Some Studies On Nash Equilibrium Problems

Nash equilibrium problem (NEP) is a fundamental concept in game theory and is an important branch in operations research. It has extensive applications in economics, political science, psychology, logic and biology, and many other areas. It has been exten-sively studied from the existence theory, numerical algorithms and applications over the past few decades. On the other hand, some elements may involve uncertain data and each player’s strategy set may depend on the rival players’ strategies in many practical prob-lems. At present the study is in its infancy and still a lot problems worth studying. Hence, in this dissertation, we mainly consider stochastic Nash equilibrium problems, stochastic generalized Nash equilibrium problems, generalized Nash equilibrium problems and gen-eralized Nash equilibrium problems with equilibrium constraints. The main results of this dissertation can be summarized as follows:Chapter2considers a class of stochastic Nash equilibrium problems. Under some mild conditions, we reformulate the stochastic Nash equilibrium problems as a stochastic mixed complementarity problem. We apply the well-known sample average approximation method to solve the stochastic mixed complementarity problem. We further introduce a semismooth Newton method to solve the sample average approximation problems. Com-prehensive convergence analysis is given as well. In addition, we demonstrate the proposed approach on a stochastic Nash equilibrium model in the wholesale gas-oil markets.Chapter3studies a class of stochastic generalized Nash equilibrium problems. Under some mild conditions, we reformulate the stochastic generalized Nash equilibrium prob-lems as a stochastic mixed complementarity problem. We apply the well-known sample average approximation method to solve the stochastic mixed complementarity problem. Even under very restrictive assumptions, since multiplier is typically unbounded such that an accumulation point not exists. We further introduce an interior point method to solve the sample average approximation problems. Comprehensive convergence analysis is given as well. We demonstrate the proposed approach on a stochastic generalized Nash equilibrium model in river basin pollution game.Chapter4studies a class of generalized Nash equilibrium problems which is sepa- rable with positive weights, and solve its normalized equilibrium points and normalized stationary points. We reformulate this kind of generalized Nash equilibrium problems to a standard optimization problem by imposing some additional conditions. In addition, we demonstrate the proposed approach on oligopoly competition model in similar products market.Chapter5studies a class of generalized Nash equilibrium problems with equilibrium constraints which is completely separable, and solve its normalized stationary points where the multipliers of the leaders on the shared constraints are proportionable. We reformulate this kind of generalized Nash equilibrium problems with equilibrium constraints to a stan-dard Mathematical programs with equilibrium constraints. In addition, we demonstrate the proposed approach on Multi-Leader-Follower games in similar products market.