Refinements Of Equilibria For Multiobjective Population Games

Population games(PGs)provides a simple and unified framework for describing strategic interactions among large numbers of small anonymous agents.The classical population games describe the rational behavior of agents in a society.Thus,multiobjective population games(MPGs)have attracted much attention.Hence,the refinements of equilibria are theoretically and practically significant for(MPGs).This thesis studies(MPGs),which is better to describe the rational behavior of agents in the society than(PGs).This is mainly devoted to the refinements of equilibria for(MPGs).It consists of six chapters.Chapter 1 provides an introduction of the background,the significance and the current status of this research.Chapter 2 presents some necessary mathematical concepts and related results for the later chapters.In chapter 3,based on the idea of multiobjective or vector optimization problems,we construct the general models of(MPGs),multiobjective potential games and the weighted population games.Furthermore,we introduce the concepts of(weakly)Pareto-Nash equilibrium,properly Pareto-Nash equilibrium and weighted Nash equilibrium.We show that for a given weight combination: if it comprises all strictly positive weight vectors,then a weighted Nash equilibrium is a properly Pareto-Nash one and then a ParetoNash one;and if all positive weight vectors does not equal zero,then a weighted Nash equilibrium is a weakly Pareto-Nash one.Furthermore,from the view of refinements,a properly Pareto-Nash equilibrium is a refinement of a Pareto-Nash equilibrium,and then a Pareto-Nash equilibrium is also a refinement of a weakly Pareto-Nash equilibrium.Chapter 4 deals with the existence of all the above equilibria and their property descriptions for(MPGs).Firstly,it is shown the existence of Nash equilibria by Brouwer’s fixed points for(PGs)with continuous payoff functions.This is a new proof.For(MPGs)with continuous payoff functions,we investigate the existence of all the above equilibria one by one.Secondly,we prove the existence of weighted Nash equilibria by Kakutani’s fixed point theorem,variational inequality and a constructive proof referred to the Nash’s mapping,respectively.Thirdly,based on the existence result of weighted Nash equilibria,we further establish that of(weakly)Pareto-Nash equilibria and properly Pareto-Nash equilibria.Besides,the existence of Pareto-Nash equilibria and that of weakly Pareto-Nash equilibria are obtained by vector-valued variational inequality and Ky Fan’s inequality with vector-valued functions separately.At the end of chapter 4,for multiobjective potential games,by introducing the notions of a strong Kuhn-Tucker(S-KT)state and a weak one(W-KT)of its potential functions,we establish that a(SKT)(or(W-KT))state is a Pareto-Nash equilibrium(or a weakly one).In particular,the converse is also true for multiobjective potential games with 2 strategies.Chapter 5 concentrates on the generic stability of equilibria,which is considered as a refinement of equilibria for(MPGs).Namely,a sensible equilibrium should be stable against slight perturbations of payoff functions and other parameters.Firstly,we verify the stability of weighted Nash equilibria against perturbations of payoff functions and weight combinations.Secondly,the stability of weakly Pareto-Nash equilibria is proven with the perturbed payoff functions.Under the same perturbed case,the stability of Pareto-Nash equilibria is obtained by seeking an upper semi-continuous sub-mapping.Chapter 6 continues the refinements of equilibria for(MPGs).On the one hand,by constructing a rational model with an abstract rational function,we prove the stability of Nash equilibria for(PGs)and that of weighted Nash equilibria for(MPGs),respectively.The latter leads to the existence of stable(weakly)Pareto-Nash equilibria under a certain rational model.Besides,parametric optimization problems is viewed as a general model for(PGs),hence the stability of their solutions is also investigated under two different rational models separately.On the other hand,based on the idea of making mistake,we introduce the notions of(weakly)Pareto perfect equilibria and proper ones.Furthermore,it shows that the former is a refinement of(weakly)Pareto-Nash equilibria and then the latter is a further refinement of the former.Finally,throughout this thesis,we declare that a Pareto proper equilibrium is different from a properly Pareto-Nash equilibrium in their respective backgrounds.