Game Theory

This article sketches the basic concepts of the theory of games in
order to discuss some of their philosophical implications and
problems.

Consider the following situation: when two hunters set out to hunt a
stag and lose track of each other in the process, each hunter has to
make a decision. Either she continues according to plan, hoping that
her partner does likewise (because she cannot bag a deer on her own),
and together they catch the deer; or she goes for a hare instead,
securing a prey that does not require her partner's cooperation, and
thus abandoning the common plan. Each hunter prefers a deer shared
between them to a hare for herself alone. But if she decides to hunt
for deer, she faces the possibility that her partner abandons her,
leaving her without deer or hare. So, what should she do? And, what
will she do?

Situations like this, in which the outcome of an agent's action
depends on the actions of all the other agents involved, are called
interactive. Two people playing chess is the archetypical example of
an interactive situation, but so are elections, wage bargaining,
market transactions, the arms race, international negotiations, and
many more. Game theory studies these interactive situations. Its
fundamental idea is that an agent in an interactive decision should
and does take into account the deliberations of her opponents, who, in
turn, take into account her deliberations. A rational agent in an
interactive situation should therefore not ask: "What can I do, given
what is likely to happen?" but rather: "What can I do in response to
what they do, given that they have a belief about what I will do?"
Based on this perspective, game theory recommends rational choices for
these situations, and predicts agents' behavior in them.

This article presents the basic tenets of game theory in a non-formal
way. It then discusses two broad philosophical issues arising from the
theory. First, whether the rationality concept employed by the theory
is justifiable – whether it is intuitively rational to choose as the
theory prescribes. Second, whether the theory can in principle be a
good predictive theory of human behavior – whether it has empirical
content, whether it is testable and whether there are good reasons to
believe that it is true or false.

1. Sketch of the Theory

Game theory belongs to a family of theories often subsumed under the
umbrella term Rational Choice Theory. All these theories (in
particular, decision theory, game theory and social choice theory)
discuss conditions under which agents' actions, or at least their
decision to act, can be said to be rational. Depending on how these
conditions are interpreted, Rational Choice theory may have a positive
or a normative function: it may contribute to the prediction and
explanation of agent behavior, or it may contribute to advising agents
what they should do. Many of the purported functions of Rational
Choice theory are controversial; as a part of it, game theory is
affected by these controversies, in particular its usefulness for the
social sciences. I will address some of these general issues in
Section 3. However, game theory faces its own philosophical problems,
and these will be the focus of this article.

Decision theory, as well as game theory, assesses the rationality of
decisions in the light of preferences over outcomes and beliefs about
the likelihood of these outcomes to appear. The basic difference
between the two lies in the way they view the likelihood of outcomes.
Decision theory treats all outcomes as exogenous events, 'moves of
nature'. Game theory, in contrast, focuses on those situations in
which outcomes are determined by interactions of deliberating agents.
It proposes that agents take outcomes as determined by other agents'
reasoning, and that agent therefore assess the likelihood of an
outcome by trying to figure out how the other agents they interact
with will reason. The likelihoods of outcomes therefore becomes
"endogenous" in the sense that players take their opponents' payoffs
and rationality into account when figuring out the consequences of
their strategies.

We are familiar with such reasoning from card and board games. When
playing poker or chess, one must take one's opponent's reasoning into
account in order to be successful. The player who foresees her
opponent's optimal reaction to her own move will be much more
successful that the player who simply assumes that her opponent will
make a certain move with a certain probability. Theoretical reflection
about such parlor games are at the basis of game theory – for example,
James Waldegrave's discussion of the French card game Le Her in 1713,
or von Neumann's treatment 'Zur Theorie der Gesellschaftsspiele'
('Towards a Theory of Parlor Games') from 1928 – but today game theory
has little to do with these games, and instead discusses a wide
variety of social interactions. (Game theory is also applied to
problems in biology and even in logic – these applications will not be
discussed in this article).

The formal theory defines a game as consisting of two or more players,
a set of pure strategies for each player and the players' payoff
functions. A player's pure strategy specifies her choice for each time
she has to choose in the game (which may be more than once). Players
have to have at least two strategies to choose between, otherwise the
game would be trivial. All players of a game together determine a
consequence. Each chooses a specific strategy, and their combination
(called strategy profiles) yields a specific consequence. The
consequence of a strategy profile can be a material prize – for
example money – but it can also be any other relevant event, like
being the winner, or feeling guilt. Game theory is really only
interested in the players' evaluations of this consequence, which are
specified in each players' so-called payoff or utility function.

The part of the theory that deals with situations in which players'
choice of strategies cannot be enforced is called the theory of
non-cooperative games. Cooperative game theory, in contrast, allows
for pre-play agreements to be made binding (e.g. through legally
enforceable contracts). This article will not discuss cooperative game
theory. More specifically, it will focus – for reasons of simplicity –
on non-cooperative games with two players, finite strategy sets and
precisely known payoff functions.

Game theory uses two means to represent games formally: strategic form
and extensive form. Commonly (though not necessarily!), these two
methods of representation are associated with two different kinds of
games. Extensive form games represent dynamic games, where players
choose their actions in a determined temporal order. Strategic form
games represent static games, where players choose their actions
simultaneously.
a. Static Games

Static two-person games can be represented by m-by-n matrices, with m
rows and n columns corresponding to the players' strategies, and the
entries in the squares representing the payoffs for each player for
the pair of strategies (row, column) determining the square in
question. As an example, figure 1 is a possible representation of the
stag-hunt scenario described in the introduction.

(Figure 1)
Figure 1: The stag hunt

The 2-by-2 matrix of figure 1 determines two players, Row and Col, who
each have two pure strategies: R1 and C1 (go deer hunting) and R2 and
C2 (go hare hunting). Combining the players' respective strategies
yields four different pure strategy profiles, each associated with a
consequence relevant for both players: (R1,C1) leads to them catching
a deer, (R2,C1) leaves Row with a hare and Col with nothing, (R2,C2)
gets each a hare and (R1,C2) leaves Row empty-handed and Col with a
hare. Both players evaluate these consequences of each profile. Put
informally, players rank consequences as 'better than' or 'equally
good as'. In the stag-hunt scenario, players have the following
ranking:

Figure 2: The hunters' respective rankings of the strategy profiles

This ranking can be quite simply represented by a numerical function
u, according to the following two principles:

1. For all consequences X, Y: X is better than Y if and only if u(X) > u(Y
2. For all consequences X, Y: X is equally good as Y if and only if
u(X) = u(Y)

A function that meets these two principles (and some further
requirements that are not relevant here) is called an ordinal utility
function. Utility functions are used to represent players' evaluations
of consequences in games (for more on preferences and utility
functions, see Grüne-Yanoff and Hansson 2006). Convention has it that
the first number represents Row's evaluation, while the second number
represents Col's evaluation. It is now easy to see that the numbers of
the game in figure 1 represent the ranking of figure 2.

Note, however, that the matrix of figure 1 is not the only way to
represent the stag-hunt game. Because the utilities only represent
rankings, there are many ways how one can represent the ranking of
figure 2. For example, the games in figure 3 are identical to the game
in figure 1.

Figure 3: Three versions of the stag hunt

In (a) of figure 3, all numbers are negative, but they retain the same
ranking of consequences. And similarly in (b), only that here the
proportional relations between the numbers (which don't matter) are
different. This should also make clear that utility numbers only
express a ranking for one and the same player, and do not allow to
compare different players' evaluations. In (c), although the numbers
are very different for the two players, they retain the same ranking
as in figure 1. Comparing, say, Row's evaluation of (R1,C1) with Col's
evaluation of (R1,C1) simply does not have any meaning.

Note that in the stag-hunt game, agents do not gain if others lose.
Everybody is better off hunting deer, and losses arise from lack of
coordination. Games with this property are therefore called
coordination games. They stand in stark contrast to games in which one
player's gain is the other player's loss. Most social games are of
this sort: in chess, for example, the idea of coordination is wholly
misplaced. Such games are called zero-sum games. They were the first
games to be treated theoretically, and the pioneering work of game
theory, von Neumann and Morgenstern's (1944) The Theory of Games and
Economic Behavior concentrates solely on them. Today, many of the
games discussed are of a third kind: they combine coordination aspects
with conflicting aspects, so that players may at times gain from
coordinating, but at other times from competing with the other
players. A famous example of such a game is the Prisoners' Dilemma, to
be discussed shortly.

Players can create further strategies by randomizing over pure
strategies. They can choose a randomization device (like a dice) and
determine for each chance result which of their pure strategies they
will play. The resultant probability distribution over pure strategies
is called a mixed strategy σ. For example, Row could create a new
strategy that goes as follows: toss a (fair) coin. Play R1 if heads,
and R2 if tails. Because a fair coin lands heads 50% of the time, such
a mixed strategy is denoted σR = (0.5,0.5). As there are no limits to
the number of possible randomization devices, each player can create
an infinite number of mixed strategies for herself. The players'
evaluation of mixed strategies profiles is represented by the expected
values of the corresponding pure-strategy payoffs. Such an expected
value is computed as the weighted average of the pure-strategy
payoffs, and the weights are the probabilities with which each
strategy is played. For example, if Row in figure 1 plays her mixed
strategy σR = (0.5,0.5), and Col plays a strategy σC = (0.8,0.2), then
Row's expected utility will be computed by:

uR(σR,σC) = 0.5(0.8×2 + 0.2×0) + 0.5(0.8×1 + 0.2×1) = 1.3

With the same mixed strategies, Col's expected utility, uC(σR,σC) = 1.
For the payoffs of mixed strategy to be computable, the utility
function has to carry cardinal information. Now it is also important
how much a player prefers a consequence X to a consequence Y, in
comparison to another pair of consequences X and Z. Because mixed
strategies are a very important concept in game theory, it is
generally assumed that the utility functions characterizing the
payoffs are cardinal. However, it is important to note that cardinal
utilities also do not allow making interpersonal comparisons. In fact,
such interpersonal comparisons play no role in standard game theory at
all.

Solution Concepts

Representing interactive situations in these highly abstract games,
the objective of game theory is to determine the outcome or possible
outcomes of each game, given certain assumptions about the players. To
do this is to solve a game. Various solution concepts have been
proposed. The conceptually most straightforward solution concept is
the elimination of dominated strategies. Take the game of figure 4
(which, take note, differs from the stag-hunt game in its payoffs). In
this game, no matter what Col chooses, playing R2 gives Row a higher
payoff. If Col plays C1, Row is better off playing R2, because she can
obtain 3 utils instead of two. If Col plays C2, Row is also better off
playing R2, because she can obtain 1 utils instead of none. Similarly
for Col: no matter what Row chooses, playing C2 gives her a higher
payoff. This is what is meant by saying that R1 and C1 are strictly
dominated strategies.

