THE GRADUATE STUDENT SECTION WHAT IS... Nash Equilibrium? Rajiv Sethi and Jörgen Weibull Communicated by Cesar E. Silva In game theory, a Nash equilibrium is an array of strategies, The invention and succinct formulation of this concept, one for each player, such that no player can obtain a along with the establishment of its existence under very higher payoff by switching to a different strategy while general conditions, reshaped the landscape of research the strategies of all other players are held fixed. The in economics and other social and behavioral sciences. concept is named after John Forbes Nash Jr. Nash’s existence theorem pertains to games in which For example, if Chrysler, Ford, and GM choose produc- the strategies 푆푖 available to each player are probability tion levels for pickup trucks, a commodity whose market distributions over a finite set of alternatives. Typically, price depends on aggregate production, an equilibrium is each alternative specifies what action to take under each an array of production levels, one for each firm, such that and every circumstance that the player may encounter none can raise its profits by making a different choice. during the play of the game. The alternatives are referred Formally, an 푛-player game consists of a set 퐼 = to as pure strategies and the probability distributions {1, … , 푛} of players, a set 푆푖 of strategies for each player over these as mixed strategies. Players’ randomizations, 푖 ∈ 퐼, and a set of goal functions 푔푖 ∶ 푆1 × ⋯ × 푆푛 → ℝ according to their chosen probability distributions over that represent the preferences of each player 푖 over the their own set of alternatives, are assumed to be statistically 푛-tuples, or profiles, of strategies chosen by all players. independent. Any 푛-tuple of mixed strategies then induces A strategy profile has a higher goal-function value, or a probability distribution or lottery over 푛-tuples of payoff, than another if and only if the player prefers it to pure strategies. Provided that a player’s preferences the other. Let 푆 = 푆1 ×⋯×푆푛 denote the set of all strategy over such lotteries satisfy certain completeness and profiles, with generic element 푠, and let (푡푖, 푠−푖) denote the consistency conditions—previously identified by John strategy profile (푠1, … , 푠푖−1, 푡푖, 푠푖+1, … , 푠푛) obtained from 푠 von Neumann and Oskar Morgenstern—there exists a by switching player 푖’s strategy to 푡푖 ∈ 푆푖 while leaving all real-valued function with the 푛-tuples of pure strategies other strategies unchanged. An equilibrium point of such as its domain such that the expected value of this function a game is a strategy profile 푠∗ ∈ 푆 with the property that, represents the player’s preferences over 푛-tuples of mixed for each player 푖 and each strategy 푡푖 ∈ 푆푖, strategies. Given only this restriction on preferences, Nash was able to show that every game has at least one 푔 (푠∗) ≥ 푔 (푡 , 푠∗ ). 푖 푖 푖 −푖 equilibrium point in mixed strategies. That is, a strategy profile is an equilibrium point if no Emile Borel had a player can gain from a unilateral deviation to a different precursory idea, con- strategy. Nash equilibrium cerning symmetric pure reshaped the conflicts of interest be- Rajiv Sethi is professor of economics at Barnard College, Columbia tween two parties with University, and external professor at the Santa Fe Institute. His landscape of very few alternatives email address is [email protected]. at hand. In 1921 he Jörgen Weibull is professor at the Stockholm School of Economics. research in defined the notion of He is also affiliated with the KTH Royal Institute of Technology, a finite and symmet- Stockholm, and with the Institute for Advanced Study in Toulouse. economics. ric zero-sum two-player His email address is [email protected]. game. In such a game For permission to reprint this article, please contact: each player has the same number of pure strategies, the [email protected]. gain for one player equals the loss to the other, and DOI: http://dx.doi.org/10.1090/noti1375 they both have the same probability of winning whenever 526 Notices of the AMS Volume 63, Number 5 THE GRADUATE STUDENT SECTION they use the same pure strategy. Borel also formalized convex-valued, since a convex combination of countering the concept of a mixed strategy, and for games in which 푛-tuples must itself be a countering 푛-tuple. And since the each player has three pure strategies, proved the exis- payoff functions are all continuous (in fact, polynomial) tence of what would later come to be called a maxmin functions with closed domain, the correspondence has pair of mixed strategies. This is