Playing games in many possible worlds

In traditional game theory, players are typically endowed with exogenously given knowledge of the structure of the game--either full omniscient knowledge or partial but fixed information. In real life, however, people are often unaware of the utility of taking a particular action until they perform research into its consequences. In this paper, we model this phenomenon. We imagine a player engaged in a question and- answer session, asking questions both about his or her own preferences and about the state of reality; thus we call this setting "Socratic" game theory. In a Socratic game, players begin with an a priori probability distribution over many possible worlds, with a different utility function for each world. Players can make queries, at some cost, to learn partial information about which of the possible worlds is the actual world, before choosing an action. We consider two query models: (1) an unobservable-query model, in which players learn only the response to their own queries, and (2) an observable-query model, in which players also learn which queries their opponents made.The results in this paper consider cases in which the underlying worlds of a two-player Socratic game are either constant-sum games or strategically zero-sum games, a class that generalizes constant-sum games to include all games in which the sum of payoffs depends linearly on the interaction between the players. When the underlying worlds are constant sum, we give polynomial-time algorithms to find Nash equilibria in both the observable- and unobservable-query models. When the worlds are strategically zero sum, we give efficient algorithms to find Nash equilibria in unobservablequery Socratic games and correlated equilibria in observablequery Socratic games.