Figure 4: The Prisoners' Dilemma

More generally, a player A's pure strategy is strictly dominated if
there exists another (pure or mixed) strategy for A that has a higher
payoff for each of A's opponent's strategies. To solve a game by
eliminating all dominated strategies is based on the assumption that
players do and should choose those strategies that are best for them,
in this very straightforward sense. In cases like in figure 4, where
each player has only one non-dominated strategy, the elimination of
dominated strategies is a straightforward and plausible solution
concept. However, there are many games, which do not have any
dominated strategies, as for example the stag-hunt game or the
zero-sum game of figure 5.

Recall that in a zero sum game, one player's payoff is exactly the
inverse of that of the other player. For example, figure 5 shows Row's
payoffs, while Col's payoffs are the negative of Row's payoffs.

Figure 5: A zero-sum game

Von Neumann and Morgenstern argued for the Minimax Rule as the
solution concept for zero-sum games. In these games, they suggest,
each player makes the following consideration: 'my adversary tries to
get out of the play as much as possible. Her gain is my loss. So I
better look for how much I minimally get out of each option and try to
make this amount as large as possible. If this is reasonable, then my
adversary will do the same. Since my maximizing my minimum is best
against her maximizing her minimum, I should stick to my choice'. The
minimax solution therefore recommends that Row choose the strategy
with the highest minimum, while Col choose a strategy with the lowest
maximum. Thus, in figure 5, Row chooses R2, as it has the highest
minimal payoff for her, and Col chooses C2, as it has the lowest
maximal payoff for Row (and hence the highest minimal payoff for her).

Unfortunately, there are many non-zero-sum games without dominated
strategies, for example the game of figure 6.

Figure 6: A game without dominated strategies

For these kinds of games, the Nash equilibrium solution concept offers
greater versatility than dominance or maximin (as it turns out, all
maximin solutions are also Nash equilibria). In contrast to dominated
strategy elimination, the Nash equilibrium applies to strategy
profiles, not to individual strategies. Roughly, a strategy profile is
in Nash equilibrium if none of the players can do better by
unilaterally changing her strategy. Take the example of matrix 6.
Consider the strategy profile (R1,C1). If Row knew that Col would play
C1, then she would play R2 because that's the best she can do against
C1. On the other hand, if Col knew that Row would play R1, he would
play C2 because that's the best he can do against R1. So (R1, C1) is
not in equilibrium, because at least one player (in this case both) is
better off by unilaterally deviating from it. Similarly for (R1, C3),
(R2, C1), (R2,C2) and (R2, C3): in all these profiles, one of the
players can improve her or his lot by deviating from the profile. Only
(R1, C2) is a pure strategy Nash equilibrium – neither player is
better off by unilaterally deviating from it.

There are games without a pure strategy Nash equilibrium, as matrix 7
shows. The reader can easily verify that each player has an incentive
to deviate, whichever pure strategy the other chooses.

Figure 7: Matching pennies

However, there is an equilibrium involving mixed strategies.
Randomizing between the two strategies, assigning equal probability to
each, yields a payoff of 0.5(0.5×1+0.5x-1)+0.5(0.5×1+0.5x-1) = 0 for
both players. As mutually best responses, these mixed strategies
constitute a Nash equilibrium. As one of the fundamental results of
game theory, it has been shown that every finite static game has a
mixed-strategy equilibrium (Nash 1950). Many games have several Nash
equilibria. Take for example figure 1. There, neither player has an
incentive to deviate from (R1, C1), nor to deviate from (R2, C2). Thus
both strategy profiles are pure-strategy Nash equilibria. With two or
more possible outcomes, the equilibrium concept loses much of its
appeal. It no longer gives an obvious answer to the normative,
explanatory or predictive questions game theory sets out to answer.
The assumption that one specific Nash equilibrium is played relies on
there being some mechanism or process that leads all the players to
expect the same equilibrium.

Schelling's (1960) theory of focal points suggests that in some
"real-life" situations players may be able to coordinate on a
particular equilibrium by using information that is abstracted away by
the strategic form. Examples of information that has such focal power
may be the names of strategies or past common experiences of the
players. Little systematic work exists on the "focalness" of various
strategies, as they depend on the players' cultural and personal
backgrounds. Mainstream game theory has never incorporated these
concepts into the formal structure of the theory (for exceptions, see
Bacharach 1993, Sugden 1995).

A focal point that might evade such context-dependence is
Pareto-dominance, if pre-play communication is allowed. An equilibrium
is Pareto-dominant over another if it makes everybody at least as well
off and makes at least one person better off. This is the case in the
game of figure 1: (R1, C1) makes both players better off than (R2,
C2). The intuition for this focal point is that, even though the
players cannot commit themselves to play the way they claim they will,
the pre-play communication lets the players reassure one another about
the low risk of playing the strategy of the Pareto-dominant
equilibrium. Although pre-play communication may make the
Pareto-dominant equilibrium more likely in the stag-hunt game, it is
not clear that it does so in general. Many other selection mechanisms
have been proposed that use clues derivable from the game model alone.
These mechanisms are however too complex to be discussed here.

As it will become clearer in Section 2b, the assumptions underlying
the application of the Nash concept are somewhat problematic. The most
important alternative solution concept is that of rationalizability,
which is based on weaker assumptions. Instead of relying on the
equilibrium concept, rationalizability selects strategies that are
"best" from the players' subjective point of view. Players assign a
subjective probability to each of the possible strategies of their
opponents, instead of postulating their opponents' choices and then
finding a best response to it, as in the Nash procedure. Further,
knowing their opponent's payoffs, and knowing they are rational,
players expect others to use only strategies that are best responses
to some belief they might have about themselves. And those beliefs in
turn are informed by the same argument, leading to an infinite regress
of the form: "I'm playing strategy σ1 because I think player 2 is
using σ2, which is a reasonable belief because I would play it if I
were player 2 and I thought player 1 was using σ1', which is a
reasonable thing for player 2 to expect because σ1' is a best response
to σ2'…". A strategy is rationalizable for a player if it survives
infinitely repeated selections as a best response to some rational
belief she might have about the strategies of her opponent. A strategy
profile is rationalizable if the strategies contained in it are
rationalizable for each player. It has been shown that every Nash
equilibrium is rationalizable. Further, the set of rationalizable
strategies is nonempty and contains at least one pure strategy for
each player (Bernheim 1984, Pearce 1984). The problem with
rationalizability is thus not its applicability; rather, there are too
many rationalizable strategies, so that the application of
rationalizability often does not provide a clear answer to the
advisory and predictive questions posed to game theory.
b. Dynamic Games

In static games discussed above, players choose their actions
simultaneously. Many interactive situations, however, are dynamic: a
player chooses before others do, knowing that the others choices will
be influenced by his observable choice. Players who choose later will
make their choices dependent on their knowledge of how others have
chosen. Chess is a typical example of such a dynamic interactive
situation (although one, as will be seen, that is far too complex to
explicitly model it). Game theory commonly represents these dynamic
situations in extensive form. This representation makes explicit the
order in which players move, and what each player knows when making
each of his decisions.

The extensive form consists of six elements. First, the set of players
is determined. Each player is indexed with a number, starting with 1.
Second, it is determined who moves when. The order of moves is
captured in a game tree, as illustrated in figure 8. A tree is a
finite collection of ordered nodes x (This index is for instructive
purposes only. Commonly, nodes are only indexed with the number of the
player choosing at this node). Each tree starts with one (and only
one!) initial node, and grows only 'down' and never 'up' from there.
The nodes that are not predecessors to any others are called terminal
nodes, denoted z1-z4 in figure 8. All nodes but the terminal ones are
labeled with the number of that player who chooses at this node. Each
z describes a complete and unique path through the tree from the
initial to one final node.

Figure 8: A game tree

Third, the payoffs for all players are assigned (as an list of utility
numbers: first player's utility in the first place, etc.) to the
terminal nodes. An illustration is given in figure 8. The utility
functions of each player have to satisfy the same requirements as
those in static games. Fourth, for each player at each node, a finite
set of actions is specified, labeled with capital letters in figure 8.
Each action leads to one (and only one) non-initial node. Fifth, it is
determined what each player knows about her position in the game when
she makes her choice. Her knowledge is represented by a partition of
the nodes of the tree, called the information set. If the information
set contains, say, nodes x and x', this means that the player who is
choosing an action at x is uncertain whether she is at x or at x'. To
avoid inconsistencies, information sets can contain only nodes at
which the same player chooses, and only nodes where the same player
has the same actions to choose from. In figure 9, an information set
containing more than one node is represented as a dotted line between
those nodes (Information sets that contain only one node are usually
not represented). Games that contain only singleton information sets
are called games of perfect information: all agents know where they
are at all nodes of the game. Further, it is commonly assumed that
agents have perfect recall: they neither forget what they once knew,
nor what they have chosen.

Figure 9: A game of imperfect information

Sixth and last, when a game involves chance moves, the probabilities
are displayed in brackets, as in the game of figure 10. There, a
chance move (by an imagined player called N like "Nature') determines
the payoffs for both players. Player 1 then plays L or R. Player 2
observes player 1's action, but does not know whether he is at x or x'
(if player 1 chose L) nor whether he is at y or y' (if player 1 chose
R). In other words, player 2 faces a player whose payoffs he does not
know. All he knows is that player 1 can be either of two types,
distinguished by the respective utility function.

Figure 10: A game of incomplete information

Games such as that in figure 10 will not be further discussed in this
article. They are games of incomplete information, where players do
not know their and other players' payoffs, but only have probability
distributions over them.

Extensive-form games can be represented as strategic-form games. While
in extensive-form interpretation, the players "wait" until their
respective information set is reached to make a decision, in the
strategic-form interpretation they make a complete contingency plan in
advance. Figure 11 illustrates this transformation. Player 1, who
becomes "Row', has only two strategies to choose from. Player 2, who
becomes "Col', has to decide in advance for both the case where player
1 chooses U and where she chooses D. His strategies thus contain two
moves each: for example, (L,R) means that he plays L after U and R
after D.

Figure 11: Extensive form reduced to strategic form

A similar terminology as in strategic-form games applies. A pure
strategy for a player determines her choice at each of her information
sets (That is, a strategy specifies all the past and future moves of
an agent. Seen from this perspective, one may legitimately doubt
whether extensive games really capture the dynamics of interaction to
any interesting extent). A behavior strategy specifies a probability
distribution over actions at each information set. For games of
perfect recall, behavior strategies are equivalent to mixed strategies
known from strategic-form games (Kuhn 1953).

Solution Concepts

Given that the strategic form can be used to represent arbitrarily
complex extensive-form games, the Nash equilibrium can also be applied
as a solution concept to extensive form games. However, the extensive
form provides more information than the strategic form, and on the
basis of that extra information, it is sometimes possible to separate
the "reasonable" from the "unreasonable" Nash equilibria. Take the
example from figure 11. The game has three Nash equilibria, which can
be identified in the game matrix: (U, (L,L)); (D, (L,R)) and (D,
(R,R)). But the first and the third equilibria are suspect, when one
looks at the extensive form of the game. After all, if player 2's
right information set was reached, the he should play R (given that R
gives him 3 utils while L gives him only –1 utils). But if player 2's
left information set was reached, then he should play L (given that L
gives him 2 utils, while R gives him only 0 utils). Moreover, player 1
should expect player 2 to choose this way, and hence she should choose
D (given that her choosing D and player 2 choosing R gives her 2
utils, while her choosing U and player 2 choosing L gives her only 1
util). The equilibria (U, (L,L)) and (D, (R,R)) are not "credible',
because they rely on an "empty threat" by player 2. The threat is
empty because player 2 would never wish to carry out either of them.
The Nash equilibrium concept neglects this sort of information,
because it is insensitive to what happens off the path of play.

