'll
therefore return to it below when we discuss equilibrium concepts in
game theory. For now, however, let us stay with our use of this
particular game to illustrate the difference between strategic and
extensive forms.

When people introduce the PD into popular discussions, you will sometimes hear them say that the police inspector must lock his prisoners into separate rooms so that they can't communicate with one another. The reasoning behind this idea seems obvious: if you could communicate, you'd surely see that you're both better off refusing, and could make an agreement to do so, no? This, one presumes, would remove your conviction that you must confess because you'll otherwise be sold up the river by your partner. In fact, however, this intuition is misleading and its conclusion is false.

When we represent the PD as a strategic-form game, we implicitly assume that the prisoners can't attempt collusive agreement since they choose their actions simultaneously. In this case, agreement before the fact can't help. If you are convinced that your partner will stick to the bargain then you can seize the opportunity to go scot-free by confessing. Of course, you realize that the same temptation will occur to her; but in that case you again want to make sure you confess, as this is your only means of avoiding your worst outcome. Your agreement comes to naught because you have no way of enforcing it; it constitutes what game theorists call ‘cheap talk’.

But now suppose that you do *not* move simultaneously. That
is, suppose that one of you can choose *after* observing the
other's action. This is the sort of situation that people who
think non-communication important must have in mind. Now you can see
that your partner has remained steadfast when it comes to your choice,
and you need not be concerned about being suckered. However, this
doesn't change anything, a point that is best made by
re-representing the game in extensive form. This gives us our
opportunity to introduce game-trees and the method of analysis
appropriate to them.

First, however, here are definitions of some concepts that will be helpful in analyzing game-trees:

These quick definitions may not mean very much to you until you follow them being put to use in our analyses of trees below. It will probably be best if you scroll back and forth between them and the examples as we work through them. By the time you understand each example, you'll find the concepts and their definitions quite natural and intuitive.Node: A point at which a player takes an action.

Initial node: The point at which the first action in the game occurs.

Terminal node: Any node which, if reached, ends the game. Each terminal node corresponds to anoutcome.

Subgame: Any set of nodes and branches descending uniquely from one node.

Payoff: an ordinal utility number assigned to a player at an outcome.

Outcome: an assignment of a set of payoffs, one to each player in the game.

Strategy: a program instructing a player which action to take at every node in the tree where she could possibly be called on to make a choice.

To make this exercise maximally instructive, let's suppose that you and your partner have studied the matrix above and, seeing that you're both better off in the outcome represented by the lower-right cell, have formed an agreement to cooperate. You are to commit to refusal first, at which point she will reciprocate. We will refer to a strategy of keeping the agreement as ‘cooperation’, and will denote it in the tree below with ‘C’. We will refer to a strategy of breaking the agreement as ‘defection’, and will denote it on the tree below with ‘D’. As before, you are I and your partner is II. Each node is numbered 1, 2, 3, ... , from top to bottom, for ease of reference in discussion. Here, then, is the tree:

Look first at each of the terminal nodes (those along the bottom). These represent possible outcomes. Each is identified with a assignment of payoffs, just as in the strategic-form game, with I's payoff appearing first in each set and II's appearing second. Each of the structures descending from the nodes 1, 2 and 3 respectively is a sub-game. We begin our backward-induction analysis -- using a technique called

Figure 4

What has happened here is that you realize that if you play C (refuse to confess) at node 1, then your partner will be able to maximize her utility by suckering you and playing D. (On the tree, this happens at node 3.) This leaves you with a payoff of 0 (ten years in prison), which you can avoid only by playing D to begin with. You therefore defect from the agreement.

We have thus seen that in the case of the Prisoner's Dilemma, the simultaneous and sequential versions yield the same outcome. This will often not be true, however. In particular, only finite extensive-form (sequential) games of perfect information can be solved using Zermelo's algorithm.

As noted earlier in this section, sometimes we must represent
simultaneous moves *within* games that are otherwise
sequential. (As we said above, in all such cases the game as a whole
will be one of imperfect information, so we won't be able to solve it
using Zermelo's algorithm.) We represent such games using the device
of *information sets*. Consider the following tree:

The oval drawn around nodes

Figure 5

It's useful to start the discussion here from the case of the
Prisoner's Dilemma because it's unusually simple from the
perspective of these puzzles. What we referred to as its
‘solution’ is the unique *Nash equilibrium* of the
game. (The ‘Nash’ here refers to John Nash, the Nobel
Prize-winning mathematician who in
Nash (1950)
did most to extend and generalize von Neumann &
Morgenstern's pioneering work.) Nash equilibrium (henceforth
‘NE’) applies (or fails to apply, as the case may be) to
whole *sets* of strategies, one for each player in a game. A
set of strategies is a NE just in case no player could improve her
payoff, given the strategies of all other players in the game, by
changing her strategy. Notice how closely this idea is related to the
idea of strict dominance: no strategy could be a NE strategy if it is
strictly dominated. Therefore, if iterative elimination of strictly
dominated strategies takes us to a unique outcome, we know we have
found the game's unique NE. Now, almost all theorists agree that
avoidance of strictly dominated strategies is a *minimum*
requirement of rationality. This implies that *if *a game has an
outcome that is a unique NE, as in the case of joint confession in the
PD, that must be its unique solution. This is one of the most
important respects in which the PD is an ‘easy’ (and
atypical) game.

We can specify one class of games in which NE is always not only
necessary but *sufficient* as a solution concept. These are
finite perfect-information games that are also *zero-sum*. A
zero-sum game (in the case of a game involving just two players) is
one in which one player can only be made better off by making the
other player worse off. (Tic-tac-toe is a simple example of such a
game: any move that brings me closer to winning brings you closer to
losing, and vice-versa.) We can determine whether a game is zero-sum
by examining players' utility functions: in zero-sum games these will
be mirror-images of each other, with one player's highly ranked
outcomes being low-ranked for the other and vice-versa. In such a
game, if I am playing a strategy such that, given your strategy, I
can't do any better, and if you are *also* playing such a
strategy, then, since any change of strategy by me would have to make
you worse off and vice-versa, it follows that our game can have no
solution compatible with our mutual rationality other than its unique
NE. We can put this another way: in a zero-sum game, my playing a
strategy that maximizes my minimum payoff if you play the best you
can, and your simultaneously doing the same thing, is just
*equivalent* to our both playing our best strategies, so this
pair of so-called ‘maximin’ procedures is guaranteed to find the
unique solution to the game, which is its unique NE. (In tic-tac-toe,
this is a draw. You can't do any better than drawing, and neither can
I, if both of us are trying to win and trying not to lose.)

However, most games do not have this property. It won't be possible,
in this one article, to enumerate *all* of the ways in which
games can be problematic from the perspective of their possible
solutions. (For one thing, it is highly unlikely that theorists have
yet discovered all of the possible problems!) However, we can try to
generalize the issues a bit.

First, there is the problem that in most non-zero-sum games, there is more than one NE, but not all NE look equally plausible as the solutions upon which strategically rational players would hit. Consider the strategic-form game below (taken from Kreps (1990), p. 403):

Figure 6

This game has two NE: s1-t1 and s2-t2. (Note that no rows or columns
are strictly dominated here. But if Player I is playing s1 then Player
II can do no better than t1, and vice-versa; and similarly for the
s2-t2 pair.) If NE is our only solution concept, then we shall be
forced to say that either of these outcomes is equally persuasive as a
solution. However, if game theory is regarded as an explanatory and/or
normative theory of strategic reasoning, this seems to be leaving
something out: surely rational players with perfect information would
converge on s1-t1? (Note that this is *not* like the situation
in the PD, where the socially superior situation is unachievable
because it is not a NE. In the case of the game above, both players
have every reason to try to converge on the NE in which they are
better off.)

This illustrates the fact that NE is a relatively (logically)
*weak* solution concept, often failing to predict intuitively
sensible solutions because, if applied alone, it refuses to allow
players to use principles of equilibrium selection that, if not
*demanded* by rationality, are at least not
*irrational*. Consider another example from
Kreps (1990), p. 397:

Figure 7

Here, no strategy strictly dominates another. However, Player I's top
row, s1, *weakly* dominates s2, since I does *at least as
well* using s1 as s2 for any reply by Player II, and on one reply
by II (t2), I does better. So should not the players (and the analyst)
delete the weakly dominated row s2? When they do so, column t1 is then
strictly dominated, and the NE s1-t2 is selected as the unique
solution. However, as Kreps goes on to show using this example, the
idea that weakly dominated strategies should be deleted just like
strict ones has odd consequences. Suppose we change the
payoffs of the game just a bit, as follows:

Figure 8

s2 is still weakly dominated as before; but of our two NE, s2-t1 is
now the most attractive for both players; so why should the analyst
eliminate its possibility? (Note that this game, again, does
*not* replicate the logic of the PD. There, it makes sense to
eliminate the most attractive outcome, joint refusal to confess,
because both players have incentives to unilaterally deviate from it,
so it is not an NE. This is not true of s2-t1 in the present game. You
should be starting to clearly see why we called the PD game
‘atypical’.) The argument *for* eliminating weakly dominated
strategies is that Player 1 may be nervous, fearing that Player II is
not completely *sure* to be rational (or that Player II fears
that Player I isn't completely rational, or that Player II fears that
Player I fears that Player II isn't completely rational, and so on ad
infinitum) and so might play t2 with some positive probability. If the
possibility of departures from rationality is taken seriously, then we
have an argument for eliminating weakly dominated strategies: Player I
thereby insures herself against her worst outcome, s2-t2. Of course,
she pays a cost for this insurance, reducing her expected payoff from
10 to 5. On the other hand, we might imagine that the players could
communicate before playing the game and agree play *correlated
strategies *so as to *coordinate* on s2-t1, thereby removing
some, most or all of the uncertainty that encourages elimination of
the weakly dominated row s1, and eliminating s1-t2 as a viable NE
instead!

