Abstract
Game Theory is the study of rational behavior in a multi-player environment, which we call a game. The players of the game are rational entities that aim to maximize their expectations from this interaction by properly choosing their actions. As those strategic interactions increase in frequency, and the players are learning from their past experience their behavior reassembles a dynamical system, which we call game dynamics. A Nash equilibrium is a game outcome in which no player has an incentive to unilaterally deviate from their chosen action in order to improve their own expected gain. In this thesis, we are going to examine the problem of equilibrium selection from the prism of game dynamics, i.e., the identification of optimal Nash equilibria from the perspective of the players or the game designer. We are going to concern ourselves with questions like: (a) How do we compare the performance of different game dynamics? (b) Are there game dynamics that are optimal for a given metric for all games in some class? (c) Can we identify how players are learning from past experience, i.e., can we learn the game dynamics? (d) Can we design game dynamics that lead to optimal outcomes even in highly complex games? In our attempt to answer these questions, we are going to define an average performance metric that can measure the optimality of different game dynamics, and we are going to establish conditions that dictate the relative performance of well-known dynamical systems. Furthermore, we are going to define a framework that can discover the learning behavior of the players from a limited number of observations and knowledge about their strategic intricacies. Finally, we are going to define a class of highly complex games that hide a monotone structure, which we are going to exploit through a preconditioning scheme to construct game dynamics that converge to the game’s equilibrium points at optimal rates.