Two-player stochastic games I: A reduction

Abstract

This paper is the first step in the proof of existence of equilibrium payoffs for two-player stochastic games with finite state and action sets. It reduces the existence problem to the class of so-called positive absorbing recursive games. The existence problem for this class is solved in a subsequent paper

On Nash Equilibria in Stochastic Games

“…Our construction is based on a general construction from repeated games. The basic idea is that player i plays optimal strategies in the zero sum game against all other players, and any deviation by player i is punished indefinitely by the other players by playing -optimal spoiling strategies in the zero sum game against player i (see, e.g., [19,28] Note that the construction above for probabilistic Borel games guarantees only -optimality. Example 1 shows that there are two player turn-based probabilistic zero sum games where only -optimal strategies exist, and optimal strategies do not exist.…”

“…For limit average payoffs, exact Nash equilibria need not exist even for two-player games [1]. Recently the existence of -Nash equilibria for all > 0 was proved in [28,29] for the two-player case, and the general case remains an important open question. For games with payoffs defined by Borel sets, surprisingly little is known.…”

Communication, correlation and cheap-talk in games with public information

“…Vieille (2000aVieille ( , 2000b proved that any two-player undiscounted stochastic game (with a nite number of states) admits an equilibrium payo (without any communication). Whether this holds true for stochastic games with more than two players is an open problem.…”

Subgame-Perfection in Quitting Games with Perfect Information and Differential Equations

“…Depending on the exact class of games that is studied, one constructs a nonstationary -equilibrium in which players play mainly the limit stationary strategy profile, and perturb to other actions with small probability, while monitoring the actions of their opponents to detect deviations. This approach, which was initiated by Mertens and Neyman [12] to prove the existence of the uniform value in two-player zero-sum stochastic games, was later exploited in numerous studies (see, e.g., Vrieze and Thuijsman [30]; Flesch et al [5]; Thuijsman and Raghavan [26]; Solan [20]; Vieille [28,29]; Rosenberg and Vieille [14]; Rosenberg et al [15,16]; Solan and Vohra [24]). …”