Any proposed principle for solving games that may have the effect of
eliminating one or more NE from consideration is referred to as a
*refinement* of NE. In the case just discussed, elimination of
weakly dominated strategies is one possible refinement, since it
refines away the NE s2-t1, and correlation is another, since it
refines away the other NE, s1-t2, instead. So which refinement is more
appropriate as a solution concept? People who think of game theory as
an explanatory and/or normative theory of strategic rationality have
generated a substantial literature in which the merits and drawbacks
of a large number of refinements are debated. In principle, there
seems to be no limit on the number of refinements that could be
considered, since there may also be no limits on the set of
philosophical intuitions about what principles a rational agent might
or might not see fit to follow or to fear or hope that other players
are following.

Behaviorists take a dim view of much of this activity. They see the
job of game theory as being to predict outcomes *given* some
distribution of strategic dispositions, and some distribution of
expectations about the strategic dispositions of others, that have
been shaped by institutional processes and / or evolutionary
selection. (See
Section 7
for further discussion.) On this view, which NE are viable in a
game is determined by the underlying dynamics that equipped players
with dispositions *prior to* the commencement of a game. The
strategic natures of players are thereby treated as a set of
exogenous inputs to the game, just as utility functions are.
Behaviorists are thus less inclined to seek *general*
refinements of the equilibrium concept itself, at least insofar as
these involve the modeling of *more sophisticated *expressions
of rationality over and above merely consistent maximization of
utility. Behaviorists are often inclined to doubt that the goal of
seeking a *general* theory of rationality makes sense as a
project. Institutions and evolutionary processes build many
environments, and what counts as rational procedure in one
environment may not be favoured in another. Economic rationality
requires only that agents have consistent preferences, that is, that
they not prefer *a* to *b* and *b* to *c*
**and** *c* to *a*. A great many arrangements of
strategic dispositions are compatible with this minimal requirement,
and evolutionary or institutional processes might generate games in
any of them. On this view, NE is a robust equilibrium concept because
if players evolve their strategic dispositions in settings that are
competitive, those who don't do what's optimal given the
strategies of others *in that specific environment* will be
outcompeted, and so selection will either eliminate them or encourage
the learning of new dispositions. There is no more
‘refined’ concept of rationality of which this can be
argued to be true *in general*; and so, according to
behaviorists, refinements of NE based on refinements of rationality
are likely to be of merely occasional interest.

This does not imply that behaviorists abjure all ways of restricting sets of NE to plausible subsets. In particular, they tend to be sympathetic to approaches that shift emphasis from rationality itself onto considerations of the informational dynamics of games. We should perhaps not be surprised that NE analysis alone often fails to tell us much of interest about strategic-form games (e.g., Figure 6 above), in which informational structure is suppressed. Equilibrium selection issues are often more fruitfully addressed in the context of extensive-form games.

Consider the game described by this tree:

This game is not intended to fit any preconceived situation; it is simply a mathematical object in search of an application. (L and R here just denote ‘left’ and ‘right’ respectively.)

Figure 9

Now consider the strategic form of this game:

Figure 10

(If you are confused by this, remember that a strategy must tell a
player what to do at *every* information set where that player
has an action. Since each player chooses between two actions at each
of two information sets here, each player has four strategies in
total. The first letter in each strategy designation tells each player
what to do if he or she reaches their first information set, the
second what to do if their second information set is reached. I.e., LR
for Player II tells II to play L if information set 5 is reached and R
if information set 6 is reached.) If you examine this matrix, you will
discover that (LL, RL) is among the NE. This is a bit puzzling, since
if Player I reaches her second information set (7) in the
extensive-form game, I would hardly wish to play L there; she earns a
higher payoff by playing R at node 7. Mere NE analysis doesn't notice
this because NE is insensitive to what happens *off the path of
play*. Player I, in choosing L at node 4, ensures that node 7 will
not be reached; this is what is meant by saying that it is ‘off the
path of play’. In analyzing extensive-form games, however, we
*should* care what happens off the path of play, because
consideration of this is crucial to what happens *on* the
path. For example, it is the fact that Player I *would* play R if
node 7 were reached that *would* cause Player II to play L if
node 6 were reached, and this is why Player I won't choose R at node
4. We are throwing away information relevant to game solutions if we
ignore off-path outcomes, as mere NE analysis does. Notice that this
reason for doubting that NE is a wholly satisfactory equilibrium
concept in itself has nothing to do with intuitions about rationality,
as in the case of the refinement concepts discussed in Section
2.5.

Now apply Zermelo's algorithm to the extensive form of our current example. Begin, again, with the last subgame, that descending from node 7. This is Player I's move, and she would choose R because she prefers her payoff of 5 to the payoff of 4 she gets by playing L. Therefore, we assign the payoff (5, -1) to node 7. Thus at node 6 II faces a choice between (-1, 0) and (5, -1). He chooses L. At node 5 II chooses R. At node 4 I is thus choosing between (0, 5) and (-1, 0), and so plays L. Note that, as in the PD, an outcome appears at a terminal node -- (4, 5) from node 7 -- that is Pareto superior to the NE. Again, however, the dynamics of the game prevent it from being reached.

The fact that Zermelo's algorithm picks out the strategy vector (LR,
RL) as the unique solution to the game shows that it's yielding
something other than just an NE. In fact, it is generating the game's
*subgame perfect equilibrium* (SPE). It gives an outcome that
yields a NE not just in the *whole* game but in every subgame as
well. This is an extremely persuasive solution concept because, again
unlike the refinements of Section 2.5, it does not demand ‘more’
rationality of agents, but *less*. The agents, at every node,
simply choose the path that brings them the highest payoff *in the
subgame emanating from that node*; and, then, in solving the game,
they foresee that they will all do that. Agents who proceed in this
way are said to be *modular rational*, that is, short-run
rational at each step. They do not imagine themselves, by some fancy
processes of hyper-rationality, acting against their local preferences
for the sake of some wider goal. Note that, as in the PD, this can
lead to outcomes which might be regretted from the social point of
view. In our current example, Player I would be better off, and Player
II no worse off, at the left-hand node emanating from node 7 than at
the SPE outcome. But Player I's very modular rationality, and Player
II's awareness of this, blocks the socially efficient outcome. If our
players wish to bring about the more equitable outcome (4,5) here,
they must do so by redesigning their institutions so as to change the
structures of the games they play. Merely wishing that they could be
hyper-rational in some way does not seem altogether coherent as an
approach.

This way of thinking leads to serious misunderstandings of game
theory, and so must be dispelled. Let us first introduce some
terminology for talking about outcomes. Welfare economists typically
measure social good in terms of *Pareto efficiency*. A
distribution of utility
is said to be *Pareto dominant* over another distribution
just in case from state
there is a possible redistribution of utility to
such that at least one player is better off in
than in
and no player is worse off. Failure to move to a Pareto-dominant
redistribution is *inefficient* because the existence of
as a logical possibility shows that in
some utility is being wasted. Now, the outcome (3,3) that represents
mutual cooperation in our model of the PD is clearly Pareto dominant
over mutual defection; at (3,3) *both* players are better off
than at (2,2). So it is true that PDs lead to inefficient
outcomes. This was true of our example in Section 2.6 as well.

However, inefficiency should not be associated with immorality. A
utility function for a player is supposed to represent *everything
that player cares about*, which may be anything at all. As we have
described the situation of our prisoners they do indeed care only
about their own relative prison sentences, but there is nothing
essential in this. What makes a game an instance of the PD is strictly
and only its payoff structure. Thus we could have two Mother Theresa
types here, both of whom care little for themselves and wish only to
feed starving children. But suppose the original Mother Theresa wishes
to feed the children of Calcutta while Mother Juanita wishes to feed
the children of Bogota. And suppose that the international aid agency
will maximize its donation if the two saints nominate the same city,
will give the second-highest amount if they nominate each others'
cities, and the lowest amount if they each nominate their own
city. Our saints are in a PD here, though hardly selfish or
unconcerned with the social good.

To return to our prisoners, suppose that, contrary to our
assumptions, they *do* value each other's well-being as
well as their own. In that case, this must be reflected in their
utility functions, and hence in their payoffs. If their payoff
structures are changed, they will no longer be in a PD. But all this
shows is that not every possible situation is a PD; it does
*not* show that the threat of inefficient outcomes is a special
artifact of selfishness. It is the *logic* of the
prisoners' situation, not their psychology, that traps them in
the inefficient outcome, and if that really *is* their
situation then they are stuck in it (barring further complications to
be discussed below). Agents who wish to avoid inefficient outcomes
are best advised to prevent certain games from arising; the defender
of the possibility of hyper-rationality is really proposing that they
try to dig themselves out of such games by turning themselves into
different kinds of agents.

In general, then, a game is partly *defined* by the payoffs
assigned to the players. If a proposed solution involves tacitly
changing these payoffs, then this ‘solution’ is in fact a disguised
way of changing the subject.

