On Non-Cooperative Perfect Information Semi-Markov Games

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. To avoid notational complications we took N=2 in our proof.