Large Games and the Law of Large Numbers

This paper introduces discrete large games where the set of players is a countable dense grid with a finitely additive distribution. In these games any function from player names to mixed actions is a legitimate strategy profile. No extraneous continuity or measurability conditions are assumed. Randomness can be modeled explicitly and an exact law of large numbers holds. Equilibria enjoy a strong purification property: every realization of every mixed strategy equilibrium is a pure strategy equilibrium almost surely. Every continuum-player game has a discrete large game representation that preserves the original payoffs, strategy profiles and equilibria. It is argued that strategy profiles in continuum-player games have an ambiguous meaning because measurability requirements force the smoothing out of individual variations. These variations have clear strategic meaning in finite-player games and can be expressed in discrete large games, but not when the set of players is the continuum.