Figure 11

The NE outcome here is at the single leftmost node descending from
node 8. To see this, backward induct again. At node 10, I would play L
for a payoff of 3, giving II a payoff of 1. II can do better than this
by playing L at node 9, giving I a payoff of 0. I can do better than
this by playing L at node 8; so that is what I does, and the game
terminates without II getting to move. But, now, notice the reasoning
required to support this prediction. I plays L at node 8 because she
knows that II is rational, and so would, at node 9, play L because II
knows that I is rational and so would, at node 10, play L. But now we
have the following paradox: I must suppose that II, at node 9, would
predict I's rational play at node 10 despite having arrived at a
node (9) that could only be reached if I is not rational! If I is not
rational then II is not justified in predicting that I will not play R
at node 10, in which case it is not clear that II shouldn't play
R at 9; and if II plays R at 9, then I is guaranteed of a better
payoff then she gets if she plays L at node 8. Both players must use
backward induction to solve the game; backward induction requires that
I know that II knows that I is rational; but II can solve the game
only by using a backward induction argument that takes as a premise
the irrationality of I. This is the *paradox of backward
induction*.

A standard way around this paradox in the literature is to invoke the so-called ‘trembling hand’ due to Selten (1975). The idea here is that a decision and its consequent act may ‘come apart’ with some nonzero probability, however small. That is, a player might intend to take an action but then slip up in the execution and send the game down some other path instead. If there is even a remote possibility that a player may make a mistake -- that her ‘hand may tremble’ -- then no contradiction is introduced by a player's using a backward induction argument that requires the hypothetical assumption that another player has taken a path that a rational player could not choose. In our example, II could reason about what to do at node 9 conditional on the assumption that I rationally chose L at node 8 but then slipped.

There is a substantial technical literature on this
backward-induction paradox, of which
Bicchieri (1993)
is the most comprehensive source. (Bicchieri, it should be noted,
does *not *endorse an appeal to trembling hands as the
appropriate solution. Discussing her particular proposal here would,
however, take us too far afield into technicalities. The interested
reader should study her book.) The puzzle has been introduced here
just in order to point out that refinements of the type discussed in
Section 2.6 can be encouraged by more than mere intuitions about the
concept of rationality. For if hands may tremble than merely
economically rational players *will* be motivated to worry about
the probabilities with which apparent departures from rational play
will be observed. For example, if my opponent's hand may tremble, then
this gives me good reason to avoid the weakly dominated strategy s2 in
the third example from Section 2.5.
After all, my opponent might promise to play t1 in that game, and I
may believe his promise; but if his hand then trembles and a play of
t2 results, I get my worst payoff. If I'm risk-averse, then in such
situations it would seem that I should stick to weakly dominant
strategies.

The reader won't be surprised to hear that the behaviorist has a
reply to this. He argues that when a player considers what would
happen at nodes reachable only along out-of-equilibrium paths, he need
merely refer to *possible worlds* in which the subgames beginning
at the nodes in question exist by themselves (i.e., detached from the
remainders of the games in which they occur). This might seem like an
ad hoc resort; but it is not clear that it is any *less* so than
is appeal to trembling hands, which find no independent motivation
from elsewhere in economic theory or the theory of rational choice.
In any case, appeal to possible worlds is a common strategy in science
according to many philosophers. The issue is raised here not with any
hope in mind of saying something conclusive about it, but just to give
the reader some further sense of the philosophical issues that
generate controversy in the foundations of game theory.

Suppose that we ignore rocks and cobras for a moment, and imagine
that the bridges are equally safe. In this case, the fugitive's
best course is to roll a three-sided die, in which each side
represents a different bridge (or, more conventionally, a six-sided
die in which each bridge is represented by two sides). He must then
pre-commit himself to using whichever bridge is selected by this
*randomizing device*. Against this strategy, the pursuer's
best reply is to also use a three-sided die of her own. The fugitive
now has a 2/3 probability of escaping and the pursuer a 1/3
probability of catching him. The fugitive cannot improve on these odds
if the pursuer is randomizing, since to favour one bridge merely
provides the pursuer with a pattern that can be exploited. Identical
reasoning applies to the pursuer. Therefore, the two randomizing
strategies are best replies to one another, and are therefore in Nash
equilibrium.

Now let us re-introduce the parametric factors, that is, the falling
rocks at bridge #2 and the cobras at bridge #3. Again, suppose that
bridge #3 (cobras) is four times more dangerous for the fugitive than
bridge #2 (rocks), while bridge #2 is 20% more dangerous than bridge
#1 (unobstructed). We can solve this new game *if* we make
certain assumptions about the two players' utility
functions. Suppose that Player 1, the fugitive, cares only about
living or dying (preferring life to death) while the pursuer simply
wishes to be able to report that the fugitive is dead, preferring
this to having to report that he got away. (In other words, neither
player cares about *how* the fugitive lives or dies.) In this
case, the fugitive simply takes his original randomizing formula and
weights it according to the different levels of parametric danger at
the three bridges. Each bridge should be thought of as a
*lottery* over the fugitive's possible outcomes, in which
each lottery has a different *expected payoff* in terms of the
items in his utility function.

Consider matters from the pursuer's point of view. She will be using her NE strategy when she chooses the mix of probabilities over the three bridges that makes the fugitive indifferent amongst them as crossing points. The bridge with rocks is 1.2 times more dangerous for him than the safe bridge. Therefore, he will be indifferent between the two when the pursuer is 1.2 times more likely to be waiting at the rocky bridge than the safe bridge. The cobra bridge is 4 times more dangerous for the fugitive than the rocky bridge. Therefore, he will be indifferent between these two bridges when the pursuer's probability of waiting at the rocky bridge is 4 times higher than the probability that she is at the cobra bridge. This gives us an indifference ratio amongst the bridges of 4.8:4:1. Suppose that no mixed strategy involving use of the cobra bridge is dominated by a mixed strategy involving use of only the other two bridges (i.e., that the cobra bridge is not so dangerous that the hunter can't make the fugitive indifferent between its use and that of the other at any value). Since the probabilities must sum to 100, we find the probability with which the pursuer waits at each bridge by solving the following equation:

4.8x+ 4x+x= 100

We were able to solve this game straightforwardly because we set the
utility functions in such a way as to make it *zero-sum*, or
*strictly competitive*. That is, every gain in expected utility
by one player represents a precisely symmetrical loss by the
other. However, this condition may often not hold. Suppose now that
the utility functions are more complicated. The pursuer most prefers
an outcome in which she shoots the fugitive and so claims credit for
his apprehension to one in which he dies of rockfall or snakebite; and
she prefers this second outcome to his escape. The fugitive prefers a
quick death by gunshot to the pain of being crushed or the terror of
an encounter with a cobra. Most of all, of course, he prefers to
escape. We cannot solve this game, as before, simply on the basis of
knowing the players' ordinal utility functions, since the
*intensities* of their respective preferences will now be
relevant to their strategies.

Prior to the work of
von Neumann & Morgenstern (1947),
situations of this sort were inherently baffling to analysts. This is
because utility does not denote a hidden psychological variable such
as *pleasure*. As we discussed in
Section 2.1, utility is merely a measure of
relative behavioural dispositions given certain consistency
assumptions about relations between preferences and choices. It
therefore makes no sense to imagine comparing our players'
*cardinal* -- that is, intensity-sensitive -- preferences with
one another's, since there is no independent, interpersonally
constant yardstick we could use. How, then, can we model games in
which cardinal information is relevant? After all, modeling games
requires that all players' utilities be taken simultaneously into
account, as we've seen.

A crucial aspect of
von Neumann & Morgenstern's (1947)
work was the solution to this problem. Here, we will provide a brief
outline of their ingenious technique for building cardinal utility
functions out of ordinal ones. It is emphasized that what follows is
merely an *outline*, so as to make cardinal utility
non-mysterious to you as a student who is interested in knowing about
the philosophical foundations of game theory, and about the range of
problems to which it can be applied. Providing a manual you could
follow in *building *your own cardinal utility functions would
require many pages. Fortunately, such manuals are available in many
textbooks. In any case, if you are a philosophy student you may not
wish to attempt this until you've taken a course in probability
theory.

Suppose we have an agent whose ordinal utility function is known. Indeed, suppose that it's our river-crossing fugitive. Let's assign him the following ordinal utility function:

Escape 4Now, we know that his preference for escape overDeath by shooting 3

Death by rockfall 2

Death by snakebite 1

Begin by asking our agent to pick, from the available set of
outcomes, a *best* one and a *worst*
one. ‘Best’ and ‘worst’ are defined in terms of
rational choice: a rational agent always chooses so as to maximize the
probability of the best outcome -- call this **W** -- and
to minimize the probability of the worst outcome -- call this
**L**. Now consider prizes intermediate between
**W** and **L**. We find, for a set of
outcomes containing such prizes, a lottery over them such that our
agent is indifferent between that lottery and a lottery including only
**W** and **L**. In our example, this would
be a lottery having shooting and rockfall as its possible
outcomes. Call this lottery **T** . We define a utility
function *q* = *u*(**T**) such that if
*q* is the expected prize in **T** , the agent is
indifferent between winning **T** and winning a lottery
in which **W** occurs with probability
*u*(**T**) and **L** occurs with
probability
1 *u*(**T**).

