Nash Equilibrium

Named after its inventor, John Nash (1928–), a Nash equilibrium is a combination of strategies—one for each player—such that each individual's strategy maximizes their payoff against the strategies of the other players.

For example, the game in Figure 1 represents the barroom scene occurring during the “eureka” moment in Nash's biographical film, A Beautiful Mind. John and a rival can either approach a blonde or one of her friends. If both approach the blonde, they block each other; each receives a payoff of 0, as illustrated in the northwest cell. If John approaches the blonde and his rival a friend, John's payoff is 3 and his rival's is 1 (the northeast cell). The payoffs in this cell are circled because blonde maximizes John's payoff against the rival's strategy of friend (3 > 2), and friend maximizes the rival's payoff against John's strategy of blonde (1 > 0). Consequently, the strategy combination (blonde, friend) is a Nash equilibrium.

Figure 1 Beautiful Mind Game. Note: John's payoff is listed first; his rival's is listed second.

Equivalently, any collection of strategies is a Nash equilibrium if they are mutual best replies. For example, the payoffs in the southwest cell of this game are also circled because friend is John's best reply to the rival's strategy of blonde (1 > 0). Similarly, blonde is the rival's best reply—against John's strategy of friend—because 3 > 2.

It is useful to compare Nash equilibrium with an ethical norm, one that is incorrectly identified as the Nash equilibrium in the film itself. According to the golden rule John should approach a friend (doing what he would like his rival to do). Similarly, the rival should approach a friend (because this is what he would like John to do). Yet this outcome—corresponding to the southeast cell—is not a Nash equilibrium. Either would approach the blonde if the other approaches a friend.

Nash equilibrium is the predominant solution concept for noncooperative games because one always exists under very general conditions (e.g., irrespective of the number of players or the number of strategies each player possesses). Finding Nash equilibria may require the use of mixed (probabilistic) strategies. For example, in a principal-agent game the principal (manager) monitors the agent (worker) at random intervals rather than continuously standing over his or her shoulder. The two most common interpretations of mixed strategy Nash equilibria are epistemic and mass action. The epistemic interpretation treats Nash equilibrium as the prediction of the likelihood of behavior by rational players who have common knowledge about the game. In the mass action interpretation, a Nash equilibrium is the average frequency of behavior within a population of players [Page 1468]randomly matched to play the game. This latter interpretation links Nash equilibrium to the biological solution concept of evolutionary stable strategy.

Nash equilibrium has proven to be an essential concept for auction design, such as the Federal Communication Commission's auctioning of spectrum licenses or eBay's initial public offering. The understanding of Nash behavior in competitive price setting is fundamental for detecting cartel behavior and in assessing the credibility of entry-deterring strategies such as predatory and limit pricing. Finally, in principalagent relationships a fundamental trade-off exists between Nash equilibrium, Pareto efficiency, and budget balancing of managerial incentives and revenues, thereby identifying a further role for ethics in corporate governance.

...