Game Theory
GAME THEORY
Games of Complete Information
A game is an abstract, formal description of a strategic interaction. Any strategic interaction involves two or more decision makers (players), each with two or more ways of acting (strategies), such that the outcome depends on the strategy choices of all the players. Each player has well-defined preferences among all the possible outcomes, enabling corresponding utilities (payoffs) to be assigned. A game makes explicit the rules governing players' interaction, the players' feasible strategies, and their preferences over outcomes. Game theory describes games by means of mathematical concepts (e.g., sets, functions, and relations).
normal form
A possible representation of a game is in normal form. A normal form game is completely defined by three elements that constitute the structure of the game: a list of players i = 1, …, n ; for each player i ; a finite set of pure strategies S i ; and a payoff function u i that gives player i 's payoff u i (s )n for each n-tuple of strategies (s 1, …, s n ), where ui : Xj = 1 Sj→R. A player may choose to play a pure strategy or instead to randomize over his or her pure strategies; a probability distribution over pure strategies is called a mixed strategy and is denoted by σ i . Each player's randomization is assumed to be statistically independent of that of his or her opponents, and the payoffs to a mixed strategy are the expected values of the corresponding pure strategy payoffs. A different interpretation of mixed strategies, based on the idea that players do not always randomize over their feasible actions, is that the probability distribution σ i represents other players' uncertainty about what player will do. A mixed strategy is thus thought of as other players' conjecture about a player's plans of action. The conjectures depend on the player's private information, which is left unspecified in the model. A problem with this interpretation is that if there are reasons behind the choices a player makes, they should be included in the model, since they are likely to be payoff relevant.
The two-by-two matrix in Figure 1 depicts the two-player normal form representation of the famous Prisoner's dilemma game, where C stands for cooperate and D for defect. The numbers in the cell of the matrix denote players' payoffs: the first number is the payoff for the row player, the second for the column player. Each player picks a strategy independently, and the outcome, represented in terms of players' payoffs, is the joint product of these two strategies. Notice that in the game of Figure 1, each player is better off defecting no matter what the other player does. For example, if the column player cooperates, the row player gets a payoff of 3 by defecting and a payoff of 2 by cooperating, while if the column player defects, the row player gains a payoff of 1 by defecting and of 0 by cooperating. When, regardless of what other players do, a strategy yields a player a (strictly) inferior
payoff than some other strategy, it is called a dominated strategy. When a strategy yields the same payoff of another undominated strategy, but it has an inferior payoff against at least one opponent's strategy, it is called a weakly dominated strategy.
The game of Figure 1 is one of complete information, in that the players are assumed to know the rules of the game (which include players' strategies) and other players' payoffs. If players are allowed to enter into binding agreements before the game is played, one can say that the game is cooperative. Noncooperative games instead make no allowance for the existence of an enforcement mechanism that would make the terms of the agreement binding on the players. What strategies should rational players choose? What could be rightly called the central dogma of game theory states that rational players will always jointly maximize their expected utilities, or play a Nash equilibrium (compare Nash 1996). Informally, a Nash equilibrium specifies players' actions and beliefs such that (1) each player's action is optimal given his or her beliefs about other players' choices; (2) players' beliefs are correct. Thus, an outcome that is not a Nash equilibrium requires either that a player chooses a suboptimal strategy or that some players misperceive the situation.
More formally, a Nash equilibrium is a vector of strategies (σ *1, …, σ *n ), one for each of the n players in the game, such that each σ *i is optimal given (or is a best reply to) σ *−i. That is
u i (σ *i , σ *−i) ≥ u i (σ i , σ *−i ) for all mixed strategies of player i σ i
Note that optimality is only conditional on a fixed? σ−i , not on all possible σ −i . A strategy that is a best reply to a
given combination of the opponents' strategies may fare poorly vis-à-vis another strategy combination.
In a game like the one depicted in Figure 2 the row player gains a payoff of 1 if the toss of two coins results in two heads or two tails and loses 1 otherwise, and vice versa for the column player.