We now construct a *compound lottery* **T*** over
the outcome set {**W**, **L**} such that the
agent is indifferent between **T** and
**T***. A compound lottery is one in which the prize in
the lottery is another lottery. This makes sense because, after all,
it is still **W** and **L** that are at
stake for our agent in both cases; so we can then analyze
**T*** into a simple lottery over **W** and
**L**. Call this lottery **r**. It follows
from transitivity that **T** is equivalent to
**r**. (Note that this presupposes that our agent does
not gain utility from the complexity of her gambles.) The rational
agent will now choose the action that maximizes the probability of
winning **W**. The mapping from the set of outcomes to
*u*(**r**) is a *von Neumann-Morgenstern
utility function* (VNMuf).

What exactly have we done here? We've simply given our agent
choices over lotteries, instead of over prizes directly, and observed
how much extra risk he's willing to run to increase the chances
of winning escape over snakebite relative to getting shot or clobbered
with a rock. A VNMuf yields a *cardinal*, rather than an
ordinal, measure of utility. Our choice of endpoint-values,
**W** and **L**, is arbitrary, as before;
but once these are fixed the values of the intermediate points are
determined. Therefore, the VNMuf *does* measure the relative
preference intensities of a single agent. However, since our
assignment of utility values to **W** and
**L** *is* arbitrary, we can't use VNMufs to
compare the cardinal preferences of one agent with those of
another. Furthermore, since we are using a *risk-metric* as our
measuring instrument, the construction of the new utility function
depends on assuming that our agent's *attitude to risk
itself* stays constant from one comparison of lotteries to
another. This seems reasonable for a single agent in a single
game-situation. However, two agents in one game, or one agent under
different sorts of circumstances, may display very different attitudes
to risk. Perhaps in the river-crossing game the pursuer, whose life
is not at stake, will enjoy gambling with her glory while our fugitive
is cautious. In general, a *risk-averse* agent prefers a
guaranteed prize to its equivalent expected value in a lottery. A
*risk-loving* agent has the reverse preference. A
*risk-neutral* agent is indifferent between these options. In
analyzing the river-crossing game, however, we don't *have
to* be able to compare the pursuer's cardinal utilities with
the fugitive's. Both agents, after all, can find their NE
strategies if they can estimate the probabilities each will assign to
the actions of the other. This means that each must know both VNMufs;
but neither need try to comparatively value the outcomes over which
they're gambling.

We can now fill in the rest of the matrix for the bridge-crossing
game that we started to draw in Section 2. If all that the fugitive
cares about is life and death, but not the manner of death, and if all
the hunter cares about is preventing the fugitive from escaping, then
we can now interpret both utility functions cardinally. This permits
us to assign expected utilities, expressed by multiplying the original
payoffs by the relevant probabilities, as outcomes in the
matrix. Suppose that the hunter waits at the cobra bridge with
probability *x* and at the rocky bridge with probability
*y*. Since her probabilities across the three bridges must sum
to 1, this implies that she must wait at the safe bridge with
probability 1
(*x* + *y*). Then, continuing to assign the fugitive a
payoff of 0 if he dies and 1 if he escapes, and the hunter the
reverse payoffs, our complete matrix is as follows:

Figure 12

We can now read the following facts about the game directly from the matrix. No rows or columns strictly or weakly dominate any others. Therefore, the game's NE must be in mixed strategies.

Here is an odd feature of our analysis of the river-crossing game
above. In the situation as we have imagined it, the fugitive knows
that the hunter is least likely to cross at the cobra bridge. Yet we
have told her to select a bridge at which to wait by flipping a
coin. This process gives a non-zero probability of selecting the cobra
bridge. Yet, surely, if the coin selects the cobra bridge it cannot be
rational for the hunter to do as it directs. Won't she feel like a bit
of an idiot standing there, knowing that the fugitive is probably
waltzing across the safe bridge, or the relatively safe rocky bridge?
Note that she will *not* feel torn in this situation if she plays
this game with fugitives on a regular basis (and they know this). In
that case, it is perfectly reasonable that she must *sometimes*
wait at the cobra bridge, lest all fugitives be able to do better than
their odds at NE by using the cobra bridge more often. Now, as far as
the behaviorist is concerned this ends the matter. If the hunter and
the fugitive have regularly played games that structurally
*resemble* this river-crossing game, then selection pressures
will have encouraged habits in them that lead them both to play its NE
strategies *and to sincerely rationalize doing so* by means of
some satisfying story or other. If neither party has ever been in a
situation like this, and if their biological and/or cultural
ancestors haven't either, and if neither is concerned with revealing
information to opponents in expected future situations of this sort,
then their behavior should be predicted not by a game theorist but by
friends of theirs who are familiar with their personal
idiosyncracies. Behaviorists are happy to recognize that game theory
isn't useful for every decision problem, or even every strategic
decision problem, that comes along.

However, the philosopher who wants game theory to serve as a
descriptive and/or normative theory of strategic rationality cannot
rest content with this answer. He must find a satisfying line of
advice for the players even when their game is alone in the universe
of strategic problems. No such advice can be given that is
*uncontroversially* satisfactory -- behaviorists, after all, are
often behaviorists *because* they aren't satisfied by any
available approach here -= but there is a way of handling the matter
that many game theorists have found worthy of detailed pursuit. This
involves the computation of *equilibria in beliefs*.

In fact, the behaviorist needs the concept of equilibrium in beliefs too, but for different purposes. As we've seen, the concept of NE sometimes doesn't go deep enough as an analytical instrument to tell us all that we think might be important in a game. Thus even behaviorists who aren't impressed with the project of refinements make frequent use of the concept of subgame-perfect equilibrium (SPE) as discussed in Section 2.6. But now consider the three-player imperfect-information game below (taken from Kreps (1990), p. 426):

Figure 13

One of the NE of this game is Lr_{2}l_{3}. This is
because if Player I plays L, then Player II playing r_{2} has
no incentive to change strategies because her only node of action, 12,
is off the path of play. But this NE seems to be purely technical; it
makes little sense as a solution. This reveals itself in the fact that
if the game beginning at node 14 could be treated as a subgame,
Lr_{2}l_{3} would not be an SPE. Whenever she
*does* get a move, Player II should play l_{2}. But if
Player II is playing l_{2} then Player I should switch to
R. In that case Player III should switch to r_{3}, sending
Player II back to r_{2}. And here's a new,
‘sensible’, NE: Rr_{2}r_{3}. I and II in
effect play ‘keepaway’ from III; and so that's what
we'll name this game.

This NE is ‘sensible’ in just the same way that a SPE outcome in a perfect-information game is more sensible than other non-SPE NE. However, we can't select it by applying Zermelo's algorithm. Because nodes 13 and 14 fall inside a common information set, Keepaway has only one subgame (namely, the whole game). We need a ‘cousin’ concept to SPE that we can apply in cases of imperfect information, and we need a new solution procedure to replace Zermelo's algorithm for such games.

Notice what Player III in Keepaway is wondering about as he selects
his strategy. "Given that I get a move," he asks himself,
"was my action node reached from node 11 or from node 12?"
What, in other words, are the *conditional probabilities *that
III is at node 13 or 14 given that he has a move? Now, if conditional
probabilities are what III wonders about, then what Players I and II
must make conjectures about when they select *their* strategies
are III's *beliefs* about these conditional
probabilities. In that case, I must conjecture about II's beliefs
about III's beliefs, and III's beliefs about II's
beliefs and so on. The relevant beliefs here are not merely strategic,
as before, since they are not just about what players will *do*
given a set of payoffs and game structures, but about what they think
makes sense given some understanding or other of conditional
probability.

What beliefs about conditional probability is it reasonable for
players to expect from each other? The normative theorist might insist
on whatever the best mathematicians have discovered about the
subject. Clearly, however, if this is applied then a theory of games
that incorporated it would not be descriptively true of most
people. The behaviorist will insist on imposing only behavioral habits
that a process of natural selection might build into its
products. Perhaps some actual or possible creatures might observe
habits that respect *Bayes's rule*, which is the minimal
true generalization about conditional probability that an agent could
know if it knows any such generalizations at all. Adding more
sophisticated knowledge about conditional probability amounts to
refining the concept of equilibrium-in-belief, just as some game
theorists like to refine NE. You can imagine what behaviorists think
of *that* project!

Here, we will restrict our attention to the least refined equilibrium-in-belief concept, that obtained when we require players to reason in accordance with Bayes's rule. Bayes's rule tells us how to compute the probability of an event F given information E (written ‘pr(F/E)’):

pr(F/E) = [pr(E/F) X pr(F)] / pr(E)

We will henceforth assume that players do not hold beliefs inconsistent with this equality.

We may now define a *sequential equilibrium*. A SE has two
parts: (1) a strategy profile § for each player, as before, and
(2) a *system of beliefs*
for each player.
assigns to each information set *h* a probability distribution
over the nodes *x* in *h*, with the interpretation that
these are the beliefs of player *i(h)* about where in his
information set he is, given that information set *h* has been
reached. Then a sequential equilibrium is a profile of strategies
§ and a system of beliefs
consistent with Bayes's rule such that starting from every
information set *h* in the tree player *i(h)* plays
optimally from then on, given that what he believes to have transpired
previously is given by
(*h*)
and what will transpire at subsequent moves is given by
§.

