Worst-Case Analysis for a Leader-follower Partially Observable Stochastic Game

Abstract

Partially observable stochastic games provide a rich mathematical paradigm for modeling multi-agent dynamic decision making under uncertainty and partial information. However, they generally do not admit closed-form solutions and are notoriously difficult to solve. Also, in reality, each agent often does not have complete knowledge of the other agent. This paper studies a leader-follower partially observable stochastic game where the leader has little knowledge of the adversarial follower's reward structure, level of rationality, and process for gathering and transmitting data relevant for decision making. We introduce the worst-case analysis to the partially observable stochastic game to cope with this lack of knowledge and determine the best worst-case value function of the leader. The resulting problem from the leader's perspective has a simple sufficient statistic; however, different from a classical partially observable Markov decision process, the value function of the resulting problem may not be convex. We design a viable and computationally attractive solution procedure for computing a lower bound of the leader's value function as well as its associated control policy in the finite planning horizon. We illustrate the use of the proposed approach in a liquid egg production security problem.