Multi-Objective Optimization Approach to Pure Strategy Equilibria in Games With Imperfect Information

Without the expected utility hypothesis, games with imperfect information can be understood as coupled multi-objective optimizations. Within this framework, equilibria of non-dominated responses are defined and their existence is studied. One class of these equilibria is formulated as Nash equilibria of normal-form games. Another class of these equilibria is formulated with payoff constraints interpreted as robustness of the games. A concrete example is presented and solution methodologies are discussed.

I. Introduction

In control theory and game theory, the information structure of a problem has a decisive effect on the properties of the solution [1], [2]. In practice, agents often receive discrete information about the state of the system, such as classification results or answers to yes-no questions. Discretization of information also facilitates computation in machine learning [3]. This kind of information is typically modeled with information sets in game theory. However, the studies of information sets are usually limited in finite or countable state spaces [4], whereas uncertain information in more general state spaces such as Euclidean spaces are typically modeled with random variables using probability theory. The probabilistic approach makes strong assumptions about how information is perceived and processed by agents, particularly the expected utility hypothesis. This letter studies information sets in general topological state spaces by proposing a solution concept called non-dominated equilibria. This novel framework also incorporates the idea of robustness in games.