This game has no Nash equilibrium in pure strategies. Nash proved that—provided certain restrictions are imposed on strategy sets and payoff functions—every game has at least one equilibrium in mixed strategies. In a mixed strategy equilibrium, the equilibrium strategy of each player makes the other indifferent between the strategies on which he or she is randomizing. In particular, the game in Figure 2 has a unique Nash equilibrium in which both players randomize between their strategies with probability ½. Then, if the first player plays σ 1 = (½ H, ½ T), his or her expected payoff is ½ 1 + ½ − 1 = 0 regardless of the strategy of the second player.
The players (and the game theorist) can predict that a specific equilibrium will be played just in case they have enough information to infer players' choices. The standard assumptions in game theory are:
CK1. The structure of the game is common knowledge
CK2. The players are rational (i.e., they are expected utility maximizers) and this is common knowledge
The concept of common knowledge was introduced by David K. Lewis (1969) in his study on convention, which is arguably the first major philosophical work in which game theory plays a central role as a modeling tool. Simply put, the idea of common knowledge is that a certain proposition p is common knowledge among two players if both of them know p, both of them know that they know p, and so on ad infinitum. The previous assumptions may allow the players to predict an opponent's strategy. For example, in the prisoner's dilemma
game of Figure 1 rational players would never choose the strictly dominated strategy C. CK1 and CK2, then, allow the players to predict that the opponent will play D. However (compare Bicchieri 1993), the previous CK assumptions do not always guarantee that a prediction of play can be made. For one, even if the game has a unique equilibrium, the set of strategies that, under the assumptions CK1 and CK2, players may choose need not contain the equilibrium strategies only. Moreover, predictability is hampered by another common problem encountered in game theory: multiple Nash equilibria.
Suppose two players have to divide $100 among them. They must restrict their proposals to integers, and each has to independently propose a way to split the sum. If the total proposed by both is equal or less than $100, each gets what he or she proposed, otherwise they get nothing. This game has 101 Nash equilibria. Is there a way to predict which one will be chosen? In real life, many people would go for the fifty-fifty split. It is simple and it seems equitable. In Thomas C. Schelling's (1960) words, it is a focal point. Unfortunately, mere salience is not enough to provide a player with a reason for choice. In this example, only if it is common knowledge that the fifty-fifty split is the salient outcome does it become rational to propose $50. Game theory, however, filters out any social or cultural information regarding strategies, leaving players with the task of coordinating their actions on the sole basis of common knowledge of rationality (and of the structure of the game).
A different approach to the problem of indeterminacy is to start by considering the set of Nash equilibria and ask whether some of them should be eliminated because they are in some sense unreasonable. This is the approach taken by the refinement program (Kohlberg 1990, van Damme 1987). Consider the game in Figure 3:
The game has two Nash equilibria in pure strategies: (a,c ) and (b,d ). The equilibrium (a,c ) is Pareto dominant, since it gives both players a higher payoff than any other equilibrium in the game. However, common knowledge of rationality and of the structure of the game does not force the column player to expect the row player to eliminate the weakly dominated strategy b, nor is the row player forced to conclude that the column player will discard d. Prudence, however, may suggest that one should never be too sure of the opponents' choices. Even if the players have agreed to play a given equilibrium, some uncertainty remains. If so, one should try to model this uncertainty in the game. R. Selten's (1965) insight was to treat perfect rationality as a limit case. His "trembling hand" metaphor presupposes that deciding and acting are two separate processes, in that even if one decides to take a particular action, one may end up doing something else by mistake. An equilibrium strategy should be optimal not only against the opponents' strategies but also against some small probability ℇ > 0 that the opponents make mistakes. Such an equilibrium is trembling-hand perfect.