a pair of strategies a closed graph. The existence of a fixed point follows such that one player’s strategy maximizes his own gain from Kakutani’s theorem, and any such fixed point is while his opponent simultaneously minimizes this gain. a self-countering 푛-tuple, or an equilibrium point of the He subsequently extended this result to the case of five game. strategies per player, but seems to have doubted that A year later Nash published an alternative existence general existence results could be achieved. proof in the Annals of Mathematics that instead is based on A few years later, and apparently unaware of Borel’s Brouwer’s fixed-point theorem. Since Kakutani’s theorem partial results, von Neumann formalized the notion of is derived from Brouwer’s, Nash was more satisfied with finite zero-sum games with an arbitrary (finite) number the latter. This second proof has a touch of genius. of players, where each player has an arbitrary (finite) It is simple and intuitive in retrospect but completely number of pure strategies. For all such games involving unexpected beforehand. two players he proved the existence of a maxmin strategy In order to use Brouwer’s theorem, Nash needed to pair, presented the result in Göttingen in 1927, and construct a self-map on the space of mixed-strategy published it in 1928. profiles with the property that a strategy profile is an In comes Nash, a young doctoral student in mathemat- equilibrium point if and only if it is a fixed point of this ics at Princeton University. Nash defined a much more map. But the best-reply correspondence could not be used general class of games and a more general equilibrium for this purpose, since it need not be single-valued and concept. He allowed for any (finite) number of players, does not permit a continuous selection in general. each having an arbitrary (finite) number of pure strategies This is how he did it. Consider any 푛-tuple of mixed at his or her disposal and equipped with any goal function. strategies 푠, and recall that the payoff to a player 푖 at this In particular, players may be selfish, altruistic, spiteful, strategy profile is 푔푖(푠). Let 푔푖ℎ(푠) denote the payoff that moralistic, fair-minded, or have any goal function whatso- player 푖 would receive if he were to switch to the pure ever. His definitions and his existence result contain those strategy ℎ while all other players continued to use the of Borel and von Neumann as special cases. Previously strategies specified in 푠. Define the continuous function restricted to pure conflicts of interest, game theory could 휙푖ℎ(푠) = max{0, 푔푖ℎ(푠) − 푔푖(푠)}. now be addressed to any (finite) number of parties with Each function value 휙 (푠) represents the “excess payoff” arbitrary goal functions in virtually any kind of strategic 푖ℎ obtained by pure strategy ℎ ∈ 푆 , as compared with the interaction. Nash published this in a one-page article in 푖 payoff obtained under strategy profile 푠. Letting 푠 denote the Proceedings of the National Academy of Sciences in 푖ℎ the probability with which pure strategy ℎ is played under 1950. 푠, the function 휙 may be used to obtain a new 푛-tuple of His existence proof—merely sketched in this short mixed strategies, 푠′, from 푠 by setting paper—is based upon Kakutani’s fixed-point theorem 푠 + 휙 (푠) (established some years earlier). Kakutani’s theorem states 푠′ = 푇 (푠) = 푖ℎ 푖ℎ . 푚 푖ℎ 푖ℎ that if a subset 푋 of ℝ is nonempty, compact and convex, 1 + ∑ℎ 휙푖ℎ(푠) and a (set-valued) correspondence Γ ∶ 푋 ⇉ 푋 is nonempty- This defines a self-map 푇 on the space of mixed strategy valued, convex-valued and has a closed graph, then there profiles. As long as there exists a pure strategy with exists 푥 ∈ 푋 such that 푥 ∈ Γ(푥). That is, there exists a positive excess payoff, 푇 lowers the probabilities with fixed point of the correspondence. Nash’s existence proof which pure strategies having zero excess payoff are relies on the construction of what today is called the played. It is clear that if 푠 is an equilibrium point, it must best-reply correspondence, which can then be shown to be a fixed point of 푇, since no pure strategy ℎ can yield satisfy the conditions of Kakutani’s theorem. player 푖 a higher payoff, forcing 휙푖ℎ(푠) = 0 for all 푖 and ℎ. Given any 푛-tuple of mixed strategies, Nash defined a It is easily verified that the converse is also true: if 푠 is a countering 푛-tuple as a mixed-strategy profile that obtains fixed point of 푇, so that 휙푖ℎ(푠) = 0 for all 푖 and ℎ, then 푠 for each player the highest payoff given the strategies must be an equilibrium point of the game. chosen by other players in the original, countered 푛-tuple.