Analysis of partially observed asymmetric zero-sum stochastic games

Abstract

Shapley introduced stochastic games in [1]. After this pioneering work a large number of papers were written on this topic. A good survey on zero-sum stochastic games can be found in Vrieze [2]. Stochastic games generalize Markov decision processes, in the sense that the latter may be treated as one player stochastic games. Most of the available literature in this area falls into the category of stochastic games with complete state information, that is, at each stage the state of the game is completely known to the players. Though there is considerable literature on Markov decision processes with partial information (see, e.g., Bertsekas and Shreve [3], Dynkin and Yushkevich [4], Hernandez-Lerma [5]), the corresponding results on stochastic games seem to be rather sparse. A two person zero-sum stochastic game with partial observation on general (uncountable) state and action spaces with discounted payo was studied in Ghosh et. al. [6]. But a general theory on stochastic games with partial observation do not seem to be available in the literature. In this work, we propose to study the existence of value and saddle-point equilibria of a two-person zero-sum stochastic game when one of the players (say, minimizer) observes only a subspace of the total space of observations while the other (say, maximizer) has a full observation. Such a informational asymmetry is relevant in various business problems when one of the agents in the market has informational advantage i.e., when one is playing against the market maker. Such an example has come up recently in one of our projects with a large corporate company dealing in procurement and processing of certain commodities. To the best of our knowledge the study of such games under complete information asymmetry is novel and not much studied in the literature.