Stochastic Games with General Payoff Functions

TL;DR

This paper extends the theory of multiplayer stochastic games by establishing the existence of approximate equilibria and strategies under general payoff functions, using advanced measure-theoretic techniques.

Contribution

It generalizes existing results by proving the existence of various approximate equilibria in stochastic games with broad payoff functions.

Findings

Players can guarantee their maxmin payoffs within epsilon in every subgame.

Existence of epsilon-equilibria in the entire game and subgames.

Strategies can be constructed to approximate minmax and maxmin values.

Abstract

We consider multiplayer stochastic games in which the payoff of each player is a bounded and Borel-measurable function of the infinite play. By using a generalization of the technique of Martin (1998) and Maitra and Sudderth (1998), we show four different existence results. In each stochastic game, it holds for every $\epsilon>0$ that (i) each player has a strategy that guarantees in each subgame that this player's payoff is at least her maxmin value up to $\epsilon$, (ii) there exists a strategy profile under which in each subgame each player's payoff is at least her minmax value up to $\epsilon$, (iii) the game admits an extensive-form correlated $\epsilon$-equilibrium, and (iv) there exists a subgame that admits an $\epsilon$-equilibrium.