Nash Equilibrium

BIBLIOGRAPHY

Nash equilibrium is a fundamental concept in the theory of games and the most widely used method of predicting the outcome of a strategic interaction in the social sciences. A game (in strategic or normal form) consists of the following three elements: a set of players, a set of actions (or pure-strategies) available to each player, and a payoff (or utility) function for each player. The payoff functions represent each player’s preferences over action profiles, where an action profile is simply a list of actions, one for each player. A pure-strategy Nash equilibrium is an action profile with the property that no single player can obtain a higher payoff by deviating unilaterally from this profile.

This concept can best be understood by looking at some examples. Consider first a game involving two players, each of whom has two available actions, which we call A and B. If the players choose different actions, they each get a payoff of 0. If they both choose A, they each get 2, and if they both choose B, they each get 1. This “coordination” game may be represented as in Figure 1, where player 1 chooses a row, player 2 chooses a column, and the resulting payoffs are listed in parentheses, with the first component corresponding to player 1’s payoff. The action profile (B,B ) is an equilibrium, since a unilateral deviation to A by any one player would result in a lower payoff for the deviating player. Similarly, the action profile (A, A ) is also an equilibrium.

As another example, consider the game “matching pennies,” which again involves two players, each with two actions. Each player can choose either heads (H ) or tails (T ); player 1 wins a dollar from player 2 if their choices are the same, and loses a dollar to player 2 if they are not. This game is shown in Figure 2 and has no pure-strategy Nash equilibria.

In some cases, instead of simply choosing an action, players may be able to choose probability distributions over the set of actions available to them. Such randomizations over the set of actions are referred to as mixed strategies. Any profile of mixed strategies induces a probability distribution over action profiles in the game. Under certain assumptions, a player’s preferences over all such lotteries can be represented by a function (called a von Neumann-Morgenstern utility function ) that assigns a real number to each action profile. One lottery is preferred to another if and only if it results in a higher expected value of this utility function, or expected utility. A mixed strategy Nash-equilibrium is then a mixed strategy profile with the property that no single player can obtain a higher value of expected utility by deviating unilaterally from this profile.

The American mathematician John Nash (1950) showed that every game in which the set of actions available to each player is finite has at least one mixed-strategy

equilibrium. In the matching pennies game, there is a mixed-strategy equilibrium in which each player chooses heads with probability 1/2. Similarly, in the coordination game of the above example, there is a third equilibrium in which each player chooses action A with probability 1/3 and B with probability 2/3. Such multiplicity of equilibria arises in many economically important games, and has prompted a large literature on equilibrium refinements with the purpose of identifying criteria on the basis of which a single equilibrium might be selected.

Nash equilibria can sometimes correspond to outcomes that are inefficient, in the sense that there exist alternative outcomes that are both feasible and preferred by all players. This is the case, for instance, with the equilibrium (B, B ) in the coordination game above. An even more striking example arises in the prisoner’s dilemma game, in which each player can either “cooperate” or “defect,” and payoffs are as shown in Figure 3.

The unique Nash equilibrium is mutual defection, an outcome that is worse for both players than mutual cooperation. Now consider the game that involves a repetition of the prisoner’s dilemma for n periods, where n is commonly known to the two players. A pure strategy in this repeated game is a plan that prescribes which action is to be taken at each stage, contingent on every possible history of the game to that point. Clearly the set of pure strategies is very large. Nevertheless, all Nash equilibria of this finitely repeated game involve defection at every stage. When the number of stages n is large, equilibrium payoffs lie far below the payoffs that could have been attained under mutual cooperation.

It has sometimes been argued that the Nash prediction in the finitely repeated prisoner’s dilemma (and in many other environments) is counterintuitive and at odds with experimental evidence. However, experimental tests of the equilibrium hypothesis are typically conducted with monetary payoffs, which need not reflect the preferences of subjects over action profiles. In other words, individual preferences over the distribution of monetary payoffs may not be exclusively self-interested. Furthermore, the equilibrium prediction relies on the hypothesis that these preferences are commonly known to all subjects, which is also unlikely to hold in practice.

To address this latter concern, the concept of Nash equilibrium has been generalized to allow for situations in which players are faced with incomplete information. If each player is drawn from some set of types, such that the probability distribution governing the likelihood of each type is itself commonly known to all players, then we have a Bayesian game. A pure strategy in this game is a function that associates with each type a particular action. A Bayes-Nash equilibrium is then a strategy profile such that no player can obtain greater expected utility by deviating to a different strategy, given his or her beliefs about the distribution of types from which other players are drawn.

Allowing for incomplete information can have dramatic effects on the predictions of the Nash equilibrium concept. Consider, for example, the finitely repeated prisoner’s dilemma, and suppose that each player believes that there is some possibility, perhaps very small, that his or her opponent will cooperate in all periods provided that no defection has yet been observed, and defect otherwise. If the number of stages n is sufficiently large, it can be shown that mutual defection in all stages is inconsistent with equilibrium behavior, and that, in a well-defined sense, the players will cooperate in most periods. Hence, in applying the concept of Nash equilibrium to practical situations, it is important to pay close attention to the information that individuals have about the preferences, beliefs, and rationality of those with whom they are strategically interacting.

SEE ALSO Game Theory; Multiple Equilibria; Noncooperative Games; Prisoner’s Dilemma (Economics)

BIBLIOGRAPHY

Cournot, A. A. 1838. Recherches sur les principes mathématiques de la théorie des richesses. Paris: L. Hachette.

Fudenberg, Drew, and Jean Tirole. 1991. Game Theory. Cambridge, MA: MIT Press.

Harsanyi, John C. 1967–1968. Games with Incomplete Information Played by Bayesian Players. Management Science 14 (3): 159–182, 320–334, 486–502.

Harsanyi, John C., and Reinhard Selten. 1998. A General Theory of Equilibrium Selection in Games. Cambridge, MA: MIT Press.

Kreps, David, Paul Milgrom, John Roberts, and Robert Wilson. 1982. Rational Cooperation in the Finitely Repeated Prisoner’s Dilemma. Journal of Economic Theory 27: 245–252.

Nash, John F. 1950. Equilibrium Points in N-Person Games. Proceedings of the National Academy of Sciences 36 (1): 48–49.

Osborne, Martin J., and Ariel Rubinstein. 1994. A Course in Game Theory. Cambridge, MA: MIT Press.

von Neumann, John, and Oskar Morgenstern. 1944. Theory of Games and Economic Behavior. Princeton, NJ: Princeton University Press.

Rajiv Sethi