Existence of Equilibrium in Bayesian Games with Infinitely Many Players

Differential information in large games with strategic complementarities

“…In the paper, we study large games with differential information as in Kim and Yannelis (1997), but with strategic complementarities. For our games, we begin by considering the question of existence and characterization of distributional Bayesian Nash equilibria and then turn to Bayesian Nash equilibria in the sense of Schmeidler (1973).…”

“…Since the seminal papers of Schmeidler (1973) and Mas-Colell (1984), on equilibria in games with continuum of players, as well as their various generalizations including games with incomplete information in the tradition of Harsanyi (1967), and games with differential information in the tradition of Balder and Rustichini (1994) and Kim and Yannelis (1997), the framework of large games has become of the central interest in both game theory and economics. The technical and conceptual issues raised in the extensions of the Schmeidler/Mas-Colell frameworks to the incomplete information or differential games raise few technical issues.…”

“…1 For example, in large games with incomplete information, exact laws of large numbers (ELLN) for a continuum of random variables are typically used (e.g., see Feldman and Gilles 1985;Judd 1985 for an early discussion of this technical issue, as well as Alós-Ferrer 1998). 2 Alternatively, Balder and Rustichini (1994) and Kim and Yannelis (1997) consider games with differential information, but still the conditions for the existence of equilibria are quite distinct from the finite number of players case. In such games with differential information, only a single state of the game is drawn, but it is observed by every player with respect to a private sub σ -field that can differ across agents and, hence, characterizes the private information structure in the game.…”

“…In this paper, we provide a unified set of results concerning the existence, comparison, and computation of Bayesian Nash equilibria in a broad class of large games with differential information in the spirit of Balder and Rustichini (1994) and Kim and Yannelis (1997). 5 As we focus on the subclass of large games with differential information that also possess strategic complementarities, we extend the existing literature on GSC with a finite number of players (e.g., Athey 2002, 2001or Reny 2011Vives and Zandt 2007) to a settings with a continuum of players.…”

“…8 In our definition of equilibria, distributional equilibria are not equivalent to equilibria in the sense of Schmeidler (1973). In particular, in the case of equilibria in the sense of 1973, we require all agents to behave optimally (as e.g., in Balder andRustichini 1994 or Kim andYannelis 1997). See our discussion later in the paper for the importance of this difference.…”