Constrained discounted stochastic games

TL;DR

This paper develops a new framework for analyzing constrained non-cooperative stochastic Markov games with countable states, establishing stationary Nash equilibria under weaker assumptions than previous methods.

Contribution

It introduces a novel approach to define a constrained static game that induces stationary Nash equilibria in Markov games, extending to unbounded costs with weaker conditions.

Findings

Defined a constrained static game for Markov games with bounded costs.

Extended the approach to unbounded costs using approximation and uniform integrability.

Provided weaker assumptions than previous weighted norm methods.

Abstract

In this paper, we consider a large class of constrained non-cooperative stochastic Markov games with countable state spaces and discounted cost criteria. In one-player case, i.e., constrained discounted Markov decision models, it is possible to formulate a static optimisation problem whose solution determines a stationary optimal strategy (alias control or policy) in the dynamical infinite horizon model. This solution lies in the compact convex set of all occupation measures induced by strategies, defined on the set of state-action pairs. In case of n-person discounted games the occupation measures are induced by strategies of all players. Therefore, it is difficult to generalise the approach for constrained discounted Markov decision processes directly. It is not clear how to define the domain for the best-response correspondence whose fixed point induces a stationary equilibrium in the Markov game. This domain should be the Cartesian product of compact convex sets in locally convex topological vector spaces. One of our main results shows how to overcome this difficulty and define a constrained non-cooperative static game whose Nash equilibrium induces by a stationary Nash equilibrium in the Markov game. This is done for games with bounded cost functions and positive initial state distribution. An extension to a class of Markov games with unbounded costs and arbitrary initial state distribution relies on approximation of the unbounded game by bounded ones with positive initial state distributions. In the unbounded case, we assume the uniform integrability of the discounted costs with respect to all probability measures induced by strategies of the players, defined on the space of plays (histories) of the game. Our assumptions are weaker than those applied in earlier works on discounted dynamic programming or stochastic games using so-called weighted norm approaches.