Mathematical Game Theory

Game theory is a branch of mathematics where the interactions between rational decision makers (players) are modelled and analysed. Game theory can broadly be divided into the study of noncooperative games and cooperative games. For noncooperative games we study two-player games, games in extensive form, games of perfect and imperfect information, games with complete and incomplete information, games with chance moves, repeated games, and Bayesian games. To analyse these games we introduce the concepts of Nash equilibria, evolutionary stable strategies, subgame perfect equilibria, and Harsanyi games. For cooperative games we study coalitional games with transferable utility, and introduce the concepts of coalitions, characteristic functions, the core, the Shapley value, and the nucleolus. We discuss in detail the well known Bonderava-Shapley theorem which gives conditions for the nonemptyness of the core. This subject provides a rigorous mathematical treatment of game theory, and will include applications selected from queueing theory, biology, population dynamics, resource allocation, auction theory, political science, and military applications.