To identify "reasonable" Nash equilibria, game theorists have employed
equilibrium refinements. The simplest of these is the
backward-induction solution that applies to finite games of perfect
information. Its rational was already used in the preceding paragraph.
"Zermelo's algorithm" (Zermelo 1913) specifies its procedure more
exactly: Since the game is finite, it has a set of penultimate nodes –
i.e. nodes whose immediate successors are terminal nodes. Specify that
the player, who can move at each such node, chooses whichever action
that leads to the successive terminal node with the highest payoff for
him (in case of a tie, make an arbitrary selection). So in the game of
figure 11, player 2's choices R if player 1 chooses U and L if player
1 chooses D can be eliminated:

Figure 11a: First step of backward induction

Now specify that each player at those nodes, whose immediate
successors are penultimate nodes, choose the action that maximizes her
payoff over the feasible successors, given that the players at the
penultimate nodes play as we have just specified. So now player 1's
choice U can be eliminated:

Figure 11b: Second step of backward induction

Then roll back through the tree, specifying actions at each node (not
necessary for the given example anymore, but one gets the point). Once
done, one will have specified a strategy for each player, and it is
easy to check that these strategies form a Nash equilibrium. Thus,
each finite game of perfect information has a pure-strategy Nash
equilibrium.

Backward induction fails in games with imperfect information. In a
game like in figure 12, there is no way to specify an optimal choice
for player 2 in his second information set, without first specifying
player 2's belief about the previous choice of player 1. Zermelo's
algorithm is inapplicable because it presumes that such an optimal
choice exists at every information set given a specification of play
at its successors.

Figure 12: A game not solvable by backward induction

However, if one accepts the argument for backward induction, the
following is also convincing. The game beginning at player 1's second
information set is a simultaneous-move game identical to the one
presented in figure 7. The only Nash equilibrium of this game is a
mixed strategy with a payoff of 0 for both players (as argued in
Section 1a). Using the equilibrium payoff as player 2's payoff to
choose R, it is obvious that player 2 maximizes his payoff by choosing
L, and that player 1 maximizes her payoff by choosing R. More
generally, an extensive form game can be analyzed into proper
subgames, each of which satisfies the definition of extensive-form
games in their own right. Games of imperfect information can thus be
solved by replacing a proper subgame with one of its Nash equilibrium
payoffs (if necessary repeatedly), and performing backward induction
on the reduced tree. This equilibrium refinement technique is called
subgame perfection.

Repeated Games

Repeated games are a special kind of dynamic game. They proceed in
temporal stages, and players can observe all players' play of these
previous stages. At the end of each stage, however, the same structure
repeats itself. Let's recall the static game of figure 4. As discussed
in the previous section, it can be equivalently represented as an
extensive game where player 2 does not know at which node he is.

Figure 13: Equivalent Static and Extensive Games

All that changed from the original game from figure 4 is the
nomination of the players (Row becomes 1 and Col becomes 2) and the
strategies (C and D). A repetition of this game is shown in figure 14.
Instead of the payoff matrices, at each of terminal nodes of the
original static game, the same game starts again. The payoffs
accumulate over this repetition: while in the static game, the
strategy profile (D,D) yields (1,1), the strategy profile
((D,D),(D,D)) in figure 14 yields (2,2). These payoffs are written at
the terminal nodes of the last repetition game.

Figure 14: A repeated Prisoners' Dilemma

If a repeated game ends after a finite number of stages, it is solved
by subgame perfection. Starting with the final subgames, the Nash
equilibrium in each is (D,D). Because the payoffs are accumulative,
the preference structure within each subgame does not change (recall
from Section 1a that only ordinal information is needed here). Thus
for each subgame at any stage of the repetition, the Nash equilibrium
will be (D,D). Therefore, for any number of finite repetitions of the
game from figure 4, subgame perfection advises both players to always
play D.

All this changes dramatically, if the game is repeated indefinitely.
The first thing that needs reinterpretation are the payoffs. Any
positive payoff, whether large or small, would be infinitely large if
they just summed up over indefinitely many rounds. This would
obliterate any interesting concept of an indefinitely repeated game.
Fortunately, there is an intuitive solution: people tend to value
present benefits higher than those in the distant future. In other
words, they discount the value of future consequences by the distance
in time by which these consequences occur. Hence in indefinitely
repeated games, players' payoffs are specified as the discounted sum
of what each player wins at each stage.

The second thing that changes with indefinitely repeated game is that
the solution concept of subgame perfection does not apply, because
there is no final subgame at which the solution concept could start.
Instead, the players may reason as follows. Player 1 may tell player 2
that she is well disposed towards him, relies on his honesty, and will
trust him (i.e. play C) until proven wrong. Once he proves to be
untrustworthy, she will distrust him (i.e. play D) forever. In the
infinitely repeated game, player 2 has no incentive to abuse her trust
if he believes her declaration. If he did, he would make a momentary
gain from playing D while his opponent plays C. This would be
followed, however, by him forever forgoing the extra benefit from
(C,C) in comparison to (D,D). Unless player 2 has a very high discount
rate, the values the present gain from cheating will be offset by the
future losses from non-cooperating. More generally, the folk theorem
shows that in infinitely repeated games with low enough discounting of
the future, any strategy that give each player more than the worst
payoff the others can force him to take is sustainable as an
equilibrium (Fudenberg and Maskin 1986). It is noteworthy that for the
folk theorem to apply, it is sufficient that players do not know when
a repeated game ends, and have a positive belief that it may go on
forever. It is more appropriate to speak of indefinitely repeated
games, than of infinitely repeated ones, as the former does not
conflict with the intuition that humans cannot interact infinitely
many times.
c. The Architecture of Game Theory

From a philosophy of science perspectives, game theory has an
interesting structure. Like many other theories, it employs highly
abstract models, and it seeks to explain, to predict and to advice on
phenomena of the real world by a theory that operates through these
abstract models. What is special about game theory, however, is that
this theory does not provide a general and unified mode of dealing
with all kinds of phenomena, but rather offers a 'toolbox', from which
the right tools must be selected.

As discussed in the two preceding sections, game theory consists of
game forms (the matrices and trees), and a set of propositions (the
'theory proper') that defines what a game form is and provides
solution concepts that solve these models. Game theorists often focus
on the development of the formal apparatus of the theory proper. Their
interest lies in proposing alternative equilibrium concepts or proving
existing results with fewer assumptions, not in representing and
solving particular interactive situations. "Game theory is for proving
theorems, not for playing games" (Reinhard Selten, quoted in Goeree
and Holt 2001, 1419).

Although they habitually employ labels like 'players', 'strategies' or
'payoffs', the game forms that the theory proper defines and helps
solving are really only abstract mathematical objects, without any
link to the real world. After all, only very few social situations
come with labels like 'strategies' or 'rules of the game' attached.
What is needed instead is an interpretation of a real-world situation
in terms of the formal elements provided by the theory proper, so that
it can be represented by a game form. In many cases, this is where all
the hard work lies: to construct a game form that captures the
relevant aspects of a real social phenomenon. To acquire an
interpretation, the game forms are complemented with an appropriate
story (Morgan 2005) or model narrative. As regularly exemplified in
textbooks, this narrative may be purely illustrative: it provides (in
non-formal terms) a plausible account of an interactive situation,
whose salient features can be represented by the formal tools of game
theory. A good example of such a narrative is given by Selten when
discussing the 'Chain Store Paradox':

A chain store, also called player A, has branches in 20 towns,
numbered from 1 to 20. In each of these towns there is a potential
competitor, a small businessman who might raise money at the local
bank in order to establish a second shop of the same kind. The
potential competitor at town k is called player k. [...]
Just now none of the 20 small businessmen has enough owned capital
to be able to get a sufficient credit from the local bank but as time
goes on, one after the other will have saved enough to increase his
owned capital to the required amount. This will happen first to player
1, then to player 2, etc. As soon as this time comes for player k, he
must decide whether he wants to establish a second shop in his town or
whether he wants to use his owned capital in a different way. If he
chooses the latter possibility, he stops being a potential competitor
of player A. If a second shop is established in town k, then player A
has to choose between two price policies for town k. His response may
be 'cooperative' or 'aggressive'. The cooperative response yields
higher profits in town k, both for player A and for player k, but the
profits of player A in town k are even higher if player k does not
establish a second shop. Player k's profits in case of an aggressive
response are such that it is better for him not to establish a second
shop if player A responds in this way. (Selten 1978, 127).

The narrative creates a specific, if fictional, context that is
congruent with the structure of dynamic form games. The players are
clearly identifiable, and the strategies open to them are specified.
The story also determines the material outcomes of each strategy
combination, and the players' evaluation of these outcomes. The story
complements a game form of the following sort:

Figure 15: One step in the Chain store's paradox

Game form and model narrative together constitute a model of a
possible real-world situation. The model narrative fulfills two
crucial functions in the model. If the game form is given first, it
interprets this abstract mathematical object as a possible situation;
if a real-world phenomenon is given first, it accounts for the
phenomenon in such a way that it can be represented by a game form,
which in turn can be solved by the theory proper.

The architecture of game theory is summarized in figure 16:

Figure 16: The architecture of game theory (Grüne-Yanoff and Schweinzer 2008)

The theory proper (on the left hand side of Figure 16) specifies the
concept of a game, it provides the mathematical elements that are
needed for the construction of a game form, and it offers solution
concepts for the thus constructed game forms. The game form (left half
of the central circle) is constructed from elements of the theory
proper. The model narrative (the right half of the central circle)
provides an account of a real or hypothetical economic situation. Its
account of the situation interprets the game form.

As discussed in the previous sections, however, a specified game form
can be solved by different solution concepts. Sometimes, as in the
case of minimax and Nash equilibrium for zero-sum games, the reasoning
behind the solution concepts is different, but the result is always
the same. In other cases, in particular when equilibrium refinements
are applied, applying different solution concepts to the same game
form yields different results. As will be seen in Section 2e, this is
also the case with the chain-store paradox. The reason for this
ambiguity is that the application of many solution concepts requires
information that is not contained in the game form. Instead, the
information needed is found in an appropriate account of the situation
– i.e. in the model narrative. Thus the model narrative takes on a
third crucial function in game theory: it supports the choice of the
appropriate solution concept for a specific game (Grüne-Yanoff and
Schweinzer 2008).