Is the equilibrium (b,d ) perfect? If so, b must be optimal against c being played with probability ℇ and d being played with probability 1 − ℇ for some small ℇ > 0. But in this case the expected payoff to a is 2ε whereas the payoff to b is?. Hence for all ℇ > 0, a is a better strategy choice. The equilibrium (b,d ) is not perfect, but (a,c ) is. Therefore, a prudent player would discard (b,d ). In this simple game, checking perfection is easy, since only one mistake is possible. With many strategies, there usually are many more possible mistakes to take into account. Similarly, with many players one may need to worry about who is more likely to make a mistake.
extensive form
A different representation of a game is the extensive form. It specifies the following information: a finite set of players i = 1, …, n ; the order of moves; the players' choices at each move; and what each player knows when he or she has to choose. The order of play is represented by a game tree T, which is a finite set of partially ordered nodes t ∈ T satisfying a precedence relation <. A subgame is a collection of branches of a game such that they start from the same node and the branches and the node together form a game tree by itself. A tree representation is sequential, because it shows the order in which actions are taken by the players. It is natural to think of sequential-move games as being ones in which players choose their strategies one after the other, and of simultaneous-move games as ones in which players choose their strategies at the same time. What is important, however, is not the temporal order of events per se, but whether players know about other players' actions when they have to choose their own. In the normal form representation, players' information about other players' choices is not represented. This is the reason a normal form game could represent any one of several extensive form games. When the order of play is irrelevant to a game's outcome, then restricting oneself to the normal form is justifiable. When the order of play is relevant, however, the extensive form must be specified.
In an extensive form game the information a player has when he or she is choosing an action is explicitly represented using information sets, which partition the nodes of the tree. If an information set contains more than one node, the player who has to make a choice at that information set will be uncertain as to which node he or she is at. Not knowing at which node one is means that the player does not know which action was chosen by the preceding player. If a game contains information sets that are not singletons, the game is one of imperfect information.
A strategy for player i is a complete plan of action that specifies an action at every node at which it is i 's turn to move. Note that a strategy specifies actions even at nodes that will never be reached if that strategy is played. Consider the game in Figure 4. It is a finite game of perfect information in which player 1 moves first. If he chooses D at his first node, the game ends and player 1 nets a payoff of 1, whereas player 2 gets 0. But choosing D at the first node is only part of a strategy for player 1. For example, it can be part of a strategy that recommends "play D at your first node, and x at your last node." Another strategy may instead recommend playing D at his first node, and y at his last decision node. Though it may seem surprising that a strategy specifies actions even at nodes that will not be reached if that strategy is played, one must remember that a strategy is a full contingent plan of action. For example, the strategy Dx recommends playing D at the first node, thus effectively ending the game. It is important, however, to be able to have a plan of action in case D is not played. Player 1 may, after all, make a mistake and, because of player 2's response, find himself called to play at his last node. In that case, having a plan helps. Note that a strategy cannot be changed during the course of the game. Though a player may conjecture about several scenarios of moves and countermoves before playing the game, at the end of deliberation a strategy must be chosen and followed throughout the game.
The game of Figure 4 has two Nash equilibria in pure strategies:(Dx,d ) and (Dy,d ). Is there a way to solve the indeterminacy?
Suppose player 1 were to reach his last node. Since he is by assumption rational, he will choose x, which guarantees
him a payoff of 4. Knowing (by assumption) that player 1 is rational, player 2—if she were to reach her decision node—would play d, since by playing a she would net a lower payoff. Finally, since (by assumption) player 1 knows that player 2 is rational and that she knows that player 1 is rational, he will choose D at his first decision node. The equilibrium (Dy,d ) should therefore be ruled out, since it recommends an irrational move at the last node. In the normal form, both equilibria survive. The reason is simple: Nash equilibrium does not constrain behavior out of equilibrium. In this example, if player 1 plans to choose D and player 2 plans to choose d, it does not matter what player 1 would do at his last node, since that node will never be reached.
