On the existence of pure-strategy equilibria in large games

Over the years, several formalizations and existence results for games with a continuum of players have been given. These include those of Schmeidler [D. Schmeidler, Equilibrium points of nonatomic games, J. Stat. Phys. 4 (1973) 295-300], Rashid [S. Rashid, Equilibrium points of non-atomic games: Asymptotic results, Econ. Letters 12 (1983) 7-10], Mas-Colell [A. Mas-Colell, On a theorem by Schmeidler, J. Math. Econ. 13 (1984) 201-206], Khan and Sun [M. Khan, Y. Sun, Non-cooperative games on hyperfinite Loeb spaces, J. Math. Econ. 31 (1999) 455-492] and Podczeck [K. Podczeck, On purification of measure-valued maps, Econ. Theory 38 (2009) 399-418]. The level of generality of each of these existence results is typically regarded as a criterion to evaluate how appropriate is the corresponding formalization of large games. In contrast, we argue that such evaluation is pointless. In fact, we show that, in a precise sense, all the above existence results are equivalent. Thus, all of them are equally strong and therefore cannot rank the different formalizations of large games.Nash equilibrium Pure strategies Approximation Equilibrium distributions