Structure in the Value Function of Two-Player Zero-Sum Games of Incomplete Information

TL;DR

This paper characterizes the structure of the value function in two-player zero-sum partially observable stochastic games, revealing concavity and convexity properties that facilitate solution development and reduction to centralized models.

Contribution

It introduces a new formulation of the value function based on plan-time sufficient statistics and proves its structural properties, advancing the theoretical understanding of zs-POSGs.

Findings

Value function exhibits concavity and convexity in certain marginals.

Reduction of zs-POSGs to centralized models with shared observations.

Potential for developing solution methods exploiting the identified structure.

Abstract

Zero-sum stochastic games provide a rich model for competitive decision making. However, under general forms of state uncertainty as considered in the Partially Observable Stochastic Game (POSG), such decision making problems are still not very well understood. This paper makes a contribution to the theory of zero-sum POSGs by characterizing structure in their value function. In particular, we introduce a new formulation of the value function for zs-POSGs as a function of the "plan-time sufficient statistics" (roughly speaking the information distribution in the POSG), which has the potential to enable generalization over such information distributions. We further delineate this generalization capability by proving a structural result on the shape of value function: it exhibits concavity and convexity with respect to appropriately chosen marginals of the statistic space. This result is a key pre-cursor for developing solution methods that may be able to exploit such structure. Finally, we show how these results allow us to reduce a zs-POSG to a "centralized" model with shared observations, thereby transferring results for the latter, narrower class, to games with individual (private) observations.