Socratic Game Theory: 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 introduce Socratic game theory to 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 information about which of the possible worlds is the actual world, before choosing an action. We consider two query models: (1) an observable query model in which each player knows which query the other players made and (2) an unobservable query model in which each player learns only the response to his or her own query. The results in this paper consider cases in which the underlying worlds of a two-player Socratic game are either constantsum 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 a polynomial-time algorithm to find Nash equilibria in both the observableand unobservable-query models. When the worlds are strategically zero sum, we give efficient algorithms to find Nash equilibria in unobservable-query Socratic games and correlated equilibria in observable-query Socratic games.