Robust Design in Game Theory: Bayesian Optimization Approach to Minimax Problems with Equilibrium ConstraintsJian, Jianan (2021) Robust Design in Game Theory: Bayesian Optimization Approach to Minimax Problems with Equilibrium Constraints. Doctoral Dissertation, University of Pittsburgh. (Unpublished) This is the latest version of this item.
AbstractModern engineering systems have become increasingly complex due to the integration of human actors and advanced artificial intelligence, both of which can be interpreted as intelligent agents. Game theory is a mathematical framework that provides an explanatory model for systems constituted of those intelligent agents. It postulates that the apparent behavior of a system is an equilibrium resulting from each agent within the system individually optimizing their own objectives. Thus, designing an intelligent system is to identify a configuration such that its equilibrium is desirable with respect to some external criteria. However, equilibria are often not unique and form sets that lack topological properties on which optimization heavily relies on, e.g., convexity, connectedness, or even compactness in some cases. The unsureness nature, i.e., uncertainty, of equilibria also appeals for another common design criterion: robustness. In this context, a robust design should reach worst-case optimality to avoid sensitivity to the eventual outcome among all possible equilibria. In this dissertation, I incorporate both the game theoretical aspect and the robustness requirement of system design using the formulation of minimax problems with equilibrium constraints. The complexity of the problem structure and the non-uniqueness of potential equilibria require a new solution strategy different from traditional gradient based methods. I propose a Bayesian approach which infers the probabilistic belief of the optimality of a design given sampled objective function values. Due to the anisotropic natural of systems of independent agents, I then revisit the original Kushner’s Wiener process prior instead of radial basis kernel prior despite their popularity for other global optimization applications. I also derive theoretical results on sample maxima and their locations, develop an effective method to decompose the search space into independent regions, and design necessary adaptations to take into account the equilibrium constraints and minimax objective. Finally, I discuss a few applications of the proposed design framework. Share
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