This observation about the architecture of game theory and the role of
informal model narratives in it has two very important implications.
First, it becomes clear that game theory does not offer a universal
notion of rationality, but rather offers a menu of tools to model
specific situations at varying degrees and kinds of rationality.
Ultimately, it is the modeler who judges on her own intuitions which
kind of rationality to attributed to the interacting agents in a given
situation. This opens up the discussion about the various intuitions
that lie behind the solution concepts, the possibility of mutually
inconsistent intuitions, and the question whether a meta-theory can be
constructed that unifies all these fragmentary intuitions. Some of
these issues will be discussed in Section 2.

The second implication of this observation concerns the status of game
theory as a positive theory. Given its multi-layer architecture, any
disagreement of prediction and observation can be attributed to a
mistake either in the theory, the game form or the model narrative.
This then raises the question how to test game theory, and whether
game theory is refutable in principle. These questions will be
discussed in Section 3.
2. Game Theory as a Theory of Rationality

Game theory has often been interpreted as a part of a general theory
of rational behavior. This interpretation was already in the minds of
game theories' founders, who wrote in their Theory of Games and
Economic Behavior:

We wish to find the mathematically complete principles which
define "rational behavior" for the participants in a social economy,
and to derive from them the general characteristics of that behavior
(von Neumann and Morgenstern 1944, 31).

To interpret game theory as a theory of rationality means to give it a
prescriptive task: it recommends what agents should do in specific
interactive situations, given their preferences. To evaluate the
success of this rational interpretation of game theory means to
investigate its justification, in particular the justification of the
solution concepts it proposes. That human agents ought to behave in
such and such a way of course does not mean that they will do so;
hence there is little sense in testing rationality claims empirically.
The rational interpretation of game theory therefore needs to be
distinguished from the interpretation of game theory as a predictive
and explanatory theory. The solution concepts are either justified by
identifying sufficient conditions for them, and showing that these
conditions are already accepted as justified; or they can be justified
directly by compelling intuitive arguments.
a. Sufficient Epistemic Conditions for Solution Concepts

One way to investigate game theoretic rationality is to reduce its
solution concepts to the more intuitively understood notion of
rationality under uncertainty in decision theory. By clearly stating
the decision theoretic conditions agents have to satisfy in order to
choose in accordance with game theoretic solution concepts, we obtain
a better understanding of what game theory requires, and are thus able
to assess criticism against it more clearly.

Recall that the various solution concepts presented in Section 1
advise how to choose one's action rationally when the outcome of one's
choice depends on the actions of the other players, who in turn base
their choices on the expectation of how one will choose. The solution
concepts thus not only require the players to choose according to
maximization considerations; they also require the agent to maximize
their expected utilities on the basis of certain beliefs. Most
prominently, these beliefs include their expectations about what the
other players expect of them, and their expectations what the other
players will choose on the basis of these expectations. These
conditions are often not made explicit when people discuss game
theory; however, without fulfilling them, players cannot be expected
to choose in accord with specific solution concepts. To make these
conditions on the agent's knowledge and beliefs explicit will thus
further our understanding what is involved in the solution concepts.
In addition, if these epistemic conditions turn out to be justifiable,
one would have achieved progress in justifying the solution concepts
themselves. This line of thought has in fact been so prominent that
the interpretation of game theory as a theory of rationality has often
been called the eductive or epistemic interpretation. In the
following, the various solution concepts discussed with respect to
their sufficient epistemic conditions, and the conditions are
investigated with regard to their acceptability.

For the solution of eliminating dominated strategies, nothing is
required beyond the rationality of the players and their knowledge of
their own strategies and payoffs. Each player can rule out her
dominated strategies on the basis of maximization considerations
alone, without knowing anything about the other player. To the extent
that maximization considerations are accepted, this solution concept
is therefore justified.

The case is more complex for iterated elimination of dominated
strategies (this solution concept was not explained before, so don't
be confused. It fits in most naturally here). In the game matrix of
figure 17, only Row has a dominated strategy, R1. Eliminating R1 will
not yield a unique solution. Iterated elimination allows players to
consecutively eliminate dominated strategies. However, it requires
stronger epistemic conditions.

Figure 17: A game allowing for iterated elimination of dominated strategies

If Col knows that Row will not play R1, she can eliminate C2 as a
dominated strategy, given that R1was eliminated. But to know that, Col
has to know:

1. Row's strategies and payoffs
2. that Row knows her strategies and payoffs
3. that Row is rational

Let's assume that Col knows i.-iii., and that he thus expects Row to
have spotted and eliminated R1 as a dominated strategy. Given that Row
knows that Col did this, Row can now eliminate R3. But for her to know
that Col eliminated C2, she has to know:

1. Row's (i.e. her own) strategies and payoffs
2. that she, Row, is rational
3. that Col knows i.-ii.
4. Col's strategies and payoffs
5. that Col knows her strategies and payoffs
6. that Col is rational

Lets look at the above epistemic conditions a bit more closely. i. is
trivial, as she has to know her own strategies and payoffs even for
simple elimination. For simple elimination, she also has to be
rational, but she does not have to know it – hence ii. If Row knows i.
and ii., she knows that she would eliminate R1. Similarly, if Col
knows i. and ii., he knows that Row would eliminate R1. If Row knows
that Col knows that she would eliminate R1, and if Row also knows
iv.-vi., then she knows that Col would eliminate C2. In a similar
fashion, if Col knows that Row knows i.-vi., she will know that Row
would eliminate R3. Knowing this, he would eliminate C3, leaving
(R2,C1) as the unique solution of the game.

Generally, iterated elimination of dominated strategy requires that
each player knows the structure of the game, the rationality of the
players and, most importantly, that she knows that the opponent knows
that she knows this. The depth of one player knowing that the other
knows, etc. must be at least as high as the number of iterated
elimination necessary to arrive at a unique solution. Beyond that, no
further "he knows that she knows that he knows…" is required.
Depending on how long the chain of iterated eliminations becomes, the
knowledge assumptions may become difficult to justify. In long chains,
even small uncertainties in the players' knowledge may thus put the
justification of this solution concept in doubt.

From the discussion so far, two epistemic notions can be
distinguished. If all players know a proposition p, one says that they
have mutual knowledge of p. As the discussion of iterated elimination
showed, mutual knowledge is too weak for some solution concepts. For
example, condition iii insists that Row not only know her own
strategies, but also knows that Col knows. In the limit, this chain of
one player knowing that the other knows that p, that she knows that he
knows that she knows that p, etc. is continued ad infinitum. In this
case, one says that players have common knowledge of the proposition
p. When discussing common knowledge, it is important to distinguish of
what the players have common knowledge. Standardly, common knowledge
is of the structure of the game and the rationality of the players. As
figure 18 indictates, this form of common knowledge is sufficient for
the players to adhere to solutions provide by rationalizability.

Figure 18: Epistemic requirements for solution concepts (adapted from
Brandenburger 1992)

As figure 18 further indicates, sufficient epistemic conditions for
pure-strategy Nash equilibria are even more problematic. Common
knowledge of the game structure or rationality is neither necessary
nor sufficient, not even in conjunction with epistemic rationality.
Instead, it is required that all players know what the others will
choose (in the pure-strategy case) or what the others will conjecture
all players will be choosing (in the mixed-strategy case). This is an
implausibly strong requirement. Players commonly do not know how their
opponents will play. If there is no argument how players can reliably
obtain this knowledge from less demanding information (like payoffs,
strategies, and common knowledge thereof) then the analysis of the
epistemic conditions would put into doubt whether players will reach
Nash equilibrium. Note, however, that these epistemic conditions are
sufficient, not necessary. Formally, nobody has been able to establish
alternative epistemic conditions that are sufficient. But by
discussing alternative reasoning processes, some authors have at least
provided arguments for the plausibility that player soften reach Nash
equilibrium. Some of these arguments will be discussed in the next
section. (For further discussion of epistemic conditions of solution
concepts, see Bicchieri 1993, chapter 2).
b. Nash Equilibrium in One-Shot Games

The Nash equilibrium concept is often seen as "the embodiment of the
idea that economic agents are rational; that they simultaneously act
to maximize their utility" (Aumann 1985, 43). Particularly in the
context of one-shot games, however, doubts remain about the
justifiability of this particular concept of rationality. It seems
reasonable to claim that once the players have arrived at an
equilibrium pair, neither has any reason for changing his strategy
choice unless the other player does too. But what reason is there to
expect that they will arrive at one? Why should Row choose a best
reply to the strategy chosen by Col, when Row does not know Col's
choice at the time she is choosing? In these questions, the notion of
equilibrium becomes somewhat dubious: when scientists say that a
physical system is in equilibrium, they mean that it is in a stable
state, where all causal forces internal to the system balance each
other out and so leave it "at rest" unless it is disturbed by some
external force. That understanding cannot be applied to the Nash
equilibrium, when the equilibrium state is to be reached by rational
computation alone. In a non-metaphorical sense, rational computation
simply does not involve causal forces that could balance each other
out. When approached from the rational interpretation of game theory,
the Nash equilibrium therefore requires a different understanding and
justification. In this section, two interpretations and justifications
of the Nash equilibrium are discussed.

Self-Enforcing Agreements

Often, the Nash equilibrium is interpreted as a self-enforcing
agreement. This interpretation is based on situations in which agents
can talk to each other, and form agreements as to how to play the
game, prior to the beginning of the game, but where no enforcement
mechanism providing independent incentives for compliance with
agreements exists. Agreements are self-enforcing if each player has
reasons to respect them in the absence of external enforcement.

It has been argued that self-enforcing agreement is neither necessary
nor sufficient for Nash equilibrium. That it is not necessary is quite
obvious in games with many Nash equilibria. For example, the argument
for focal points, as discussed in Section 1a, states that only Nash
equilibria that have some extra 'focal' quality are self-enforcing. It
also has been argued that Nash equilibria are not sufficient. Risse
(2000) argues that the notion of self-enforcing agreements should be
understood as an agreement that provides some incentives for the
agents to stick to it, even without external enforcement. He then goes
on to argue that there are such self-enforcing agreements that are not
Nash equilibria. Take for example the game in figure 19.

Figure 19

Lets imagine the players initially agreed to play (R2, C2). Now both
have serious reasons to deviate, as deviating unilaterally would
profit either player. Therefore, the Nash equilibria of this game are
(R1,C2) and (R2,C1). However, in an additional step of reflection,
both players may note that they risk ending up with nothing if they
both deviate, particularly as the rational recommendation for each is
to unilaterally deviate. Players may therefore prefer the relative
security of sticking to the agreed-upon strategy. They can at least
guarantee 2 utils for themselves, whatever the other player does, and
this in combination with the fact that they agreed on (R2, C2) may
reassure them that their opponent will in fact play strategy 2. So
(R2, C2) may well be a self-enforcing agreement, but it nevertheless
is not a Nash equilibrium.

