Generalized Equilibrium Results for Games with Incomplete Information

Abstract

Milgrom and Weber (Milgrom, P. R., Weber, R. J. 1985. Distributional strategies for games with incomplete information. Math. Oper. Res. 10 619–632.) gave an existence result for a Nash equilibrium in a game with incomplete information, using their notion of a distributional strategy. Here we obtain a substantial improvement of their existence result in terms of the more traditional concept of a behavioral strategy. This improvement is reached very naturally as an application of a theory of weak convergence for transition probabilities, which is recapitulated extensively in this paper. Also, a new result on the weak convergence of product transition probabilities is included.

Equilibrium Points of Non-Cooperative Random and Bayesian Games

“…Balder [3], Mas-Collel [14], Milgrom-Weber [16] and Radner-Rosenthal [23] have provided existence of equilibrium theorems for games with incomplete information, but their approach is different than ours. We will discuss the work of the above authors in Section 5.…”

“…We now follow the original formulation by Nash [18] (and his generalizations by Fan [6] and Browder [3] among others) where random preference correspondences are replaced by random payoff functions, i.e., real valued functions defined on Q x X. …”

“…We will take up these details, however, in a subsequent paper. [3], Milgrom-Weber [16] and Radner-Rosenthal [23].…”

“…It is important to note that assumption (b) allows the above authors to express the expected utility (4.1) in a convenient way (see [16, p. 625] or [3]). …”

“…In particular, once distributional strategies are topologized with weak convergence, strategy sets are compact metric spaces, expected utility is continuous and linear and therefore the standard results of either Glicksberg, Fan or Browder (see [16] or [3] for a statement of this result) can be directly applied to prove the existence of an equilibrium. We would like to note that in our framework the expected utility is required to be only quasi-concave in each player's own strategy.…”