Rational Expectations in Games

Modern game theory was born in 1928, when John von Neumann published his Minimax Theorem. Inter alia, this theorem ascribes to all two-person zero-sum games a value—what rational players should expect to get. Almost 80 years later, strategic game theory has not gotten beyond that initial point, insofar as the basic question of value is concerned. To be sure, we do have equilibrium theories: the initial concept of John F. Nash (1951) and its various refinements and coarsenings. But when the game is not two-person zero-sum, none of these theories actually tells the players what to expect. Even when there is just one Nash equilibrium, it is not at all clear that the players “should” expect its payoff. Can one ascribe a value to each player—what she should expect—in an arbitrary nperson game? As stated, the question has no answer; the problem is underspecified. Formally, a game is defined by its strategy sets and payoff functions. But in real life, many other parameters are relevant; there is a lot more going on. Situations that substantively are vastly different may nevertheless correspond to precisely the same strategic game. For example, in a parliamentary democracy with three parties, the winning coalitions are the same whether the parties each hold a third of the seats in parliament, or, say, 49 percent, 39 percent, and 12 percent, respectively. But the political situations are quite different. The difference lies in the attitudes of the players, in their expectations about each other, in custom, and in history, though the rules of the game do not distinguish between the two situations. Another example revolves around the ultimatum game (Werner Güth, Rolf Schmittberger, and Bernd Schwarze 1982), which, when played in different cultures, leads to systematically different outcomes (Alvin E. Roth et al. 1991). Thus, if one is given only the abstract formulation of a game, one cannot reasonably hope for an expectation. Somehow, the real-life context in which the game is played must be taken into account. We are discussing not just a game, but a “game situation,” i.e., a game played in a specific context; and we should be prepared to let a player’s expectation depend upon the context—the “situation.”