On Non-Cooperative Perfect Information Semi-Markov Games

TL;DR

This paper proves the existence of pure semi-stationary Nash equilibria in N-person non-cooperative semi-Markov games under limiting ratio average pay-off, using reduction to semi-Markov decision processes and implementing an algorithm for solutions.

Contribution

It extends the existence of pure Nash equilibria to N-person semi-Markov games and provides a reduction method to solve these games via semi-Markov decision processes.

Findings

Existence of pure semi-stationary Nash equilibrium in N-person semi-Markov games.

Reduction of semi-Markov games to semi-Markov decision processes for solution.

Implementation of Mondal's algorithm to solve the reduced SMDP.

Abstract

We show that an N-person non-cooperative semi-Markov game under limiting ratio average pay-off has a pure semi-stationary Nash equilibrium. In an earlier paper, the zero-sum two person case has been dealt with. The proof follows by reducing such perfect information games to an associated semi-Markov decision process (SMDP) and then using existence results from the theory of SMDP. Exploiting this reduction procedure, one gets simple proofs of the following: (a) zero-sum two person perfect information stochastic (Markov) games have a value and pure stationary optimal strategies for both the players under discounted as well as undiscounted pay-off criteria. (b) Similar conclusions hold for N-person non-cooperative perfect information stochastic games as well. All such games can be solved using any efficient algorithm for the reduced SMDP (MDP for the case of Stochastic games). In this paper we have implemented Mondal's algorithm to solve an SMDP under limiting ratio average pay-off criterion.