Saddle functions and robust sets of equilibria

This paper introduces games with a saddle function. A saddle function is a real valued function on the set of action profiles such that, for one player, minimizing the function implies choosing her best-response, and, for the other players, maximizing it implies choosing their best-responses. We provide a new sufficient condition for robustness to incomplete information of sets of equilibria in a sense of Kajii and Morris (1997, Econometrica), Morris and Ui (2005, J. of Econ. Theory) for games with a saddle function. Our result unifies and generalizes sufficient conditions for zero-sum and best-response potential games.