Weighted Discounted Stochastic Games with Perfect Information

We consider a two-person zero-sum stochastic game with an innnite time horizon. The payoo is a linear combination of expected total discounted rewards with diierent discount factors. For a model with a countable state space and compact action sets, we characterize the set of persistently optimal (sub-game perfect) policies. For a model with nite state and action sets and with perfect information, we prove the existence of an optimal pure Markov policy, which is stationary from some epoch onward, and we describe an algorithm to compute such a policy. We provide an example which shows that an optimal policy, which is stationary after some step, may not exist for weighted discounted sequential games with nite state and action sets and without the perfect information assumption. We also provide examples of similar phenomena of nonstationary behavior for the following two classes of problems with weighted discounted criteria: (i) models with one controller and with nite state and compact action sets, (ii) nonzero-sum games with perfect information and with nite state and action sets.