Last, the argument from self-enforcing agreements does not account for
mixed strategies. In mixed equilibria all strategies with positive
probabilities are best replies to the opponent's strategy. So once a
player's random mechanism has assigned an action to her, she might as
well do something else. Even though the mixed strategies might have
constituted a self-enforcing agreement before the mechanism made its
assignment, it is hard to see what argument a player should have to
stick to her agreement after the assignment is made (Luce ad Raiffa
1957, 75).

Simulation

Another argument for one-shot Nash equilibria commences from the idea
that agents are sufficiently similar to take their own deliberations
as simulations of their opponents' deliberations.

"The most sweeping (and perhaps, historically, the most frequently
invoked) case for Nash equilibrium…asserts that a player's strategy
must be a best response to those selected by other players, because he
can deduce what those strategies are. Player i can figure out j's
strategic choice by merely imagining himself in j's position. (Pearce
1984, 1030).

Jacobsen (1996) formalizes this idea with the help of three
assumptions. First, he assumes that a player in a two-person game
imagines himself in both positions of the game, choosing strategies
and forming conjectures about the other player's choices. Second, he
assumes that the player behaves rationally in both positions. Thirdly,
he assumes that a player conceives of his opponent as similar to
himself; i.e. if he chooses a strategy for the opponent while
simulating her deliberation, he would also choose that position if he
was in her position. Jacobsen shows that on the basis of these
assumptions, the player will choose his strategies so that it and his
conjecture on the opponent's play are a Nash equilibrium. If his
opponent also holds such a Nash equilibrium conjecture (which she
should, given the similarity assumption), then the game has a unique
Nash equilibrium.

This argument has met at least two criticisms. First, Jacobsen
provides an argument for Nash equilibrium conjectures, not Nash
equilibria. If each player ends up with a multiplicity of Nash
equilibrium conjectures, an additional coordination problem arises
over and above the coordination of which Nash equilibrium to play: now
first the conjectures have to be matched before the equilibria can be
coordinated.

Secondly, when simulating his opponent, a player has to form
conjectures about his own play from the opponent's perspective. This
requires that he predict his own behavior. However, Levi (1997) raises
the objection that to deliberate excludes the possibility of
predicting one's own behavior. Otherwise deliberation would be
vacuous, since the outcome is determined when the relevant parameters
of the choice situation are available. Since game theory models
players as deliberating between which strategies to choose, they
cannot, if Levi's argument is correct, also assume that players, when
simulating others' deliberation, predict their own choices.

Concluding this section, it seems that there is no general
justification for Nash equilibria in one-shot, simultaneous-move
games. This does not mean that there is no justification to apply the
Nash concept to any one-shot, simultaneous-move game – for example,
games solvable by iterated dominance have a Nash equilibrium as their
solution. Also, this conclusion does not mean that there are no
exogenous reasons that could justify the Nash concept in these games.
However, the discussion here was concerned with endogenous reasons –
i.e. reasons that can be found in the way games are modeled. And there
the justification seems deficient.
c. Nash Equilibrium in Repeated Games

If people encounter an interactive situation sufficiently often, they
sometimes can find their way to optimal solutions by trial-and error
adaptation. In a game-theoretic context, this means that players need
not necessarily be endowed with the ability to play equilibrium – or
with the sufficient knowledge to do so – in order to get to
equilibrium. If they play the game repeatedly, they may gradually
adjust their behavior over time until there is no further room for
improvement. At that stage, they have achieved equilibrium.

Kalai and Lehrer (1993) show that in an infinitely repeated game,
subjective utility maximizers will converge arbitrarily close to
playing Nash equilibrium. The only rationality assumption they make is
that players maximize their expected utility, based on their
individual beliefs. Knowledge assumptions are remarkably weak for this
result: players only need to know their own payoff matrix and discount
parameters. They need not know anything about opponents' payoffs and
rationality; furthermore, they need not know other players'
strategies, or conjectures about strategies. Knowledge assumptions are
thus much weaker for Nash equilibria arising from such adjustment
processes than those required for one-shot game Nash solutions.

Players converge to playing equilibrium because they learn by playing
the game repeatedly. Learning, it should be remarked, is not a goal in
itself but an implication of utility maximization in this situation.
Each player starts out with subjective prior beliefs about the
individual strategies used by each of her opponents. On the basis of
these beliefs, they choose their own optimal strategy. After each
round, all players observe each other's choices and adjust their
beliefs about the strategies of their opponents. Beliefs are adjusted
by Bayesian updating: the prior belief is conditionalized on the newly
available information. On the basis of these assumptions, Kalai and
Lehrer show that after sufficient repetitions, (i) the real
probability distribution over the future play of the game is
arbitrarily close to what each player believes the distribution to be,
and (ii) the actual choices and beliefs of the players, when
converged, are arbitrarily close to a Nash equilibrium. Nash
equilibria in these situations are thus justified as potentially
self-reproducing patterns of strategic expectations.

It needs to be noted, however, that this argument depends on two
conditions that not all games satisfy. First, players must have enough
experience to learn how their opponents play. Depending on the kind of
learning, this may take more time than a given interactive situation
affords. Second, not all adjustment processes converge to a steady
state (for an early counterexample, see Shapley 1964). For these
reasons, the justification of Nash equilibrium as the result of an
adjustment process is sensitive to the game model, and therefore does
not hold generally for all repeated games.
d. Backward Induction

Backward induction is the most common Nash equilibrium refinement for
non-simultaneous games. Backward induction depends on the assumption
that rational players remain on the equilibrium path because of what
they anticipate would happen if they were to deviate. Backward
induction thus requires the players to consider out-of-equilibrium
play. But out-of-equilibrium play occurs with zero probability if the
players are rational. To treat out-of-equilibrium play properly,
therefore, the theory needs to be expanded. Some have argued that this
is best achieved by a theory of counterfactuals (Binmore 1987,
Stalnaker 1999), which gives meaning to sentences of the sort "if a
rational player found herself at a node out of equilibrium, she would
choose …". Alternatively, for models where uncertainty about payoffs
is allowed, it has been suggested that such unexpected situations may
be attributed to the payoffs' differing from those that were
originally thought to be most likely (Fudenberg, Kreps and Levine
1988).

The problem of counterfactuals cuts deeper, however, than a call for
mere theory expansion. Consider the following two-player
non-simultaneous perfect information game in figure20, called the
"centipede". For reasons of representational convenience, the game is
represented as progressing from left to right (instead of from top to
bottom as in the usual extensive-form games). Player 1 starts at the
leftmost node, choosing to end the game by playing down or to continue
the game (giving player 2 the choice) by playing right. The payoffs
are such that at each node it is best for the player who has to move
to stop the game if and only if she expects that in the event she
continues, the game will end at the next stage (by the other player
stopping the game or by termination of the game). The two zigzags
stand for the continuation of the payoffs along those lines. Now
backward induction advises to solve the game by starting at the last
node z, asking what player 2 would have done if he ended up here. A
comparison of player 2's payoffs for his two choices, given his
rationality, answers that he would have chosen down. Substituting the
payoffs of this down for node z, one now moves backwards. What would
player 1 have done had she ended up at node y? Given common knowledge
of rationality (hence the substitution of player 2's payoffs for node
z) she would have chosen down. This line of argument then continues
back to the first node.

Figure 20

For the centipede, backward induction therefore recommends player 1 to
play down at the first node; all other recommendations are
counterfactual in the sense that no rational player should ever reach
it. So what should player 2 do if he found himself at node x? Backward
induction tells him to play "down', but backward induction also told
him that if player 1 was rational, he would never face the actual
choice at node x. So either player 1 is rational, but made a mistake
('trembled" in Selten's terminology) at each node preceding x, or
player 1 is not rational (Binmore 1987). But if player 1 is not
rational, then player 2 may hope that she will not choose down at her
next choice either, thus allowing for a later terminal node to be
reached. This consideration becomes problematic for backward induction
if it also affects the counterfactual reasoning. It may be the case
that the truth of the indicative conditional "If player 2 finds
herself at x, then player 2 is not rational" influences the truth of
the counterfactual "If player 2 found herself at x, then player 2
would not be rational". Remember that for backward induction to work,
the players have to consider counterfactuals like this: "If player 2
found herself at x, and she was rational, she would choose down". Now
the truth of the first counterfactual makes false the antecedent
condition of the second: it can never be true that player 2 found
herself at x and be rational. Thus it seems that by engaging in these
sorts of counterfactual considerations, the backward induction
conclusion becomes conceptually impossible.

This is an intensely discussed problem in game theory and philosophy.
Here only two possible solutions can be sketched. The first answer
insists that common knowledge of rationality implies backward
induction in games of perfect information (Aumann 1996). This position
is correct in that it denies the connection between the indicative and
the counterfactual conditional. Players have common knowledge of
rationality, and they are not going to lose it regardless of the
counterfactual considerations they engage in. Only if common knowledge
was not immune against evidence, but would be revised in the light of
the opponents' moves, then this sufficient condition for backward
induction may run into the conceptual problem sketched above. But
common knowledge by definition is not revisable, so the argument
instead has to assume common belief of rationality. If one looks more
closely at the versions of the above argument (e.g. Pettit and Sugden
(1989)), it becomes clear that they employ the notion of common
belief, and not of common knowledge.

Another solution of the above problem obtains when one shows, as
Bicchieri (1993, chapter 4) does, that limited knowledge of
rationality and of the structure of the game suffice for backward
induction. All that is needed is that a player, at each of her
information sets, knows what the next player to move knows. This
condition does not get entangled in internal inconsistency, and
backward induction is justifiable without conceptual problems.
Further, and in agreement with the above argument, she also shows that
in a large majority of cases, this limited knowledge of rationality
condition is also necessary for backward induction. If her argument is
correct, those arguments that support the backward induction concept
on the basis of common knowledge of rationality start with a flawed
hypothesis, and need to be reconsidered.

In this section, I have discussed a number of possible justifications
for some of the dominant game theoretic solution concepts. Note that
there are many more solution concepts that I have not mentioned at all
(most of them based on the Nash concept). Note also that this is a
very active field of research, with new justifications and new
criticisms developed constantly. All I tried to do in this section was
to give a feel for some of the major problems of justification that
game theoretic solution concepts encounter.
e. Paradoxes of Rationality

In the preceding section, the focus was on the justification of
solution concepts. In this section, I discuss some problematic results
that obtain when applying these concepts to specific games. In
particular, I show that the solutions of two important games disagree
with some relevant normative intuitions. Note that in both cases these
intuitions go against results accepted in mainstream game theory; many
game theorists, therefore, will categorically deny that there is any
paradox here at all. From a philosophical point of view (as well as
from some of the other social sciences) these intuitions seem much
more plausible and therefore merit discussion.

The Chain Store Paradox

Recall the story from section 1c: a chain store faces a sequence of
possible small-business entrants in its monopolistic market. In each
period, one potential entrant can choose to enter the market or to
stay out. If he has entered the market, the chain store can choose to
fight or to share the market with him. Fighting means engaging in
predatory pricing, which will drive the small-business entrant out of
the market, but will incur a loss (the difference between
oligopolistic and predatory prices) for the chain store. Thus fighting
is a weakly dominated strategy for the chain store, and its threat to
fight the entrant is not credible.

