Rationalizability in games with a continuum of players

Abstract

The concept of rationalizability has been used in the last fifteen years to study stability of equilibria in models with a continuum of agents such as competitive markets, macroeconomic dynamics and currency attacks. However, rationalizability has been formally defined in general settings only for games with a finite number of players. We propose an exploration of rationalizability in the context of games with a continuum of players. We deal with a special class of these games, in which payoff of each player depends only on his own strategy and on an aggregate value: the state of the game, which is obtained from the complete action profile. We define the sets of point-rationalizable states and rationalizable states. For the case of continuous payoffs and compact strategy sets, we characterize these sets and prove their convexity and compactness. We provide as well results on the equivalence of point-and standard-rationalizability.

Equilibria of nonatomic anonymous games

“…The scope of this type of results is analyzed in Carmona and Podczeck [9]. Finally, in the same setting of Schmeidler [32], Jara-Moroni [16] extends the notion of rationalizability to nonatomic anonymous games while Rath [28] investigates the issue of existence of perfect, proper, and persistent equilibria, being all of them refinements of Nash equilibrium. three concepts of equilibrium.…”

“…At the same time, given our assumption of anonymity and the neighborhood structure, what is relevant for t is merely the distribution of players' strategies βt , induced by βt , within the subpopulation observed. 16 With this, given δ, ε ≥ 0, we can define recursively the following sequence of sets {S k } k∈N 0 : S 0 = Σ and…”

“…In other words, the family of functions {f t (a, γ, •)} t∈T,a∈A,γ∈∆ from ∆ to [0, ∞) is equicontinuous. 16 Formally, we have that…”