The Theory of Games

Abstract

We begin with an exposition on the theory of noncooperative games. The strategic form game representation is introduced first, followed by selected equilibrium concepts based on the Nash equilibrium and some of its refinements like its undominated variant, the Coalition-Proof and the Strong Nash equilibrium. The second part on noncooperative games covers their representation in extensive form, where we introduce another variant, the subgame perfect Nash equilibrium Next comes the consideration of cooperative games. We first define the notion of a cooperative game as compared to a noncooperative one and introduce the characteristic function which assigns the gains from cooperation to each possible coalition of players. Depending on whether there even exist overall gains from cooperation and how these gains specifically arise and change within and over different coalitions, we can classify cooperative games and assign certain properties to them. The last part of this chapter is devoted to some of the most prominent solution concepts for cooperative games. These concepts determine how the gains from cooperation are or can be distributed among the players and are generally either set- or point-valued. We begin with the classic von Neumann Morgenstern solution to which we relate the concept of the core. There, we show different conditions under which, regarding the underlying game, a solution in terms of the core exists. The section on point-valued solutions, also called allocation rules, starts with a general definition and some common properties of these rules and how the latter respond to changes to the game they are based on. The allocation rule we cover in detail is the Shapley value, followed by its weighted variant, which again is related to the core. The chapter concludes with a brief treatment of the so-called bargaining problem and corresponding bargaining solutions of Nash and Kalai–Smorodinsky. With this we include an alternative approach to allocate the gains from cooperation among a number of players.