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 \emph{a priori} probability distribution over many \emph{possible worlds}, with a different utility function for each world. Players can make \emph{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 \emph{unobservable-query} model, in which players learn only the response to their own queries, and (2) an \emph{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 \emph{constant-sum games} or \emph{\affine-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 unobservable-query Socratic games and correlated equilibria in observable-query Socratic games.