4. N-Player Games: Rationality and Equilibria

This chapter deals with N-player games (general-sum games). The standard notions of best response, weak and strict dominations are introduced. We then define the concept of rationalizable strategies and prove an existence theorem (Bernheim and Pearce). Next we present the central notion of Nash equilibria. The existence of a mixed strategy Nash equilibrium in finite games is then proved, using Brouwer’s fixed point theorem. Section 4.7 deals with generalizations to continuous games. The existence of an equilibrium is proved under topological and geometrical assumptions, for compact, continuous and quasi-concave games, and the existence of a mixed equilibrium is shown under the topological conditions. Then, the characterization and uniqueness of a Nash equilibrium are presented for smooth games where the strategy sets are convex subsets of Hilbert spaces. Section 4.8 deals with Nash and approximate equilibria in discontinuous games and notably studies Reny’s better-reply security. In Sect. 4.9 we explain that the set of mixed Nash equilibria of a finite game is semi-algebraic and define the Nash components of a game. Lastly, Sect. 4.10 discusses several notions (feasible payoffs, punishment level, threat point, focal point, Nash versus prudent behavior, common knowledge of the game) and Sect. 4.11 proves the standard fixed-point theorems of Brouwer (via Sperner’s Lemma) and Kakutani (for a multi-valued mapping from a convex compact subset of a normed vector space to itself).