Because there will only be a finite number of potential entrants, the
sequential game will also be finite. When the chain store is faced
with the last entrant, it will cooperate, knowing that there is no
further entrant to be deterred. Since the structure of the game and
the chain store's rationality are common knowledge, the last
small-business will decide to enter. But since the last entrant cannot
be deterred, it would be irrational for the chain store to fight the
penultimate potential entrant. Thus, by backward induction, the chain
store will always cooperate and the small-businesses will always
decide to enter.

Selten (1978), who developed this example, concedes that backward
induction may be a theoretically correct solution concept. However,
for the chain-store example, and a whole class of similar games,
Selten construes backward induction as an inadequate guide for
practical deliberation. Instead, he suggests that the chain store may
accept the backward induction argument for the last x periods, but not
for the time up to x. Then, following what Selten calls a deterrence
theory, the chain store responds aggressively to entries before x, and
cooperatively after that. He justifies this theory (which, after all,
violates the backward induction argument, and possibly the dominance
argument) by intuitions about the results:

…the deterrence theory is much more convincing. If I had to play a
game in the role of [the chain store], I would follow the deterrence
theory. I would be very surprised if it failed to work. From my
discussion with friends and colleagues, I get the impression that most
people share this inclination. In fact, up to now I met nobody who
said that he would behave according to the [backwards] induction
theory. My experience suggests that mathematically trained persons
recognize the logical validity of the induction argument, but they
refuse to accept it as a guide to practical behavior. (Selten 1978,
132-3)

Various attempts have been made to explain the intuitive result of the
deterrence theory on the basis of standard game theory. Most of these
attempts are based on games of incomplete information, allowing the
chain store to exploit the entrants' uncertainty about its real
payoffs. Another approach altogether takes the intuitive results of
the deterrence theory and argues that standard game should be
sensitive to limitations of the players' rationality. Some of these
limitations are discussed under the heading of bounded rationality in
Section 2f.

The One-Shot Prisoners' Dilemma

The prisoners' dilemma has attracted much attention, because all
standard game theoretic solution concepts unanimously advise each
player to choose a strategy that will result in a Pareto-dominated
outcome.

Figure 21: Prisoners' Dilemma

Recall that the unique Nash equilibrium, as well as the dominant
strategies, in the prisoners' dilemma game is (R2,C2) – even though
(R1,C1) yields a higher outcome for each player. In Section 1b, the
case of an infinitely repeated prisoners' dilemma yielded a different
result. Finite repetitions, however, still yield the result (R2,C2)
from backward induction. That case is structurally very similar to the
chain store paradox, whose implausibility was discussed above. But
beyond the arguments for (R1,C1) derived from these situations, many
also find the one-shot prisoners' dilemma result implausible, and seek
a justification for the players to play (R1,C1) even in that case.
Gauthier (1986) has offered such a justification based on the concept
of constrained maximization. In Gauthier's view, constrained
maximization bridges the gap between rational choice and morality by
making moral constraints rational. As a consequence, morality is not
to be seen as a separate sphere of human life but as an essential part
of maximization.

Gauthier envisions a world in which there are two types of players:
constrained maximizers (CM) and straightforward maximizers (SM). An SM
player plays according to standard solution concepts; A CM player
commits herself to choose R1 or C1 whenever she is reasonably sure she
is playing with another CM player, and chooses to defect otherwise. CM
players thus do not make an unconditional choice to play the dominated
strategy; rather, they are committed to play cooperatively when faced
with other cooperators, who are equally committed not to exploit one
another's good will. The problem for CM players is how to verify this
condition. In particular in one-shot games, how can they be reasonably
sure that their opponent is also CM, and thus also committed to not
exploit? And how can one be sure that opponents of type CM correctly
identify oneself as a CM type? With regards to these questions,
Gauthier offers two scenarios, which try to justify a choice to become
a CM. In the case of transparency, players' types are common
knowledge. This is indeed a sufficient condition for becoming CM, but
the epistemic assumption itself is obviously not well justified,
particularly in one-shot games – it simply "assumes away" the problem.
In the case of translucency, players only have beliefs about their
mutual types. Players' choices to become CM will then depend on three
distinct beliefs. First, they need to believe that there are at least
some CMs in the population. Second, they need to believe that players
have a good capacity to spot CMs, and third that they have a good
capacity to spot SMs. If most players are optimistic about these
latter two beliefs, they will all choose CM, thus boosting the number
of CMs, making it more likely that CMs spot each other. Hence they
will find their beliefs corroborated. If most players are pessimistic
about these beliefs, they will all choose SM and find their beliefs
corroborated. Gauthier, however, does not provide a good argument of
why players should be optimistic; so it remains a question whether CM
can be justified on rationality considerations alone.
f. Bounded Rationality in Game Players

Bounded rationality is a vast field with very tentative delineations.
The fundamental idea is that the rationality which mainstream
cognitive models propose is in some way inappropriate. Depending on
whether rationality is judged inappropriate for the task of rational
advice or for predictive purposes, two approaches can be
distinguished. Bounded rationality which retains a normative aspect
appeals to some version of the "ought implies can" principle: people
cannot be required to satisfy certain conditions if in principle they
are not capable to do so. For game theory, questions of this kind
concern computational capacity and the complexity-optimality
trade-off. Bounded rationality with predictive purposes, on the other
hand, provides models that purport to be better descriptions of how
people actually reason, including ways of reasoning that are clearly
suboptimal and mistaken (for an overview of bounded rationality, see
Grüne-Yanoff 2007). The discussion here will be restricted to the
normative bounded rationality.

The outmost bound of rationality is computational impossibility.
Binmore (1987) discusses this topic by casting both players in a
two-player game as Turing machines. A Turing machine is a theoretical
model that allows for specifying the notion of computability. Very
roughly, if a Turing machine receives an input, performs a finite
number of computational steps (which may be very large), and gives an
output then the problem is computable. If a Turing machine is caught
in an infinite regress while computing a problem, however, then the
problem is not computable. The question Binmore discusses is whether
Turing machines can play and solve games. The scenario is that the
input received by one machine is the description of another machine
(and vice versa), and the output of both machines determines the
players' actions. Binmore shows that a Turing machine cannot predict
its opponent's behavior perfectly and simultaneously participate in
the action of the game. Roughly put, when machine 1 first calculates
the output of machine 2 and then takes the best response to its
action, and machine 2 simultaneously calculates the output of machine
1 and then takes the best response to its action, the calculations of
both machines enter an infinite regress. Perfect rationality,
understood as the solution to the outguessing attempt in "I thank that
you think that I think…" is not computable in this sense.

Computational impossibility, however, is very far removed from the
realities of rational deliberation. Take for example the way people
play chess. Zermelo (1913) long ago showed that chess has a solution.
Despite this result, chess players cannot calculate the solution of
the game and choose their strategies accordingly. Instead, it seems
that they typically "check out" several likely scenarios and that they
entertain some method to evaluate the endpoint of each scenario (e.g.
by counting the chess pieces). People differ in the depth of their
inquiry, the quality of the "typical scenarios" selected, and the way
they evaluate their endpoint positions.

The justification for such "piecemeal" deliberation is that computing
the solution of a game can be very costly. Deliberation costs reduce
the value of an outcome; it may therefore be rational to trade the
potential gains from a full-blown solution with the moderate gains
from "fast and frugal" deliberation procedures that are less costly
(the term "fast and frugal" heuristics was coined by the ABC research
group. Compare Gigerenzer et al 1999). Rubinstein (1998) formalizes
this idea by extending the analysis of a repeated game to include
players' sensitivity to the complexity of their strategies. He
restricts the set of strategies to those that can be executed by
finite machines. He then defines the complexity of a strategy as the
number of states of the machine that implements it. Each player's
preferences over strategy profiles increase with her payoff in the
repeated game, and decrease with the complexity of her strategy's
complexity (He considers different ranking methods, in particular
unanimity and lexicographic preferences). Rubinstein shows that the
set of equilibria for complexity-sensitive games is much smaller than
that of the regular repeated game.
3. Game Theory as a Predictive Theory

We now turn from the use of game theory as a normative theory to its
use as a scientific theory of human behavior. Game theory, as part of
Rational Choice Theory, is an important social scientific method.
There is, however, considerable controversy about the usefulness of
Rational Choice Theory for the purposes of the social sciences. Some
of this controversy arises along disciplinary boundaries: while
Rational Choice Theory is considered mainstream in economics (to the
extent that no one even bothers using this label), sociologists and
political scientists are more divided. A long debate in those
disciplines reached its peak with the publications of Green and
Shapiro's (1994) Pathologies of Rational Choice Theory. In this book,
they make two major claims about the scientific usefulness of Rational
Choice Theory. First, they argue that Rational Choice Theory is
empirically empty: that it has produced virtually no new propositions
about politics that have been carefully tested and not found wanting.
Second, they argue that the perceived universality claim of Rational
Choice Theory is misguided: that even if an empirically successful
Rational Choice Theory were to emerge, it would not be any more
universal than the middle-level theories that they advocate.

These two claims have been challenged on various fronts. First, it has
been pointed out that Green and Shapiro employ inappropriate standards
for testing Rational Choice Theory, standards that not even successful
theories of the hard sciences would survive (Diermeier 1995). Second,
defenders of Rational Choice Theory have argued that Green and
Shapiro's argument relies on a biased selection of rational choice
literature to survey; and further, that even in the literature they
selected, there are interesting and empirically confirmed propositions
that satisfy their minimum requirements (Cox 1999). Third, one can
argue that Green and Shapiro's criticism of Rational Choice Theory as
a universal theory goes amiss. In Section 1c, I argued that game
theory is in fact not a universal theory of rationality, but rather
offers a menu of tools to model specific situations. At least with
respect to game theory, therefore, they attack the wrong target: game
theory is useful because it is a widely applicable method, which works
well in certain circumstances, rather than a universal substantive
theory of human behavior.

Although game theory cannot be dismissed as not useful for prediction
just because it is part of Rational Choice Theory, game theory has a
number of problems of its own that need to be discussed in depth. The
first issue is to what extent the role of game theory as a theory of
rationality is relevant here. I contrast this possibility with a brief
sketch of evolutionary game theory, which abandons the rationality
notion altogether. In the consecutive section, I discuss the problems
of specifying the payoffs in a game, and thus giving a game model
empirical content. Last, I discuss the possibility whether game theory
can be tested at all, and investigate a recent claim that indeed game
theory has been tested, and refuted.

