Existence and stability of Nash equilibrium
The author presents some problems related to existence and stability of Nash equilibria in continuous normal form games. The results discussed in the book are mainly based on the author's works and therefore reflects mainly his understanding of the problems. The book starts with general conditions of the existence of pure Nash strategies in the case of continuous and quasiconvex payoff functions. The line of reasoning is based on a fixed point theorem used in Nash's original paper. Then the existence of Nash equilibria in mixed strategies for non-quasiconcave functions is considered. Stability is understood by the author in the sense of robustness of Nash equilibria to the changes in payoff functions. The Nash equilibrium correspondence mapping games into their Nash equilibria is considered and found to be upper hemicontinuous. Then games with discontinuous payoff functions are considered in the context of better-reply security. The author presents some sufficient conditions for a game to be generalized better-reply secure. Variations and extensions of better-reply security including multiplayer well-behaved security, diagonal transfer continuity, generalized L-security, lower single deviation properly and generalized weak transfer continuity are also discussed. Yet another concept of Nash equilibria considered in the book are the \(\varepsilon\)-equilibria and they are analyzed by two forms of limit results. The existence and stability results are also presented for games with an endogenous sharing rule and once more limit results are used to derived existence theorems. The last class of games considered in the book is defined by continuum of players which interact in an anonymous way. The correspondence between an equilibrium distribution for such game and the Nash equilibria of associated finite-player game is found. As mentioned before the book reflects strongly the author's idea on the problem of existence and stability of Nash equilibria. For example such important notions as evolutionary stable states and strategies are not present in the book at all. Nevertheless a potential reader interested in problems related to properties of Nash equilibria and their applications will be satisfied by rigorous treatment presented in the book.
scientific article; zbMATH DE number 6047071