205.docx

The definition of Nash equilibrium does not require equilibrium choices to be strictly better than other available choices. More important, a Nash equilibrium does not have to be jointly best for the players. In a Nash equilibrium, each player chooses her “best response” to the other’s choice. But t...

The definition of Nash equilibrium does not require equilibrium choices to be strictly better than other available choices. More important, a Nash equilibrium does not have to be jointly best for the players. In a Nash equilibrium, each player chooses her “best response” to the other’s choice. But the two choices are made simultaneously. How can one respond to something that has not yet happened, at least when one does not know what has happened? People play simultaneous-move games all the time and do make choices. To do so, they must find a substitute for actual knowledge or observation of the others’ actions. Players could make blind guesses and hope that they turn out to be inspired ones, but luckily there are more systematic ways to try to figure out what the others are doing. One method is experience and observation—if the players play this game or similar games with similar players all the time, they may develop a pretty good idea of what the others do. Then choices that are not best will be unlikely to persist for long. Another method is the logical process of thinking through the others’ thinking. We introduced the concept of strategic uncertainty. Even when all the rules of a game—the strategies available to all players and the payoffs for each as functions of the strategies of all—are known without any uncertainty external to the game, such as weather, each player may be uncertain about what actions the others are taking at the same time. We can define Nash equilibrium in an alternative and equivalent way: it is a set of strategies, one for each player, such that (1) each player has correct beliefs about the strategies of the others and (2) the strategy of each is the best for herself, given her beliefs about the strategies of the others. This way of thinking about Nash equilibrium has two advantages. First, the concept of “best response” is no longer logically flawed. Each player is choosing her best response, not to the as yet unobserved actions of the others, but only to her own already formed beliefs about their actions. Second, in Chapter 7, where we allow mixed strategies, the randomness in one player’s strategy may be better interpreted as uncertainty in the other players’ beliefs about this player’s action.