The sequential procedure one has used to conclude that only (Dx,d ) is a reasonable solution is known as backward induction. In finite games of perfect information with no ties in payoffs, backward induction always identifies a unique equilibrium. The premise of the backward induction argument is that mutual rationality and the structure of the game are common knowledge among the players. It has been argued by Ken Binmore (1987), Cristina Bicchieri (1989, 1993), and Philip J. Reny (1992) that under certain conditions common knowledge of rationality leads to inconsistencies. For example, if player 2 were to reach her decision node, would she keep thinking that player 1 is rational? How would she explain player 1's move? If player 1's move is inconsistent with common knowledge of rationality, player 2 will be unable to predict future play; as a corollary, what constitutes an optimal choice at her node remains undefined. As a consequence of the previous criticisms, the usual premises of backward induction arguments have come to be questioned (compare Pettit and Sugden 1989, Basu 1990, Bonanno 1991). There are a number of further equilibrium refinements for games in extensive form. Their multiplicity makes it impossible to delve into details here. The interested reader can consult Bicchieri (1993, chapter 3).
Games of Incomplete Information
In games of incomplete information certain elements of the game are not common knowledge among the players. The knowledge and beliefs of the players have to be incorporated into the game-theoretic model, as one usually does in extensive form games, and an appropriate equilibrium concept has to be devised. The approach is based on the seminal work of John C. Harsanyi (1968). In the Bayesian approach adopted by Harsanyi, a player's uncertainty about variables that are relevant for his or her decision ought to be made explicit by means of probability distributions representing his or her beliefs. Moreover, second-order beliefs (beliefs about other players' beliefs) can be represented by further probability distributions, and third-order beliefs about second-order ones, and so on. The flexibility of Harsanyi's model allows one to incorporate all such infinite sequence of higher-order beliefs without an explicit representation of it.
The main idea is that the payoffs associated to each strategy profile depend on certain parameters θ 1, …, θ n , one for each player 1, …, n. Each parameter is drawn from a set Θ i = (a i , b i , … ) associated with each player i. The composition of the sets Θ i is known, yet the true value of the parameter θ i is not (at least for one of the players). The parameter θ i is called i 's type and the set Θ i represents, intuitively, the other player's ignorance about i 's characteristics. A type amounts to a specification of certain variables: a player's strategy set, a player's preferences and payoff function, and so on, that make up the private information of a player. Although it is convenient to refer to "the type a i of player i " as if it was a separate individual, one should keep in mind that types represent players' knowledge (and uncertainty about others) only. As mentioned earlier, in a Bayesian approach uncertainties are represented by probability distributions. Hence, each player has an initial probability distribution μ i = (μ i (a −i ), μ i (b -i ), … ) over the types of any other player other than i. Since in a Bayesian game the choices of a player depend on his or her type, the concept of Nash equilibrium has to be generalized accordingly.
Note that all that a player knows, except from the game itself (and the priors), is his own type, and the fact that the other players do not know his own type as well. As their best responses depend on the players' actual types, a player must see himself through his opponents' eyes and plan a best reply against the possible strategies of his opponents for each potential type of his own. Thus, a strategy in a Bayesian game of incomplete information must map each possible type of each player into a plan of actions. Then, since the other players' types are unknown, each player forms a best reply against the expected strategy of each opponent, where he averages over the (well-specified) reactions of all possible types of an opponent, using his prior probability measure on the type space. Such a profile of type-dependent strategies which are unilaterally unimprovable in expectations over the competing types' strategies forms a Bayesian Nash equilibrium. In other words, a Bayesian Nash equilibrium is a Nash equilibrium "at the interim stage" where each player selects a best response against the average best responses of the competing players.
In the framework provided by Harsanyi (1968) it is possible to reduce a game of incomplete information to one of imperfect information. "Nature" is called to make the first move of the game, as if it was an actual player. Nature's random moves determine the type of each player, with a fixed probability that represents the prior probability attached to the events that player i is of type θ i . Priors are assumed to be common knowledge, and players observe their own type only. Players then pick their strategies in this extended game, and it is possible to show that the equilibrium of such a game corresponds to the Bayesian Nash equilibrium of the game with incomplete information. In particular, the choice function s i yields the action σ i (θ i ) if and only if (iff) that is the action that player i chooses in the game with Nature when she observes her type θ i .