Game theory may be useful in predicting human behavior for two
distinct reasons. First, it may be the case that game theory is a good
theory of rationality, that agents are rational and that therefore
game theory predicts their behavior well. If game theory was correct
for this reason, it could reap the additional benefit of great
stability. Many social theories are inherently unstable, because
agents adjust their behavior in the light of its predictions. If game
theory were a good predictive theory because it was a good theory of
rationality, this would be because each player expected every other
player to follow the theory's prescriptions and had no incentive to
deviate from the recommended course of action. Thus, game theory would
already take into account that players' knowledge of the theory has a
causal effect on the actions it predicts (Bicchieri 1993, chapter
4.4). Such a self-fulfilling theory would be more stable than a theory
that predicts irrational behavior. Players who know that their
opponents will behave irrationally (because a theory tells them) can
improve their results by deviating from what the theory predicts,
while players who know that their opponents will behave rationally
cannot. However, the prospects for game theory as a theory where
prescription and prediction coincide are not very good; evidence from
laboratory experiments, as well as from casual observations, often
puts doubt on it.

Second, and independently of the question of whether game theory is a
good theory of rationality, game theory may be a good theory because
it offers the relevant tools to systematize and predict interactive
behavior successfully. This distinction may make sense when separating
our intuitions about how agents behave rationally from a systematic
account of our observations of how agents behave. Aumann for example
suggests that

philosophical analysis of the definition [of Nash equilibrium]
itself leads to difficulties, and it has its share of counterintuitive
examples. On the other hand, it is conceptually simple and attractive,
and mathematically easy to work with. As a result, it has led to many
important insights in the applications, and has illuminated and
established relations between many different aspects of interactive
decision situations. It is these applications and insights that lend
it validity. (Aumann 1985, 49).

These considerations can lead one to accept the view that the
principles of game theory provide an approximate model of human
deliberation, which sometimes provides insights into real phenomena
(this seems to be Aumann's position). Philosophy of Science discusses
various ways of how approximate models can relate to real phenomena;
each has its specific problems, which cannot be discussed here.

Aumann's considerations can also lead one to seek an alternative
interpretation of the Nash concept that does not refer to human
rationality, but retains all the formally attractive properties.
Evolutive approaches of game theory offer such an interpretation
(Binmore 1987 proposed this term in order to distinguish it from the
eductive approaches discussed in Section 2). Its proponents claim that
the economic, social and biological evolutionary pressure directs
human agents, who have no clear idea what is going on, to behavior
that is in accord with the solution concepts of game theory.
a. The Evolutive Interpretation

The evolutive interpretation seeks to apply techniques, results, and
justifications of assumptions from evolutionary game theory to game
theory as a predictive theory of human behavior. Evolutionary game
theory was developed in biology; it studies the appearance, robustness
and stability of behavioral traits in animal populations. This article
cannot do justice even to the basics of this very vibrant and
expanding field (for a concise and formal introduction, see Maynard
Smith 1982 and Weibull 1995), but instead presents only some aspects
relevant to two questions; namely (i), to what extend can standard
game theory elements be based on evolutionary game theory? And (ii),
does this reinterpretation help in the prediction of human behavior?

Evolutionary game theory studies games that are played over and over
again by players drawn from a populations. These players do not have a
choice between strategies, but rather are "programmed" to play only
one strategy. It is thus often said that the strategies themselves are
the players. Success of a strategy is defined in terms of the number
of replications that a strategy will leave of itself to play in games
of future generations. Rather than determining equilibrium as the
consequence of strategic reasoning by rational players, evolutionary
game theory determines the stability of a strategy distribution in a
population either as the resistance to mutant invasions, or as the
result of a dynamic process of natural selection. Its equilibrium
concept is thus much closer to the stable state concept of the natural
sciences, where different causal factors balance each other out, than
the eductive interpretation is.

Evolutionary game theory can be distinguished into a static and into a
dynamic approach. The static approach specifies strategies that are
evolutionary stable against a mutant invasion. Imagine a population of
players programmed to play one (mixed or pure) strategy A. Imagine
further that a small fraction of players "mutate" – they now play a
strategy B different from A. A strategy is an evolutionary stable
strategy (ESS) if for every possible mutant strategy B different from
A, the payoff of playing A against the A is higher than the payoff of
playing B against A – or, if both payoffs are equal, then the payoff
of playing A against B is higher than playing B against B. Note that
ESS is a robustness test only against a single mutation at a time. It
is assumed that the population that plays an ESS has time to adjust
back to status quo before the next mutant invasion begins. It follows
from this definition that every ESS is a strategy that is in Nash
equilibrium with itself. However, not every strategy that is Nash
equilibrium with itself is an ESS.

The dynamic approach of evolutionary game theory considers a selection
mechanism that favors some strategies over others in a continuously
evolving population. Imagine a population whose members are programmed
to play different strategies. Pairs of players are drawn at random to
play against each other. Their payoff consists in an increase or
decrease in fitness, measured as the number of offspring per time
unit. Each offspring inherits the parent's strategy. Reproduction
takes place continuously over time, with the birthrate depending on
fitness, and the death rate being uniform for all players. Long
continuations of tournaments between players then may lead to stable
states in the population, depending on the initial population
distribution. This notion of dynamic stability is wider than that of
evolutionary stability: while all evolutionary stable strategies are
also dynamically stable, not all dynamically stable strategies are
evolutionary stable. It has been shown that in the long run, all
strictly dominated and all iteratively strictly dominated strategies
are eliminated from the population. The relation between stable states
and Nash equilibria is more complex, and would require specifications
that go beyond the scope of this brief sketch.

Evolutionary game theory provides interesting concepts and techniques
that are quite compatible with the solution concepts of standard game
theory discussed in Section 1 (however, it focuses mainly on
two-person static games; dynamic games and game repetitions are less
investigated). Clearly, evolutionary game theory is more concerned
with discovering conditions of stability and robustness of strategies
in populations, than with finding the equilibria of a single game. The
question that remains is whether it competes in its explanatory
efforts with eductive game theory, or whether it deals instead with
different (although maybe related) phenomena.

Those who claim that explanatory efforts between these two
interpretations do compete hope that evolutionary concepts will
replace players' rationality – better even, that they will explain why
we sometimes think that players are rational. This hope is well
illustrated at the hand of Binmore's evolutive model and the criticism
directed against it. Binmore's approach starts with the concept of a
meme – "a norm, an idea, a rule of thumb, a code of conduct –
something that can be replicated from one head to another by imitation
or education, and that determines some aspects of the behavior of the
person in whose head it is lodged" (Binmore 1994, 20). Players are
mere hosts to these memes, and their behavior is partly determined by
them. Fitness is a property of the meme and its capacity to replicate
itself to other players. Expected utility maximization is then
interpreted as a result of evolutionary selection:

People who are inconsistent [in their preferences] will
necessarily be sometimes wrong and hence will be at a disadvantage
compared to those who are always right. And evolution is not kind to
memes that inhibit their own replication. (Binmore 1994, 27)

This is of course a version of the dynamic approach discussed above.
To that extent, the theory of the fittest memes becoming relatively
more frequent is an analytic truth, as long as "fitness" is no more
than high "rate of replication". But Binmore then transfers the
concept of strategy fitness to player rationality. Critics have
claimed that this theory of meme fitness cannot serve as the basis for
the claim that the behavior of human individuals as hosts of memes
will tend towards a rational pattern. The error occurs, Sugden (2001)
argues, when Binmore moves from memes fitness to fitness of players.
In the analogous biological case – which is based on genes instead of
memes – the reproductive success of phenotype depends on the
combination of genes that carry it. Genes have positive consequences
in combination with some genes while bad consequences in combination
with others. A gene pool in equilibrium therefore may contain genes
which, when brought together in the same individual by a random
process of sexual reproduction, have bad consequences for that
individual's survival and reproduction. Therefore, genes may be
subject to natural selection, but there may be a stable proportion of
unfit phenotypes produced by them in the population. It is thus not
necessarily the case that natural selection favors phenotype survival
and reproduction. The same argument holds for memes: unless it is
assumed that an agent's behavior is determined by one meme alone,
natural selection on the level of memes does not guarantee that
agents' behavioral patterns are rational in the sense that they are
consistent with expected utility theory. It therefore remains an
empirical question whether people behave in accord with the principles
game theory proposes. The evolutive interpretation cannot determine a
priori that players will play Nash equilibrium.
b. The Problem of Alternative Descriptions

While intuitions about rational behavior may be teased out in
fictional, illustrative stories, the question of whether prediction is
successful is answerable only on the basis of people's observed
behavior. Behavioral game theory observes how people behave in
experiments in which their information and incentives are carefully
controlled. With the help of these experiments, and drawing on further
evidence from psychology, it hopes to test game-theoretic principles
for their correctness in predicting behavior. Further, in cases where
the tests do not yield positive results, it hopes that the experiments
suggest alternative principles that can be included in the theory (for
more details on Behavioral Game Theory, their experimental methods and
results, see Camerer 2003). To test game theory, the theory must be
made to predict particular behavior. To construct specific
experimental setups, and to make the theory predict such particular
behavior, however, particular interactive phenomena need to be modeled
as games, so that the theory's solution concepts can be applied. This
brings with it the problem of interpretation discussed in Section 1c.
The most contentious aspect of a game modeling lies in the payoffs.
The exemplary case is the disagreement over the relevant evaluations
of the players in the Prisoners' Dilemma.

Some critics of the defect/defect Nash equilibrium solution have
claimed that players would cooperate because they would not only
follow their selfish interests, but also take into account non-selfish
considerations. They may cooperate, for example, because they care
about the welfare of their opponents, because they want to keep their
promises out of feelings of group solidarity or because they would
otherwise suffer the pangs of a bad conscience. To bring up these
considerations against the prisoners' dilemma, however, would expose a
grave misunderstanding of the theory. A proper game uses the players'
evaluation, captured in the utility function, of the possible
outcomes, not the material payoff (like e.g. money). The evaluated
outcome must be described with those properties the players find
relevant. Thus either the non-selfish considerations are already
included in the players' payoffs (altruistic agents, after all, also
have opposing interest – e.g. which charitable cause to benefit); or
the players will not be playing the Prisoners' Dilemma. They will be
playing some other game with different payoffs.

Incorporating non-material interests in the payoffs has been
criticized for making game theory empirically empty. The critics argue
that with such a broad interpretation of the payoffs, any anomaly in
the prediction of the theory can be dissolved by a re-interpretation
of the agents' evaluations of the consequences. Without constraints on
re-interpretation, the critics claim, the theory cannot be held to any
prediction.

To counter this objection, many economists and some game theorists
claim to work on the basis of the revealed preference approach. At a
minimum, this approach requires that the preferences – and hence the
utility function – of an agent are exclusively inferred from that
agent's choices (for a discussion of the revealed preference account,
see Grüne 2004). This ostensibly relieves game modelers to engage in
"psychologizing" when trying to determine the players' subjective
evaluations.

However, it has been argued that the application of the revealed
preference concept either trivializes game theory or makes it
conceptually inconsistent. The first argument is that the revealed
preference approach completely neglects the importance of beliefs in
game theory. An equilibrium depends on the players' payoffs and on
their beliefs of what the other players believe and what they will do.
In the stag hunt game of figure 1, for example, Row believes that if
Col believed that Row would play R2, then he would play C2. But if the
payoff numbers represented revealed preferences, Hausman (2000)
argues, then they would say how individuals would choose, given what
the other chose, period. The payoffs would already incorporate the
influence of belief, and belief would play no further role. Game
theory as a theory of rational deliberation would have lost its job.

