A Lattice-Theoretical Optimization Approach to Nash Equilibria in Two-Person Games

Abstract

We propose a functional formulation of Nash equilibrium based on the optimization approach: the set of Nash equilibria, if it is nonempty, is identical to the set of optimizers of a real-valued function. Combining this characterization with lattice theory, we revisit the interchangeability and monotone properties of Nash equilibria in two-person games. We show that existing results on (i) zero-sum games and (ii) supermodular games can be derived in a unified fashion, by the sublattice structure on optimal solutions.

Keywords: Nash equilibrium, optimization, lattice, interchangeability, supermodularity