Equilibrium Points of Non-Cooperative Random and Bayesian Games

Abstract

: We provide random equilibrium existence theorems for non-cooperative random games with a countable number of players.Our results give as corollaries generalized random versions of the ordinary equilibrium existence result of Nash [18]. Moreover, they can be used to obtain equilibrium existence results for games with incomplete information, and in particular Bayesian games.In view of recent work on applications of Bayesian games and Bayesian equilibria, the latter results seem to be quite useful since they delineate conditions under which such equilibria exist.

On the Existence of Nash Equilibrium in Bayesian Games

“…In this framework, a strategy is an A i -measurable map from Ω into actions. Yannelis and Rustichini [79] prove the existence of Bayes-Nash equilibrium in the state-space setup assuming that the state space is a measurable space, payoffs are jointly measurable in state-action profiles, and continuous in action profiles. In a state space model with countably many states, He and Yannelis [34] provide existence results for discontinuous payoffs satisfying two variations of Monteiro and Page's [51] uniform payoff security.…”

Equilibria in Random and Bayesian Games with a Continuum of Players

“…Hence, it follows from the Fan-Glicksberg fixed point theorem [Fan (1952), Glicksberg (1952) Balder (1988), Harsanyi (1967), MilgromWeber (1985) and Yannelis-Rustichini (1988 …”

Equilibrium points of random generalized games

“…Kim et al [12] also proved the existence of equilibria in abstract economy with measure space of gents and with an inllnite-dimensionM strategy spce by random fixed point theorems. In particular, Tan and Yuan [21] and Yannelis and Rustichini [29] [7] in deterministic case; and if in addition, A B for each E I, our definition of an equilibrium point coincides with the standard definition in the deterministic case; e.g., see Borglin and Keiding [3], Tulces [27] It is e that eve coespondence f 8s Ls, is Ls,-majoed. We note that our notions of the coespondence bng of 8s Ls, d Ls,-majoed coespondence genere the concepts of correspondence of ses L; d ;-msjozed troduced by D et [7], w t genere the notions of G C(X, Y, 0) d C-majored coespondence rpectively troduced by cea [27] A measurable space (f,E) is a pair where f is a set and E is a r-algebra of subsets of f. If X is a set, A C X, and :D is a non-empty family of subsets of X, we shall denote by/P N A the family {D N A D G 9} and by rx(:D) the smallest r-algebra on X generated by/P.…”