The second criticism claims that it is conceptually impossible that
games can be constructed on the basis of revealed preferences. Take as
an example the simple game in figure 22.

Figure 22: A game tree

How can a modeler determine the payoff pairs z1-z4 for both players
according to the revealed preference method? Let's start with player
2. Could one construct two choice situations for player 2 in which he
chooses between z1 and z2 and between z3 and z4 respectively? No,
argues Hausman (2000): two thus constructed choice situation exactly
differ from the game in figure 22 in that they are not preceded by
player 1's choice. Hence there is no reason why it could not be the
case that player 2 chooses z1 over z2 in the game but chooses z2 over
z1 in the constructed choice situation. More problematically still,
player 2 must be able to compare z1 with z3 and z2 with z4. But it is
logically impossible that she will ever face such a choice, as player
1 will choose either U or D. Last, turning to player 1, she never
faces a choice between the outcomes of this game, only between U and
D. So the revealed preference theorist cannot assign preferences over
outcomes to player 1 at all, and to player 2 only partially. With the
preferences that he can assign – to player 2's played strategy, and to
player 1's choices – prediction is only possible at the pain of
trivializing game theory. The only prediction that the revealed
preference theorist now can offer is that the players play whatever
action they revealed prefer – that is, they do what they do.

These problems may have contributed to a widespread neglect of the
problem of preference ascription in game theoretic models. As Weibull
(2002) observes:

While experimentalists usually make efforts to carefully specify
to the subject the game form … they usually do not make much effort to
find the subject's preferences, despite the fact that these
preferences constitute an integral part of the very definition of a
game. Instead, it is customary to simply hypothesize subjects'
preferences. (Weibull 2002, 2)

The problem of preference identification has been insufficiently
addressed in rational choice theory in general and in game theory in
particular. But it is not unsolvable. One solution is to find a
criterion for outcome individuation. Broome offers such a criterion by
justifiers: "outcomes should be distinguished as different if and only
if they differ in a way that makes it rational to have a preference
between them" (Broome 1991, 103). This criterion, however, requires a
concept of rationality independent of the principles of rational
choice. A rational choice is no longer based on preferences alone, but
preferences themselves are now based on the rationality concept. This
constitutes a radical departure of how most rational choice theorists,
including game theorists, regard the concept of rationality. Another
option that Hausman (2005) suggests is that economists can use game
theoretic anomalies to study the factors influencing preferences. By
altering features of the game forms and, in particular, by
manipulating the precise beliefs each player has about the game and
about others' conjectures, experimenters may be able to make progress
in understanding what governs choices in strategic situations and
hence what games people are playing.
c. Testing Game Theory

Whether game theory can be tested depends on whether the theory makes
any empirical claims, and whether it can be immunized against
predictive failure.

Does the theory make testable claims? At first, it does not seem so.
The theory as discussed in Sections 1a-1b mainly takes the form of
theorems. Theorems are deductive conclusions from initial assumptions.
So to test game theory, these assumptions need to be tested for their
empirical adequacy. In this vein, Hausman (2005) claims that game
theory is committed to contingent and testable axioms concerning human
rationality, preferences, and beliefs. This claim remains
controversial. Many economists believe that theories should not be
tested with regard to their assumptions, but only with respect to
their predictions (a widespread view that was eloquently expressed by
Friedman 1953). But the theory only makes empirical claims in
conjunction with its game models.

Further, testing game theory through its predictions is difficult as
such tests must operate through the mediation of models that represent
an interactive situation. Here the issue of interpreting the modeled
situation (see Section 1c) and of model construction drives a wedge
between the predicting theory and the real world phenomena, so that
predictive failures can often be attributed to model misspecification
(as discussed in section 3b).

Guala (2005) recently pointed to a specific element of game theory
that seems to make an empirical claim all by itself, and independent
of auxiliary hypotheses. For this purpose, he discusses the phenomenon
of reciprocity. Agents reciprocate to other agents who have exhibited
"trust" in them because they want to be kind to them. Reciprocation of
an agent 1 to another agent 2 is necessarily dependent on 2 having
performed an action that led 1 to reciprocate. Reciprocation is thus
clearly delineated from general altruism or justice considerations.

The question that Guala raises is whether reciprocity can be accounted
for in the payoff matrix of a game. The 'kindness' of an action
depends on what could have been chosen: I think that you are kind to
me because you could have harmed me for your benefit, but you elected
not to. This would mean that the history of chosen strategies would
endogenously modify the payoffs, a modeling move that is explicitly
ruled out in standard game theory. Guala shows that the exclusion of
reciprocity is connected right to the core of game theory: to the
construction of the expected utility function. All existing versions
of the existence proof of expected utility theory rely on the
so-called rectangular field assumption. It assumes that decision
makers form preferences over every act that can possibly be
constructed by combining consequences with states of the world.
However, if reciprocity has to be modeled in the consequences, and
reciprocity depends on others' acts that in turn depend on the
players' own acts, then it is conceptually impossible to construct
acts in accord with the rectangular field assumption, because the act
under question would be caught in an infinite regress.

If Guala's argument is correct, it seems impossible to model
reciprocity in the payoffs, and game theory is not flexible enough to
accommodate reciprocity considerations into its framework. But that
would mean that game theory claims that reciprocity does not exist in
general. With this claim, game theory would be testable, and – if
reciprocity were indeed a relevant factor in strategic decisions, as
the evidence seems to suggest – would be refuted.
4. Conclusion

Game theory, this survey showed, does not provide a general and
unified theory of interactive rationality; nor does it provide a
positive theory of interactive behavior that can easily be tested.
These observations have many implications of great philosophical
interest, some of which were discussed here. Many of the questions
that arise in these discussions are still left unanswered, and
therefore require more attention from philosophers than they currently
receive.

This article could only sketch the basic concepts of game theory in
order to discuss some of their philosophical implications and
problems. Wherever possible, it abstained from presenting any formal
detail. To fully understand game theory, however, a formal treatment
is inevitable. A good and fun introduction that also points out some
philosophical issues is Binmore (1991). A textbook that puts more
emphasis on the mathematical proofs is Osborne and Rubinstein (1994);
a thorough and technical treatment (including excellent
bibliographies) is Fudenberg and Tirole (1991). Some of the graphs
found in this article were taken from that book."

See also the discussions of game theory in these articles: Law and
Economics, Egoism, Libertarianism, and Social Contract Theory.
5. References and Further Reading

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Frontiers of Economics. Ed. K Arrow and S. Honkapojah. Oxford:
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* Aumann, Robert. "Backward Induction and Common Knowledge of
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* Bacharach, Michael. "Variable Universe Games," Frontiers of Game
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* Brandenburger, Adam. "Knowledge and Equilibrium in Games,"
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* Bernheim, D. "Rationalizable Strategic Behavior," Econometrica
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* Bicchieri, Christina. Rationality and Coordination. Cambridge:
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* Binmore, Ken. "Modeling Rational Players: Part I," Economics and
Philosophy 3 (1987): 179-214.
* Binmore, Ken Fun and Games, D.C. Heath, 1991.
* Binmore, Ken. Game Theory and the Social Contract. Volume I:
Just Playing. Cambridge, Mass.: MIT Press, 1994.
* Broome, John. Weighting Goods. Oxford: Oxford University Press,
1991. Camerer, Colin F. Behavioral Game Theory. Princeton NJ:
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* Cox, Gary W. "The Empirical Content of Rational Choice Theory: A
Reply to Green and Shapiro," Journal of Theoretical Politics 11(1999):
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* Diermeier, Daniel "Rational Choice and the Role of Theory in
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* Ellsberg, Daniel. "Theory of the Reluctant Duelist," The
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* Friedman, Milton. "The methodology of positive economics," in
Essays in Positive Economics. Chicago: University of Chicago Press,
1953, pp. 3-43.
* Fudenberg, Kreps and Levine. "On the Robustness of Equilibrium
Refinements," Journal of Economic Theory 44 (1988): 354-380.
* Fudenberg and Maskin. "The Folk Theorem with Discounting and
with Incomplete Information," Econometrica 54 (1986): 533-554.
* Fudenberg, Drew and Jean Tirole. Game Theory. Cambridge, Mass.:
MIT Press, 1991.
* Gauthier, David. Morals by Agreement. Oxford: Clarendon Press, 1986
* Gigerenzer, G., Todd, P. & the ABC Research Group (1999). Simple
heuristics that make us smart. New York: Oxford University Press..
* Goeree Jacob K. and Charles A. Holt "Ten Little Treasures of
Game Theory and Ten Intuitive Contradictions," American Economic
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* Green, Donald and Ian Shapiro. Pathologies of Rational Choice
Theory, New Haven, CT: Yale University Press, 1994.
* Grüne, Till. "The Problems of Testing Preference Axioms on the
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* Grüne-Yanoff, Till "Bounded Rationality," Philosophy Compass,
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* Grüne-Yanoff, Till and Sven Ove Hansson "Preferences," The
Stanford Encyclopedia of Philosophy, Edward N. Zalta (ed.),
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* Grüne-Yanoff, Till and Paul Schweinzer "The Role of Stories in
Applying Game Theory," Journal of Economic Methodology, 2008.
* Guala, Francesco (2006) "Has Game Theory Been Refuted?" The
Journal of Philosophy 103 (55): 239-263.
* Hausman, Danniel M. "Revealed Preferences, Belief and Game
Theory," Economics and Philosophy 16 (2000): 99-115.
* Hausman, Daniel M. "'Testing' Game Theory," Journal of Economic
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* Jacobsen, Hans Jørgen. "On the Foundations of Nash Equilibrium,"
Economics and Philosophy 12 (1996): 67-88.
* Kalai, Ehud and Ehud Lehrer. "Rational Learning Leads to Nash
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and A. W. Tucker. Princeton, N.J.: Princeton University Press, 1953.
* Levi, Isaac. "Prediction, Deliberation, and Correlated
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* Morgan, Mary. "The Curious Case of the Prisoner's Dilemma: Model
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* von Neumann, John and Oskar Morgenstern. The Theory of Games and
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* Osborne, Martin and Ariel Rubinstein. A Course in Game Theory.
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* Pearce, David G. "Rationalizable Strategic Behavior and the
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* Rubinstein, Ariel Modeling Bounded Rationality, MIT Press, 1998.
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* Stalnaker, Robert. "Knowledge, Belief and Counterfactual
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* Sugden, Robert. "The Evolutionary Turn in Game Theory," Journal
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* Weibull, Jörgen W. Evolutionary Game Theory. Cambridge, Mass.:
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* Weibull, Jörgen W. "Testing Game Theory," Mimeo, 2004.
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* Zermelo, Ernst. "Über eine Anwendung der Mengenlehre auf die
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