Epistemic Foundations of Game Theory
An important development of game theory is the so-called epistemic approach. In the epistemic approach to game theory strategic reasoning is analyzed on the basis of hypothesis about what players know about the game, about other players' knowledge, and about other players' rationality. Since Robert J. Aumann's (1976) formalization, the idea of common knowledge, and the analysis of what players choose depending on what their beliefs about each other are, began to play an increasingly important role in game theory. In particular, one can evaluate solution concepts by examining the epistemic assumptions and hypotheses from which they can be derived (compare Battigalli and Bonanno 1999). Such epistemic hypotheses are treated formally using the tools provided by interactive epistemology (compare Aumann 1999).
To formalize players' knowledge states, one considers a space set Ω whose elements are possible worlds. An event is then represented by a subset of Ω. For example, the proposition "it is sunny in Philadelphia" is represented by the set of all possible worlds in which it is sunny in Philadelphia. For each player, there exists an information function that partitions the space set. Intuitively, a player cannot distinguish among worlds belonging to the same cell of his or her information partition. Thus, in a possible world ω, player i knows an event E iff the set E (of possible worlds in which E obtains) includes the cell of his or her information partition containing ω. The intuition behind this is that if a player cannot distinguish among all the worlds in which E is true, then he or she knows that E is the case. It is possible to define a knowledge function K i for each player i so that, when given E as an argument, it returns as a value the set of those worlds such that, for each one of them, the cell of i 's information partition that contains it is a subset of E. That is to say, K i E is the event that i knows E.
By imposing certain conditions on the K i 's, one can force the epistemic functions to possess certain properties. For example, by requiring that K i E be a subset of E, one requires that what players know is true, since in every possible world in which K i E obtains, E obtains as well; similarly, by requiring that K i KiE be a subset of KiE, one establishes that players know what they know, and by requiring that Ki−KiE be a subset of −K iE that they know what they do not know (where − is the usual set-theoretical operation of complementation). The first condition is often referred to as the truth axiom, the second as the positive introspection axiom, and the third as the negative introspection axiom. Note that this setup has an equivalent formulation in terms of modal logics (compare Fagin et al. 1995, Meyer and van der Hoek 2004). To see the equivalence of the two approaches, consider that modal formulas express propositions whose semantic interpretation is given in terms of Kripke structures of possible worlds. It is then possible to establish a correspondence between formulas of the modal logic and events in the approach described earlier. In a Kripke model, then, an event corresponds to the set of those possible worlds that satisfy the formula expressing the proposition associated to that event.
Knowledge functions can be iterated, thus they can represent mutual and higher-order knowledge, and Aumann (1976) provides a mathematical definition of the idea of common knowledge in the setup sketched earlier. A proposition p is common knowledge between, say, two players i and j iff the set of worlds representing p includes the cells of i 's and j 's partitions meet that contain p, where the meet of two partitions is the finest common coarsening of them. An application of the definition is the theorem proved in the same article, in which it is shown that if players have common priors, and their posteriors are common knowledge, then the posteriors are equal, even if the players derived them by conditioning on different information. Or, in other words, that one cannot "agree to disagree." As mentioned earlier, Aumann formalized Lewis's (1969) definition of common knowledge. However, it is currently debated whether Aumann's seminal definition is a faithful rendition of Lewis's informal characterization of common knowledge (compare Vanderschraaf 1998, Cubitt and Sugden 2003, Sillari 2005).