We now demonstrate the concept by application to Keepaway. Consider
again the uninteresting NE Lr_{2}l_{3}. Suppose that
Player III assigns pr(1) to her belief that if she gets a move she is
at node 13. Then Player II, given a consistent
(II),
must believe that III will play l_{3}, in which case
her only SE strategy is l_{2}. So although
Lr_{2}l_{3} is a NE, it is not a SE. This is of course
what we want.

The use of the consistency requirement in this example is somewhat trivial, so consider now a second case (also taken from Kreps (1990), p. 429):

Figure 14

Suppose that I plays L, II plays l_{2} and III plays
l_{3}. Suppose also that
(II)
assigns pr(.3) to node 16. In that case, l_{2} is not a
SE strategy for II, since l_{2} returns an expected payoff of
.3(4) + .7(2) = 2.6, while r_{2} brings an expected payoff of
3.1. Notice that if we fiddle the strategy profile for player III
while leaving everything else fixed, l_{2} could *become*
a SE strategy for II. If §(III) yielded a play of l_{3}
with pr(.5) and r_{3} with pr(.5), then if II plays
r_{2} his expected payoff would now be 2.2, so
Ll_{2}l_{3} would be a SE. Now imagine setting
(III)
back as it was, but change
(II)
so that II thinks the conditional probability of being
at node 16 is greater than .5; in that case, l_{2} is again not a SE
strategy.

The idea of SE is hopefully now clear. We can apply it to the river-crossing game in a way that avoids the necessity for the hunter to flip any coins of we modify the game a bit. Suppose now that II can change bridges twice during the fugitive's passage, and will catch him just in case she meets him as he leaves the bridge. Then the hunter's SE strategy is to divide her time at the three bridges in accordance with the proportion given by the equation in the third paragraph of Section 3 above.

It must be noted that since Bayes's rule cannot be applied to
events with probability 0, its application to SE requires that players
assign non-zero probabilities to all actions available in trees. This
requirement is captured by supposing that all strategy profiles be
*strictly mixed*, that is, that every action at every information
set be taken with positive probability. You will see that this is just
equivalent to supposing that all hands sometimes tremble. A SE is said
to be *trembling-hand perfect *if all strategies played at
equilibrium are best replies to strategies that are strictly
mixed. You should also not be surprised to be told that no weakly
dominated strategy can be trembling-hand perfect, since the
possibility of trembling hands gives players the most persuasive
reason for avoiding such strategies.

We've seen that in the one-shot PD the only NE is mutual defection. This may no longer hold, however, if the players expect to meet each other again in future PDs. Imagine that four firms, all making widgets, agree to maintain high prices by jointly restricting supply. (That is, they form a cartel.) This will only work if each firm maintains its agreed production quota. Typically, each firm can maximize its profit by departing from its quota while the others observe theirs, since it then sells more units at the higher market price brought about by the almost-intact cartel. In the one-shot case, all firms would share this incentive to defect and the cartel would immediately collapse. However, the firms expect to face each other in competition for a long period. In this case, each firm knows that if it breaks the cartel agreement, the others can punish it by underpricing it for a period long enough to more than eliminate its short-term gain. Of course, the punishing firms will take short-term losses too during their period of underpricing. But these losses may be worth taking if they serve to reestablish the cartel and bring about maximum long-term prices.

One simple, and famous (but *not*, contrary to widespread
myth, necessarily optimal) strategy for preserving cooperation in
repeated PDs is called *tit-for-tat*. This strategy tells each
player to behave as follows:

- Always cooperate in the first round.
- Thereafter, take whatever action your opponent took in the previous round.

There are two complications. First, the players must be uncertain as to when their interaction ends. Suppose the players know when the last round comes. In that round, it will be rational for players to defect, since no punishment will be possible. Now consider the second-last round. In this round, players also face no punishment for defection, since they know they will defect in the last round anyway. So they defect in the second-last round. But this means they face no threat of punishment in the third-last round, and defect there too. We can simply iterate this backwards through the game tree until we reach the first round. Since cooperation is not rational in that round, tit-for-tat is no longer a rational strategy, and we get the same outcome -- mutual defection -- as in the one-shot PD. Therefore, cooperation is only possible in repeated PDs where the expected number of repetitions is indeterminate. (Of course, this does apply to many real-life games.)

But now we introduce a second complication. Suppose that
players' ability to distinguish defection from cooperation is
imperfect. Consider our case of the widget cartel. Suppose the players
observe a fall in the market price of widgets. Perhaps this is because
a cartel member cheated. Or perhaps it has resulted from an exogenous
drop in demand. If tit-for-tat players mistake the second case for the
first, they will defect, thereby setting off a chain-reaction of
mutual defections from which they can *never* recover, since
every player will reply to the first encountered defection with
defection, thereby begetting further defections, and so on.

If players know that such miscommunication is possible, they must
resort to more sophisticated strategies. In particular, they must be
prepared to sometimes risk following defections with cooperation in
order to test their inferences. However, they mustn't be
*too* forgiving, lest other players find it rationally optimal
to exploit them through deliberate defections. In general,
sophisticated strategies have a problem. Because they are more
difficult for other players to infer, their use increases the
probability of miscommunication. But miscommunication is what causes
repeated-game cooperative equilibria to unravel in the first place!
The moral of this is that PDs, even repeated ones, are very difficult
to escape from. Rational players do best trying to *avoid*
situations that are PDs, rather than relying on cunning strategems for
trying to get out of them.

Real, complex, social and political dramas are seldom straightforward
instantiations of simple games such as PDs.
Russell Hardin (1995)
offers an analysis of two recent, very real (and very tragic)
political cases, the Yugoslavian civil war of 1991-95, and the 1994
Rwandan genocide, as PDs that were nested inside *coordination
games*. A coordination game occurs whenever the utility of two or
more players is maximized by their doing the same thing, and where
such correspondence is more important to them than *what*, in
particular, they both do. A standard example arises with rules of the
road: ‘All drive on the left’ and ‘All drive on the
right’ are both outcomes that are NEs, and neither is more
efficient than the other. In games of ‘pure’ coordination,
it doesn't even help to use more selective equilibrium criteria.
For example, suppose that we require our players to reason in
accordance with Bayes's rule (see Section 3 above). In these
circumstances, any strategy that is a best reply to any vector of
mixed strategies available in NE is said to be *rationalizable*.
That is, a player can find a set of systems of beliefs for the other
players such that any history of the game along an equilibrium path is
consistent with that set of systems. Pure coordination games are
characterized by non-unique vectors of rationalizable strategies. In
such situations, players may try to predict equilibria by searching
for *focal points*, that is, features of some strategies that
they believe will be salient to other players, and that they believe
other players will believe to be salient to them. Unfortunately, in
many of the social and political games played by people (and some
other animals), the biologically shallow properties by which people
sort themselves into racial and ethnic groups serve highly efficiently
as such features. Hardin's analysis of recent genocides relies
on this fact.

According to Hardin, neither the Yugolslavian nor the Rwandan
disasters were PDs to begin with. That is, in neither situation, on
either side, did most people begin by preferring the destruction of
the other to mutual cooperation. However, the deadly logic of
coordination, deliberately abetted by self-serving politicians,
dynamically *created *PDs. Some individual Serbs (Hutus) were
encouraged to perceive their individual interests as best served
through identification with Serbian (Hutu) group-interests. That is,
they found that some of their circumstances, such as those involving
competition for jobs, had the form of coordination games. They thus
acted so as to create situations in which this was true for other
Serbs (Hutus) as well. Eventually, once enough Serbs (Hutus)
identified self-interest with group-interest, the identification
became almost universally *correct*, because (1) the most
important goal for each Serb (Hutu) was to do roughly what every other
Serb (Hutu) would, and (2) the most distinctively *Serbian
*thing to do, the doing of which permitted coordination, was to
exclude Croats (Tutsi). That is, strategies involving such
exclusionary behavior were selected as a result of having efficient
focal points. This situation made it the case that an individual --
and individually threatened -- Croat's (Tutsi's)
self-interest was best maximized by coordinating on assertive Croat
(Tutsi) group-identity, which further increased pressures on Serbs
(Hutus) to coordinate, and so on. Note that it is not an aspect of
this analysis to suggest that Serbs or Hutus started things; the
process could have been (even if it wasn't in fact) perfectly
reciprocal. But the outcome is ghastly: Serbs and Croats (Hutus and
Tutsis) seem progressively more threatening to each other as they
rally together for self-defense, until both see it as imperative to
preempt their rivals and strike before being struck. If Hardin is
right -- and the point here is not to claim that he *is,* but
rather to point out the worldly importance of determining which games
agents are in fact playing -- then the mere presence of an external
enforcer (NATO?) would not have changed the game, pace the Hobbesian
analysis, since the enforcer could not have threatened either side
with anything worse than what each feared from the other. What was
needed was recalibration of evaluations of interests, which (arguably)
happened in Yugoslavia when the Croatian army began to decisively win,
at which point Bosnian Serbs decided that their self/group interests
were best served by the arrival of NATO peacekeepers. The Rwandan
conflict, meanwhile, drags on in the neighbouring country (the Congo)
to which military and political developments have shifted it.

Of course, it is not the case that most repeated games lead to
disasters. The biological basis of friendship in people and other
animals is probably *partly* a function of the logic of repeated
games. The importance of payoffs achievable through cooperation in
future games leads those who expect to interact in them to be less
selfish than temptation would suggest in present games. Furthermore,
cultivating shared interests and sentiments provides networks of focal
points around which coordination can be facilitated.

