Equilibrium existence and topology in some repeated games with incomplete information

Abstract:

This article proves the existence of an equilibrium in any infinitely repeated, un-discounted two-person game of incomplete information on one side where the uninformed player must base his behavior strategy on state-dependent information generated stochastically by the moves of the players and the informed player is capable of sending nonrevealing signals. This extends our earlier result stating that an equilibrium exists if additionally the information is standard. The proof depends on applying new topological properties of set-valued mappings. Given a set-valued mapping $F$ on a compact convex set $P\subset \mathbb {R} ^n$, we give further conditions which imply that every point $p_0\in P$ belongs to the convex hull of a finite subset $P _0$ of the domain of $F$ satisfying $\Cap _{x\in P_0} F(x)\ne \emptyset$.