In such a framework it is possible to investigate which strategy profiles are compatible with certain epistemic assumptions about the players. For example, CK1 and CK2 imply that players would never choose strictly dominated strategies. The first contributions in this sense are David G. Pearce (1984) and B. Douglas Bernheim (1984), in which a procedure is devised to eliminate all the players' strategies that are not rationalizable, that is, not supported by internally consistent beliefs about other players' choices and beliefs. In general, it can be proved that certain epistemic conditions are only compatible with the strategy profiles yielded by a certain solution concept, hence providing an epistemic foundation for that solution concept. For example, Aumann and Adam Brandenburger (1995) proved that, for two-person games, mutual knowledge (i.e., first-order knowledge among all the players) of the structure of the game, of rationality, and of the players' chosen strategies implies.
correlated equilibrium
So far it has been assumed that players' strategies are independent, as though each player receives a private, independent signal and chooses a (mixed) strategy after having observed his or her own signal. However, signals need not be independent. For example, players can agree to play a certain strategy according to the outcome of some external jointly observed event, for example, a coin toss. If the agreement is self-fulfilling, in that players have no incentive to deviate from it, the resulting strategy profile is an equilibrium in correlated strategies or, in short, a correlated equilibrium (compare Aumann 1974, 1987). For any Nash equilibrium in mixed strategies, a correlation device can be set such that it generates a probability distribution over the possible outcomes of the game yielding such an equilibrium profile. Note, however, that the set of correlated equilibria of a game is much larger than the corresponding set of Nash equilibria. If the correlation signal is common knowledge among the players, one speaks of perfect correlation. However, players may correlate their strategies according to different signals (less than perfect correlation). The idea is that players have information partitions whose cells include more than one possible outcome, since they ignore which signals are received by other players. To represent the fact that players receive different signals (i.e., they ignore which strategies will be chosen by other players), it is required that in every cell of the information partition of player i his or her strategy does not change. It is then possible to calculate the expected payoff of playing the strategy indicated by the correlation device versus the expected payoff obtained by playing a different strategy. If the players have no incentive to deviate from the indicated strategy, the profile yielded by the correlation device is an equilibrium. Correlation by means of private signals may generate outcomes more efficient than those obtained by playing a Nash equilibrium. An important philosophical application of correlated equilibrium is found by Peter Vanderschraaf (1998, 2001), in which conventions as defined by Lewis (1969) are shown to be correlated equilibria of coordination games.
Evolutionary Game Theory
A Nash equilibrium need not be interpreted as a unique event. If one thinks of it as an observed regularity, one wants to know by what process such an equilibrium is reached and what accounts for its stability. When multiple equilibria are possible, one wants to know why players converged to one in particular and then stayed there. An alternative way of dealing with multiple equilibria is to suppose that the selection process is made by nature.
Evolutionary theories are inspired by population biology (e.g., see Maynard Smith 1982). These theories dispense with the notion of the decision maker, as well as with best responses/optimization, and use in their place a natural selection, a "survival of the fittest" process (with mutations) to model the frequencies with which various strategies are represented in the population over time. In a typical evolutionary model players are preprogrammed for certain strategies and are randomly matched with other players in pairwise repeated encounters. The relative frequency of a strategy in a population is simply the proportion of players in that population who adopt it. The theory focuses on how the strategy profiles of populations of such agents evolve over time, given that the outcomes of current games determine the frequency of different strategies in the future.
As an example, consider the game in Figure 5 and suppose that there are only two possible behavioral types: hawk and dove.
A hawk always fights and escalates contests until it wins or is badly hurt. A dove sticks to displays and retreats if the opponent escalates the conflict; if it fights with another dove, they will settle the contest after a long time. Payoffs are expected changes in fitness due to the outcome of the game. Fitness here means just reproductive success (e.g., the expected number of offspring per time unit).
Suppose injury has a payoff in terms of loss of fitness equal to C, and victory corresponds to a gain in fitness B. If hawk meets hawk, or dove meets dove, each has a 50 percent chance of victory. If a dove meets another dove, the winner gets B and the loser gets nothing, so the average increase in fitness for a dove meeting another dove is B /2. A dove meeting a hawk retreats, so his or her fitness is unchanged, whereas the hawk gets a gain in fitness B. If a hawk meets another hawk, they escalate until one wins. The winner has a fitness gain B, the loser a fitness loss C. So the average increase in fitness is (B − C) /2. The latter payoff is negative, since one assumes the cost of injury is greater than the gain in fitness obtained by winning the contest. One can also assume that players will be randomly paired in repeated encounters, and in each encounter they will play the stage game of Figure 5.
