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.