  A traditional assumption in game theory is that players are opaque to one
another -- if a player changes strategies, then this change in strategies does
not affect the choice of other players' strategies. In many situations this is
an unrealistic assumption. We develop a framework for reasoning about games
where the players may be translucent to one another; in particular, a player
may believe that if she were to change strategies, then the other player would
also change strategies. Translucent players may achieve significantly more
efficient outcomes than opaque ones.
  Our main result is a characterization of strategies consistent with
appropriate analogues of common belief of rationality. Common Counterfactual
Belief of Rationality (CCBR) holds if (1) everyone is rational, (2) everyone
counterfactually believes that everyone else is rational (i.e., all players i
believe that everyone else would still be rational even if i were to switch
strategies), (3) everyone counterfactually believes that everyone else is
rational, and counterfactually believes that everyone else is rational, and so
on. CCBR characterizes the set of strategies surviving iterated removal of
minimax dominated strategies: a strategy $\sigma_i$ is minimax dominated for i
if there exists a strategy $\sigma'_i$ for i such that $\min_{\mu'_{-i}}
u_i(\sigma_i, \mu_{-i}') > \max_{\mu_{-i}} u_i(\sigma_i, \mu_{-i})$.