If the population were to consist predominantly of hawks, selection would favor the few doves, since hawks would meet mostly hawks and end up fighting with an average loss in fitness of (B − C )/2, and 0 > (B − C /2). In a population dominated by doves, hawks would spread, since every time they meet a dove (which would be most of the time) they would have a fitness gain of B, whereas doves on average would only get B /2. Evolutionary game theory wants to know how strategies do on average when games are played repeatedly between individuals who are randomly drawn from a large population. The average payoff to a strategy depends on the composition of the population, so a strategy may do well (in terms of fitness) in an environment and poorly in another. If the frequency of hawks in the population is q and that of doves correspondingly (1 − q ), the average increase in fitness for the hawks will be q (B − C )/2 + (1 − q )B, and (1 − q )B /2 for the doves. The average payoff of a strategy in a given environment determines its future frequency in the population. In this example, the average increase in fitness for the hawks will be equal to that for the doves when the frequency of hawks in the population is q = B /C. At that frequency, the proportion of hawks and doves is stable. If the frequency of hawks is less that B /C, then they do better than doves and will consequently spread; if their frequency
is larger than B /C, they will do worse than doves and will shrink.
Note that if C > B then (B − C )/2 < 0, so the game in Figure 5 has two pure-strategy Nash equilibria: (H, D ) and (D, H ). There is also a mixed strategy equilibrium in which hawk is played with probability q = B /C and dove is played with probability (1 − q ) = C − B /C. If the game of Figure 5 were played by rational agents who choose which behavior to display, one would be at a loss in predicting their choices. From common knowledge of rationality and of the structure of the game, the players cannot infer that a particular equilibrium will be played. In the hawk-dove example, however, players are not rational and do not choose their strategies. So if an equilibrium is attained it must be the outcome of some process very different from rational deliberation. The process at work is natural selection: High-performing strategies increase in frequency whereas low-performing strategies' frequency diminishes and eventually goes to zero.
One has seen that in a population composed mostly of doves, hawks will thrive, and the opposite would occur in a population composed mainly of hawks. So for example, if hawks dominate the population, a mutant displaying dove behavior can invade the population, since individuals bearing the dove trait will do better than hawks. The main solution concept used in evolutionary game theory is the evolutionarily stable strategy (ESS) introduced by John Maynard Smith and George R. Price (1973). A strategy or behavioral trait is evolutionarily stable if, once it dominates in the population, it does strictly better than any mutant strategy, and hence it cannot be invaded. In the hawk-dove game, neither of the two pure behavioral types is evolutionarily stable, since each can be invaded by the other. One knows, however, that a population in which there is a proportion q = B /C of hawks and (1 − q ) = C − B /C of doves is stable. This means that the type of behavior that consists in escalating fights with probability q = B /C cannot be invaded by any other type, hence it is an ESS. An ESS is a strategy that, when it dominates the population, is a best reply against itself. Therefore, an evolutionarily stable strategy such as (B /C, C − B /C ) is a Nash equilibrium. Though every ESS is a Nash equilibrium, the reverse does not hold; in our stage game, there are three Nash equilibria, but only the mixed strategy equilibrium (B /C, C − B /C ) is an ESS.
Evolutionary games provide one with a way of explaining how agents that may or may not be rational and—if so—subject to severe information and calculation restrictions, achieve and sustain a Nash equilibrium. Philosophical implications and applications can be found in the works of Brian Skyrms (1990, 1996, 2004). When there exist evolutionarily stable strategies (or states), one knows which equilibrium will obtain, without the need to postulate refinements in the way players interpret off-equilibrium moves. Yet we need to know much more about processes of cultural transmission and to develop adequate ways to represent payoffs, so that the promise of evolutionary games is actually fulfilled.
See also Decision Theory; Philosophy of Biology; Philosophy of Economics.
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Cristina Bicchieri (2005)
Giacomo Sillari (2005)