Consider the following hypothetical example (which is *not* a
PD). Suppose you own a piece of land adjacent to mine, and I'd
like to buy it so as to expand my lot. Unfortunately, you don't
want to sell at the price I'm willing to pay. If we move
simultaneously -- you post a selling price and I independently give my
agent an asking price -- there will be no sale. So I might try to
change your incentives by playing an opening move in which I announce
that I'll build a putrid-smelling sewage disposal plant on my
land beside yours unless you sell, thereby lowering your
price. I've now turned this into a sequential-move game. However,
this move so far changes nothing. If you refuse to sell in the face of
my threat, it is then not in my interest to carry it out, because in
damaging you I also damage myself. Since you know this you should
ignore my threat. My threat is *incredible*, a case of cheap
talk.

However, I could make my threat credible by *committing
*myself*.* I could sign a contract with some farmers
promising to supply them with treated sewage (fertilizer) from my
plant, but including an escape clause in the contract releasing me
from my obligation only if I can double my lot size and so put it to
some other use. Now my threat is credible: if you don't sell,
I'm committed to building the sewage plant. Since you know this,
you now have an incentive to sell me your land in order to escape its
ruination.

This sort of case exposes one of many fundamental differences between the logic of non-parametric and parametric maximization. In parametric situations, an agent can never be made worse off by having more options. But where circumstances are non-parametric, one agent's strategy can be influenced in another's favour if options are visibly restricted. Cortez's burning of his boats (see Section 1) is, of course, an instance of this, one which serves to make the usual metaphor literal.

Another example will illustrate this, as well as the applicability
of principles across game-types. Here we will build an imaginary
situation that is not a PD -- since only one player has an incentive
to defect -- but which is a social dilemma insofar as its NE in the
absence of commitment is Pareto-inferior to an outcome that is
achievable *with *a commitment device. Suppose that two of us
wish to poach a rare antelope from a national park in order to sell
the trophy. One of us must flush the animal down towards the second
person, who waits in a blind to shoot it and load it onto a
truck. You promise, of course, to share the proceeds with
me. However, your promise is not credible. Once you've got the
buck, you have no reason not to drive it away and pocket the full
value from it. After all, I can't very well complain to the
police without getting myself arrested too. But now suppose I add the
following opening move to the game. Before our hunt, I rig out the
truck with an alarm that can be turned off only by punching in a
code. Only I know the code. If you try to drive off without me, the
alarm will sound and we'll both get caught. You, knowing this,
now have an incentive to wait for me. What is crucial to notice here
is that you *prefer* that I rig up the alarm, since this makes
your promise to give me my share credible. If I don't do this,
leaving your promise *in*credible, we'll be unable to
agree to try the crime in the first place, and both of us will lose
our shot at the profit from selling the trophy. Thus, you benefit
from my binding you.

We may now combine our analysis of PDs and commitment devices in discussion of the application that first made game theory famous outside of the academic community. The nuclear stand-off between the superpowers during the Cold War was exhaustively studied by the first generation of game theorists, many of whom worked for the US military. (See Poundstone 1992 for historical details.) Both the USA and the USSR maintained the following policy. If one side launched a first strike, the other threatened to answer with a devastating counter-strike. This pair of reciprocal strategies, which by the late 1960s would effectively have meant blowing up the world, was known as ‘Mutually Assured Destruction’, or ‘MAD’. Game theorists objected that MAD was mad, because it set up a Prisoner's Dilemma as a result of the fact that the reciprocal threats were incredible. Suppose the USSR launches a first strike against the USA. At that point, the American President faces the following situation. His country is already destroyed. He doesn't bring it back to life by now blowing up the world, so he has no incentive to carry out his threat, which has now manifestly failed to achieve its point. Since the Russians know this, they should ignore the threat and strike first! Of course, the Americans are in exactly the same position. Each power will recognize this incentive on the part of the other, and so will anticipate an attack if they don't preempt it. What we should therefore expect, because it is the only NE of the game, is a race between the two powers to be the first to attack.

This game-theoretic analysis caused genuine consternation and fear on
both sides during the Cold War, and produced some rather bizarre
attempts at setting up strategic commitment devices. President Nixon,
for example, had the CIA try to convince the Russians that he was
insane, so that they'd believe that he'd launch a
retaliatory strike even when it was no longer in his interest to do
so. Similarly, the Soviet KGB leaked fabricated medical reports
exaggerating Brezhnev's senility with the same end in
mind. Ultimately, the Americans broke this deadly symmetry by using a
‘doomsday device’. They equipped a worldwide fleet of
submarines with enough missiles to destroy the USSR, and arranged
their communications technology in such a way that the President could
not be *sure *to be able to reach the submarines and cancel
their orders to attack if any Soviet missile crossed the radar
‘trigger line’. Of course, this strategy depended on making
sure that the Russians were aware of the device. In Stanley
Kubrick's classic film *Dr. Strangelove*, the world is
destroyed by accident because the Russians build a doomsday machine
*but then keep it a secret*! As a result, when a mad American
colonel launches missiles at Russia on his own accord, and the
American President tries to convince his Soviet counterpart that the
attack was unintended, the Russian President sheepishly tells him
about the secret doomsday machine. Now the two Presidents can do
nothing but watch in dismay as the world is blown up -- due to a
game-theoretic mistake.

Commitment can sometimes be secured through the value to a player of
her own *reputation*. For example, a government tempted to
negotiate with terrorists to secure the release of hostages on a
particular occasion may commit to a ‘line in the sand’
strategy for the sake of maintaining a reputation for toughness
intended to reduce terrorists' incentives to launch future
attacks. A different sort of example is provided by Qantas Airlines of
Australia. Qantas has never suffered an accident, and makes much of
this in its advertising. This means that its planes probably
*are* safer than average even if the initial advantage was
merely a statistical artifact, because the value of its ability to
claim a perfect record rises the longer it lasts, and so gives the
airline continuous incentives to incur greater costs in safety
assurance.

Certain conditions must hold if reputation effects are to underwrite
commitment. First, the game must be repeated, with uncertainty as to
which round is the last one. The repeated PD can be used to illustrate
the importance of this principle. Cooperation can be the dominant
strategy in a repeated PD because a player can gain more from his
reputation for cooperation, through inducing expectations of
cooperation in others, than he can gain through defection in a single
round. However, if the players know in advance which round will be
their last, this equilibrium unravels. In the last round reputation no
longer has a value, and so both players will defect. In the
second-last round, the players know they will defect in the last
round, so reputation becomes worthless here too and they will again
defect. This makes reputation worthless in the third-last round, and
so on. The process iterates back to the first round, so no cooperation
ever occurs. This point can be generalized to state the most basic
condition on the possibility for using reputation effects as
commitment devices: the value of the reputation must be greater to its
cultivator than the value to him of sacrificing it in *any*
particular round. Thus players may establish commitment by reducing
the value of each round so that the temptation to defect in any round
never gets high enough to make it rational. For example, parties to a
contract may exchange their obligations in small increments to reduce
incentives on both sides to renege. Thus builders in construction
projects may be paid in weekly or monthly installments. Similarly, the
International Monetary Fund often dispenses loans to governments in
small tranches, thereby reducing governments' incentives to
violate loan conditions once the money is in hand; and governments may
actually prefer such arrangements in order to remove domestic
political pressure for non-compliant use.

In 1969, for example, the philosopher
David Lewis (1969)
published *Convention*, in which the conceptual framework of
game-theory was applied to one of the fundamental issues of
twentieth-century epistemology, the nature and extent of conventions
governing semantics and their relationship to the justification of
propositional beliefs. This book stands as one of the classics of
analytic philosophy, and its stock is presently rising still further
as we become more aware of the significance of the trail it
blazed. The basic insight can be captured using a simple example. The
word ‘chicken’ denotes chickens and ‘ostrich’
denotes ostriches. We would not be better or worse off if
‘chicken’ denoted ostriches and ‘ostrich’ denoted
chickens; however, we *would* be worse off if half of us used
the pair of words the first way and half the second, or if all of us
randomized between them to refer to flightless birds generally. This
insight, of course, well preceded Lewis; but what he recognized is
that this situation has the logical form of a coordination game. Thus,
while particular conventions may be arbitrary, the interactive
structures that stabilize and maintain them are not. Furthermore, the
equilibria involved in coordinating on noun-meanings appear to have an
arbitrary element only because we cannot Pareto-rank them; but
Millikan (1984) shows implicitly that in this respect
they are atypical of linguistic coordinations. In general, the
various NE in coordination games can very often be ranked.
Ross & LaCasse (1995)
present the following example. In a city, drivers must coordinate on
one of two NE with respect to their behaviour at traffic
lights. Either all must rush yellows and pause on shifts to green, or
slow down on yellows and jump forward on shifts to green. Both
patterns are NE, in that once a community has coordinated on one of
them no individual has an incentive to deviate: those who slow down
on yellows while others are rushing them will get rear-ended, while
those who rush yellows in the other equilibrium will risk collision
with those who jump forward quickly on greens. Therefore, once a
city's traffic pattern settles on one of these equilibria it
will tend to stay there. However, the two states are not
Pareto-indifferent, since the second NE allows more cars to turn left
on each cycle (in a right-hand-drive jurisdiction), which reduces the
extent of bottlenecks and allows all drivers to expect greater
efficiency in getting about. Conventions on standards of evidence and
rationality are likely to be of this character. While various
arrangements might be NE in the social game of science, as followers
of Thomas Kuhn like to remind us, it is highly improbable that all of
these lie on a single Pareto-indifference curve. These themes,
strongly represented in contemporary epistemology, philosophy of
science and philosophy of language, are all bequests of game theory
by way (at least indirectly) of Lewis. (The reader can find a broad
sample of applications, and references to the large literature, in
Nozick (1998).)
However, Lewis restricted his attention to static game theory, in
which agents *choose* strategies given exogenously fixed
utility-functions. As a result of this restriction, his account is
able to show us why conventions are important and stable, but it
invites a difficult and perhaps ultimately fruitless quest for a
general theory of rationality. This is because, as we saw in Section 3
above, in coordination (and other) games with multiple NE, what counts
as a solution is highly sensitive to conjectures made by players about
one another's beliefs and computational ability. This has excited
a good deal of attention, especially from philosophers, on the
implications of many subtle variations in the norms of strategic
rationality. However, if game theory is to explain actual, natural
behavior and its history in the way suggested by
Gintis (2000) above,
then we need some account of what is attractive about equilibria in
games even when no analysts or rational calculators are around to
identify them. To make reference again to Lewis's topic, when
human language developed there was no external referee to care about
and arrange for Pareto-efficiency. In order to understand
Gintis's optimism about the reach of game theory, we must
therefore extend our attention to *evolutionary* games.

