Study On Some Topics From The Theory Of Learning In Games

Since the theorization in early 20th century, game theory has made great progress. And. it has been widely applied in many fields, such as military affairs, economics and political science. Nash equilibrium is a central concept in game theory. It means that under the assumption of maximizing payoff, none of rational players will change his strategy unilaterally.The theory of learning in games develops an alternative and plausible explanation for the arising or selection of Nash equilibrium. In order to reach targets, such as best response, least regret and satisfaction, players with bounded rationality modify their strategy according to collected information in each period. As a result, Nash equilibrium is the long-run consequence of the stepwise strategy modification. This dissertation will study some topics from the theory of learning in games.Chapter 1 introduces the elementary knowledge of game theory, Nash equilibrium and the theory of learning in games. And, it presents topics considered in this dissertation.A large class of the trembled adaptive learning dynamics in the theory of learning in games correspond to perturbed Markov chains in mathematics. Chapter 2 applies the theory of large deviations to analyze the intermediate or medium run behavior of the perturbed Markov chains, as the perturbation tends to zero. Decomposing the limit states of the unperturbed Markov chain into hierarchy of cycles iteratively, the most possible order of the perturbed Markov chains traversing across the limits states is studied, as well as the selection of stochastically stable equilibria.Chapter 3 studies the aspiration-based learning dynamics played in symmetric normal-form games at multiple locations. In particular, the aspiration level in one location is linked to the average performances of players in observable locations. With a decentralized information structure, the learning dynamics converge to limit states. For a large class of information structures and games, when there exists a trembling in aspiration levels, the unique stochastically stable equilibrium is achieved with the Pareto efficient symmetric outcome. In the Prisoner's Dilemma, if the probability of the trembling is small enough, both players in each location will ultimately cooperate most of the time.Chapter 4 concludes this dissertation, and it points out some future works in these areas.