Identifying behaviorally robust strategies for normal form games under varying forms of uncertainty

Abstract

Recent advances in behavioral game theory address a persistent criticism of traditional solution concepts that rely upon perfect rationality: equilibrium results are often inconsistent with empirical evidence. For normal form games, the Cognitive Hierarchy model is a solution concept based upon a sequential reasoning process, yielding accurate characterizations of experimental human game play. These characterizations are enabled by a statistically estimated parameter describing the average number of reasoning steps players utilize. If an arbitrary player were to know this parameter ex ante, they could maximize their expected payoff accordingly. However, given the nature of statistical estimation, such parameter point estimates are unknown prior to experimentation and are susceptible to error afterward. Therefore, we consider the normal form game as a decision problem from the perspective of an arbitrary player who is uncertain of opponents’ reasoning ability. Assuming such a player is confronting a set of boundedly rational opponents whose play is characterized by the Cognitive Hierarchy model, we develop a suite of six mathematical programming formulations to maximize the player’s minimum payoff, and we identify the appropriate formulation for the level of information regarding an opponent population’s reasoning ability. By leveraging robust optimization, stochastic programming, and distributionally robust optimization techniques, our set of models yields prescriptive strategies of play in a normal form game with incomplete knowledge regarding adversary rationality. A software package implementing these constructs is developed and applied to illustrative instances, demonstrating how behaviorally robust strategies vary in accordance with the underlying uncertainty.