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. © 2008 Elsevier Inc. All rights reserved. JEL classification: C7; C6; H41 E-mail address: [email protected]. URL: http://www.kellogg.northwestern.edu/faculty/alnajjar/htm/index.htm. 0899-8256/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.geb.2007.11.002 2 N.I. Al-Najjar / Games and Economic Behavior 64 (2008) 1–34 “All models are wrong; some are useful.”1