Game theory has been fruitfully applied in evolutionary biology,
where species and/or genes are treated as players, since pioneering
work by
Maynard Smith (1982)
and his collaborators. Evolutionary (or *dynamic*) game theory
now constitutes a significant new mathematical extension applicable to
many settings apart from the biological. Thus
Skyrms (1996)
uses evolutionary game theory to try to answer questions Lewis could
not even ask, about the conditions under which language, concepts of
justice, the notion of private property, and other non-designed,
general phenomena of interest to philosophers would be likely to
arise. What is novel about evolutionary game theory is that moves are
not chosen by rational agents. Instead, agents are typically
hard-wired with particular strategies, and success for a strategy is
defined in terms of the number of copies that a strategy will leave of
itself to play in the games of succeeding generations. The strategies
themselves are therefore the players, and the games they play are
dynamic rather than static.

The discussion here will closely follow Skyrms's. We begin by
introducing *the replicator dynamics*. Consider first how
natural selection works to change lineages of animals, modifying,
creating and destroying species. The basic mechanism is
*differential reproduction*. Any animal with *heritable*
features that increase its *expected number of offspring* in a
given environment will tend to leave more offspring than others so
long as the environment remains relatively stable. These offspring
will be more likely to inherit the features in question. Therefore,
the proportion of these features in the population will gradually
increase as generations pass. Some of these features may *go to
fixation*, that is, eventually take over the entire population
(until the environment changes).

How does game theory enter into this? Often, one of the most important aspects of an organism's environment will be the behavioural tendencies of other organisms. We can think of each lineage as ‘trying’ to maximize its reproductive fitness (= expected number of grandchildren) through finding strategies that are optimal given the strategies of other lineages. So evolutionary theory is another domain of application for non-parametric analysis.

In dynamic game theory, we no longer think of individuals as choosing
strategies as they move from one game to another. This is because our
interests are different. We're now concerned less with finding
the equilibria of single games than with discovering which equilibria
are stable, and how they will change over time. So we now model
*the strategies themselves* as playing against each other. One
strategy is ‘better’ than another if it is likely to leave
more copies of itself in the next generation, when the game will be
played again. We study the changes in distribution of strategies in
the population as the sequence of games unfolds.

For dynamic game theory, we introduce a new equilibrium concept, due
to
Maynard Smith (1982).
A set of strategies, in some particular proportion (e.g., 1/3:2/3,
1/2:1/2, 1/9:8/9, 1/3:1/3:1/6:1/6 -- always summing to 1) is at an
*ESS* (Evolutionary Stable Strategy) equilibrium just in case
(1) no individual playing one strategy could improve its reproductive
fitness by switching to one of the other strategies in the proportion,
and (2) no mutant playing a different strategy altogether could
establish itself (‘invade’) in the population.

The principles of evolutionary game theory are best explained through examples. Skyrms begins by investigating the conditions under which a sense of justice -- understood as a disposition to view equal divisions of resources as fair unless efficiency considerations suggest otherwise in special cases -- might arise. He asks us to consider a population in which individuals regularly meet each other and must bargain over resources. Begin with three types of individuals:

*Fairmen*always demand exactly half the resource.*Greedies*always demand more than half the resource. When a greedy encounters another greedy, they waste the resource in fighting over it.*Modests*always demand less than half the resource. When a modest encounters another modest, they take less than all of the available resource and waste some.

- Half the population is greedy and half is modest. We can calculate the average payoff here. Modest gets 1/3 of the resource in every encounter. Greedy gets 2/3 when she meets Modest, but nothing when she meets another Greedy. So her average payoff is also 1/3. This is an ESS because Fairman can't invade. When Fairman meets Modest he gets 1/2. But when Fairman meets Greedy he gets nothing. So his average payoff is only 1/4. No Modest has an incentive to change strategies, and neither does any Greedy. A mutant Fairman arising in the population would do worst of all, and so selection will not encourage the propagation of any such mutants.
- All players are Fairmen. Everyone always gets half the resource, and no one can do better by switching to another strategy. Greedies entering this population encounter Fairmen and get an average payoff of 0. Modests get 1/3 as before, but this is less than Fairman's payoff of 1/2.

We refer to equilibria in which more than one strategy occurs as
*polymorphisms*. In general, in Skyrms's game, any
polymorphism in which Greedy demands *x* and Modest demands
1-*x* is an ESS. The question that interests the student of
justice concerns the relative likelihood with which these different
equilibria arise.

This depends entirely on the proportions of strategies in the
original population state. If the population begins with more than one
Fairman, then there is some probability that Fairmen will encounter
each other, and get the highest possible average payoff. Modests by
themselves do not inhibit the spread of Fairmen; only Greedies do. But
Greedies themselves depend on having Modests around in order to be
viable. So the more Fairmen there are in the population relative to
*pairs* of Greedies and Modests, the better Fairmen do on
average. This implies a threshold effect. If the proportion of Fairmen
drops below 33%, then the tendency will be for them to fall to
extinction because they don't meet each other often enough. If
the population of Fairmen rises above 33%, then the tendency will be
for them to rise to fixation because their extra gains when they meet
each other compensates for their losses when they meet Greedies. You
can see this by noticing that when each strategy is used by 33% of the
population, all have an expected average payoff of 1/3. Therefore, any
rise above this threshold on the part of Fairmen will tend to push
them towards fixation.

This result shows that and how, given certain relatively general
conditions, justice as we have defined it *can* arise
dynamically. The news for the fans of justice gets more cheerful still
if we introduce *correlated play*.

The model we just considered assumes that strategies are not
*correlated*, that is, that the probability with which every
strategy meets every other strategy is a simple function of their
relative frequencies in the population. We now examine what happens
in our dynamic resource-division game when we introduce
correlation. Suppose that Fairmen have a slight ability to distinguish
and seek out other Fairmen as interaction partners. In that case,
Fairmen on average do better, and this must have the effect of
lowering their threshold for going to fixation.

A dynamic-game modeler studies the effects of correlation and other
parametric constraints by means of running large computer simulations
in which the strategies compete with one another, round after round,
in the virtual environment. The starting proportions of strategies,
and any chosen degree of correlation, can simply be set in the
programme. One can then watch its dynamics unfold over time, and
measure the proportion of time it stays in any one equilibrium. These
proportions are represented by the relative sizes of the *basins of
attraction* for different possible equilibria. Equilibria are
attractor points in a dynamic space; a basin of attraction for each
such point is then the set of points in the space from which the
population will converge to the equilibrium in question.

In introducing correlation into his model, Skyrms first sets the degree of correlation at a very small .1. This causes the basin of attraction for equilibrium (i) to shrink by half. When the degree of correlation is set to .2, the polymorphic basin reduces to the point at which the population starts in the polymorphism. Thus very small increases in correlation produce large proportionate increases in the stability of the equilibrium where everyone plays Fairman. A small amount of correlation is a reasonable assumption in most populations, given that neighbours tend to interact with one another and to mimic one another (either genetically or because of tendencies to deliberately copy each other), and because genetically similar animals are more likely to live in common environments. Thus if justice can arise at all it will tend to be dominant and stable.

Much of political philosophy consists in attempts to produce
deductive normative arguments intended to convince an unjust agent
that she has reasons to act justly. Skyrms's analysis suggests a
quite different approach. Fairman will do best of all in the dynamic
game if he takes active steps to preserve correlation. Therefore,
there is evolutionary pressure for both *moral approval of
justice* and *just institutions* to arise. Most people may
think that 50-50 splits are ‘fair’, and worth maintaining by
moral and institutional reward and sanction, *because* we are
the products of a dynamic game that promoted our tendency to think
this way.

The topic that has received most attention from evolutionary game
theorists is *altruism,* defined as any behaviour by an
organism that decreases its own expected fitness in a single
interaction but increases that of the other interactor. It is common
in nature. How can it arise, however, given Darwinian competition?

Skyrms studies this question using the dynamic Prisoner's Dilemma as his example. This is simply a series of PD games played in a population, some of whose members are defectors and some of whom are cooperators. Payoffs, as always in dynamic games, are measured in terms of expected numbers of copies of each strategy in future generations.

