Research On The Existence And Stability Of Cooperative Equilibrium In Population Game

On the one hand,Nash equilibrium and cooperative equilibrium are two important solution concepts in normal-form games,many scholars in the past have made a large number of studies on the existence and stability for these two solutions,and have yielded fruitful results;On the other hand,inspired by Nash's dissertation,the idea of population games has gradually been established.In 2010,Sandholm,the American professor in economics,introduced in detail the model and related theories of population games in his book?Population Games and Evolutionary Dynamics?.Scholars haven't considered cooperative behavior between different populations in their previous studies for population games,therefore,the equilibrium solutions they obtained were still some noncooperative equilibria such as Nash equilibria.In order to improve and complement the existing studies,in this thesis,we introduce the notion of cooperative equilibria for normal-form games into the model of population games,and provide the definition of cooperative equilibria for population games by considering cooperative behavior between different populations.Moreover,we investigate the existence and stability of cooperative equilibria for population games.This thesis is organized as follows.The first part,we first introduce the concept of cooperative equilibria for population games based on the model of S andholm(2010)and prove its existence theorem by Proposition 2 in Ka jii(1992).Using the theorem 2 in Fort(1951),we analyze the generic stability of cooperative equilibria for population games.Moreover,we show the existence of essential components of the cooperative equilibrium set by proving the connectivity of minimal essential sets of the cooperative equilibrium set.The second part,we introduce the model of coalitional population games with infinitely many pure strategies by generalizing the model of population games in S andholm(2010).Moreover,based on the idea of S car f(1971)and Zhao(1999a),we define the notions of NT U(Nontrans f erable Utility)core and T U(T rans f erable Utility)core for coalitional population games.We next prove the existence results for NT U core and T U core.Furthermore,as an extension of the NT U core,we introduce the notion of strong equilibria and prove the existence theorem of strong equilibria.The third part,inspired by S andholm(2010),Y ang & Y ang(2017)and Y ang et.al.(2017),we introduce the model of population games with infinitely many criteria.We first introduce the notions of weakly efficient Nash equilibria and cooperative equilibria for infinite-objective population games.Furthermore,we provide their existence theorems and give the necessary conditions for the existences of weakly efficient Nash equilibria and cooperative equilibria respectively.The fourth part,from the perspective of the bounded rationality with an abstract rational function,we study the stability of cooperative equilibria for population games with finitely many pure strategies.First,by establishing a bounded rationality model with an abstract rational function,we study the stability of NT U core and strong equilibria with respect to payoff perturbations for single-criterion population games;Second,we further investigate the stability of cooperative equilibria with respect to payoff perturbations for population games with infinitely many criteria under a bounded rationality model with an abstract rational function.