Differential Information in Large Games with Strategic Complementarities

We study equilibrium in large games of strategic complementarities (GSC) with a differential information and continuum of players. For our game, we define an appropriate notion of distributional Bayesian-Nash equilibrium in the sense of Mas-Colell (1984), and prove its existence. Further, we characterize the order-theoretic properties of the equilibrium set, provide monotone comparative statics results for ordered perturbations of the space of games, and provide explicit algorithms for computing extremal equilibrium. Our results make extensive use of the recent results on aggregating single crossing properties in Quah, Strulovici (2012). We complement these with new results on the existence of Bayesian-Nash equilibrium in the sense of Balder, Rustichini (1994) or Kim, Yannelis (1997) for large GSC, and provide analogous results for this notion of equilibrium. To obtain our results, we prove an new fixed point theorem on monotone operators in countably complete partially ordered sets. Applications of the results include riot games, "beauty contests" and common value auctions