Let **U**(*A*) be the average fitness of strategy
*A* in the population. Let **U** be the average
fitness of the whole population. Then the proportion of strategy
*A* in the next generation is just the ratio
**U**(*A*)/**U**. So if *A*
has greater fitness than the population average *A*
increases. If *A* has lower fitness than the population average
then *A* decreases.

In the dynamic PD where interaction is random (i.e., there's no correlation), defectors do better than the population average as long as there are cooperators around. This follows from the fact that, as we saw in Section 2.4, defection is always the dominant strategy in a single game. 100% defection is therefore the ESS in the dynamic game without correlation, corresponding to the NE in the one-shot static PD.

However, introducing the possibility of correlation radically changes
the picture. We now need to compute the average fitness of a strategy
*given its probability of meeting each other possible
strategy*. In the dynamic PD, cooperators whose probability of
meeting other cooperators is high do better than defectors whose
probability of meeting other defectors is high. Correlation thus
favours cooperation.

In order to be able to say something more precise about this
relationship between correlation and cooperation (and in order to be
able to relate evolutionary game theory to issues in decision theory,
a matter falling outside the scope of this article), Skyrms introduces
a new technical concept. He calls a strategy *adaptively
ratifiable* if there is a region around its fixation point in the
dynamic space such that from anywhere within that region it will go to
fixation. In the dynamic PD, both defection and cooperation are
adaptively ratifiable. The relative sizes of basins of attraction are
highly sensitive to the particular mechanisms by which correlation is
achieved. To illustrate this point, Skyrms builds several
examples.

One of Skyrms's models introduces correlation by means of a
*filter* on pairing for interaction. Suppose that in round 1 of
a dynamic PD individuals inspect each other and interact, or not,
depending on what they find. In the second and subsequent rounds, all
individuals who didn't pair in round 1 are randomly paired. In
this game, the basin of attraction for defection is large
*unless* there is a high proportion of cooperators in round
one. In this case, defectors fail to pair in round 1, then get paired
mostly with each other in round 2 and drive each other to
extinction. A model which is more interesting, because its mechanism
is less artificial, does not allow individuals to choose their
partners, but requires them to interact with those closest to
them. Because of genetic relatedness (or cultural learning by copying)
individuals are more likely to resemble their neighbours than not. If
this (finite) population is arrayed along one dimension (i.e., along a
line), and both cooperators and defectors are introduced into
positions along it at random, then we get the following
dynamics. Isolated cooperators have lower expected fitness than the
surrounding defectors and are driven locally to extinction. Members of
groups of two cooperators have a 50% probability of interacting with
each other, and a 50% probability of each interacting with a
defector. As a result, their average expected fitness remains smaller
than that of their neighbouring defectors, and they too face probable
extinction. Groups of three cooperators form an unstable point from
which both extinction and expansion are equally likely. However, in
groups of four or more cooperators at least one encounter of a
cooperator with a cooperator sufficient to at least replace the
original group is guaranteed. Under this circumstance, the
cooperators as a group do better than the surrounding defectors and
increase at their expense. Eventually cooperators go *almost*
to fixation -- but nor quite. Single defectors on the periphery of the
population prey on the cooperators at the ends and survive as little
‘criminal communities’. We thus see that altruism can not
only be maintained by the dynamics of evolutionary games, but, with
correlation, can even spread and colonize originally non-altruistic
populations.

Darwinian dynamics thus offers qualified good news for
cooperation. Notice, however, that this holds only so long as
individuals are stuck with their natural or cultural programming and
can't re-evaluate their utilities for themselves. If our agents
get too smart and flexible, they may notice that they're in PDs
and would each be best off defecting. In that case, they'll
eventually drive themselves to extinction - unless they develop
stable, and effective, moral norms that work to reinforce
cooperation. But, of course, these are just what we would expect to
evolve in populations of animals whose average fitness levels are
closely linked to their capacities for successful social
cooperation. Even given this, these populations will go extinct unless
they care about future generations for some reason. But there's
no rational reason as to why agents *should* care about future
generations if each new generation wholly replaces the preceding one
at each change of cohorts. For this reason, economists use
‘overlapping generations’ models when modeling distribution
games. Individuals in generation 1 who will last until generation 5
save resources for the generation 3 individuals with whom they'll
want to cooperate; and by generation 3 the new individuals care about
generation 6; and so on.

An enormous range of further applications of both static and dynamic game theory have been developed, but we have perhaps now provided enough to convince the reader of the tremendous utility of this analytical tool. The reader whose appetite for more has been thoroughly aroused should find that she now has sufficient grasp of fundamentals to be able to work through the large literature, of which some highlights are listed below.

Game theory has countless applications, of which this article has
been able to suggest only a few. Readers in search of more, but not
wishing to immerse themselves in mathematics, can find a number of
good sources.
Dixit and Nalebuff (1991) is especially strong on
political and social examples.
McMillan (1991) emphasizes business
applications. The great historical breakthrough is
von Neumann and Morgenstern (1947), which those
with scholarly interest in game theory should read with classic
papers of
John Nash (1950a, 1950b, 1951). For a
contemporary mathematical treatment that is unusually philosophically
sophisticated,
Binmore (1992) (**) is in a class by itself. The second
half of
Kreps (1990) (**) is the best available
starting point for a tour of the philosophical worries surrounding equilibrium
selection for non-behaviorists.
Koons (1992)
takes these issues further.
Fudenberg and Tirole (1991) is the most thorough
and complete mathematical text available.
Gintis (2000) (**) has provided a
new text crammed with terrific problem exercises, which is also unique
in that it treats evolutionary game theory as providing the
foundational basis for game theory *in general. *This likely
represents the wave of the future. Recent developments in fundamental
theory are well represented in
Binmore, Kirman and Tani (1993).

The philosophical foundations of the basic game-theoretic concepts as economists understand them are presented in LaCasse and Ross (1994). Ross and LaCasse (1995) outline the relationships between games and the axiomatic assumptions of microeconomics and macroeconomics. Philosophical puzzles at this foundational level are critically discussed in Bicchieri (1993) (**). Lewis (1969) (**) puts game-theoretic equilibrium concepts to wider application in philosophy, a program that is carried a good deal further in Skyrms (1996) (**). (See also Nozick [1998].) Gauthier (1986) launches a literature not surveyed in this article, in which the possibility of game-theoretic foundations for contractarian ethics is investigated. This work is critically surveyed in Vallentyne (1991), and extended into a dynamic setting in Danielson (1992). Binmore (1994, 1998) (**), however, effectively demolishes this project. Philosophers will also find Hollis (1998) to be of interest.

Hardin (1995) is one of many examples of the application of game theory to problems in applied political theory. Baird, Gertner and Picker (1994) review uses of game theory in legal theory and jurisprudence. Mueller (1997) surveys applications in political economy. Ghemawat (1997) does the same in business strategy. Poundstone (1992) provides a lively history of the Prisoner's Dilemma and its use by Cold War strategists. Durlauf and Young (2001) is a good collection on applications to social structures and social change.

Evolutionary game theory owes its explicit genesis to Maynard Smith (1982) (**). For a text that integrates game theory directly with biology, see Sigmund (1993). The most exciting applications of evolutionary game theory to a range of philosophical issues, on which this article has drawn heavily, is Skyrms (1996) (**). These issues and others are critically discussed from various angles in Danielson (1998). Mathematical foundations for dynamic games are presented in Weibull (1995), and pursued further in Samuelson (1997) and Fudenberg and Levine (1998). As noted above, Gintis (2000) (**) now provides an introductory textbook that takes evolutionary modeling to be foundational to all of game theory. Many philosophers will also be interested in Binmore (1994, 1998) (**), which shows that application of game-theoretic analysis can underwrite a loosely Rawlsian theory of justice that does not require recourse to Kantian presuppositions about what rational agents would desire behind a veil of ignorance concerning their identities and social roles. (In addition, Binmore offers excusions into a vast range of other issues both central and peripheral to both the foundations and the frontiers of game theory; these books are a tour de force.) And almost everyone will be interested in Frank (1988) (**), where evolutionary game theory is used to illuminate basic features of human nature and emotion.

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*Game Theory and the Law*. Cambridge, MA: Harvard University Press. - Bicchieri, C. (1993).
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Pure Theory of Consumers' Behaviour.’
*Econimica*5:61-71. - Selten, R. (1975). ‘Re-examination of
the Perfectness Concept for Equilibrium Points in Extensive
Games.’
*International Journal of Game Theory*4:22-55. - Sigmund, K. (1993).
*Games of Life*. Oxford: Oxford University Press. - Skyrms, B.(1996).
*Evolution of the Social Contract*. Cambridge: Cambridge University Press. - Vallentyne, P. (ed.). (1991).
*Contractarianism and Rational Choice*. Cambridge: Cambridge University Press. - von Neumann, J., and Morgenstern, O., (1947).
*The Theory of Games and Economic Behavior*. Princeton: Princeton University Press, 2nd edition. - Weibull, J. (1995).
*Evolutionary Game Theory*. Cambridge, MA: MIT Press.

- History of Game Theory
- Principia Cybernetica entry: Game Theory
- University of Rochester Economics Department: Game Theory
- TU Wroclaw IMath -- Game Theory
- What is Game Theory?
- Al Roth's Game Theory and Experimental Economics Page

University of Cape Town

*First published: January 25, 1997*

*Content last modified: September 11, 2001*