Research On Solutions To Two-type Games And Related Game Problem

The biform game is a two-stage game model.In the first stage,the players are involved in a noncooperative game,and in the second stage,the players are involved in the cooperative games.It is widely used in various fields,playing an important role in particular in the applications of corporate asset integration,supply chain networks,and bilateral link formation networks.If there is no pre-determined noncooperative game for the players in the first stage,the biform game can also be considered as a more general game related to the strategic choices and coalition profits of the players,referred to as the “n-person strategic-coalition form” game.The change of coalition profits on strategy combinations(coalition function)makes us pay attention not only to the allocation of outcomes,but also to the impact of strategy choice on coalition profits.Therefore,for the solution of n-person strategic-coalition form game,how to determine the characteristic function of the coalitional game by the coalition function? How do players rationally choose their strategies? All of these have become critical research topics in the study ofn-person strategic-coalition form games.In this thesis,based on the coalition function,the payoff function of players is generated through the allocation of coalition profit,and then the solution is obtained by analyzing the noncooperative game,or the characteristic function of the coalitional game is constructed through the behavior of players in coalition and complementary coalition-,and then the solution is analyzed from the perspective of a cooperative game.Moreover,the stability of the solution is further investigated.The main research contents of this thesis are the following.1.Nash equilibria of the biform game.Starting mainly from the coalition function,the payoff function of the players is formed by allocating the coalition profits through the Shapley value and Owen value.The solutions of the biform game are given in terms of the Nash equilibria of the noncooperative game,and the stability of the solutions is discussed.Include:(1)Nash Equilibria based on Shapley value.In response to the problem that the core in the cooperative stage of the biform game may be an empty set and the pure strategy Nash equilibrium may not exist,the mixed strategy is introduced to define the expected coalition function and the expected Shapley value allocation function,to establish the biform game based on the Shapley value,and the game solution is given by the existence of Nash equilibrium under the mixed strategy.The Shapley value allocation function is discussed axiomatically in terms of collective rationality,anonymity,linearity,and dummy player property.(2)Nash equilibrium based on Owen value.Mainly in strategy-related supply chains,for the problem that members and coalitions are restricted by order,precedence constraint is introduced to characterize the order relationship between precedence coalitions and their internal members,and the investment strategies of supply chain members are applied to define the coalition function,thus defining the Owen value allocation function,and an n-person noncooperative game model based on Owen value is established,which gives the solution of the biform game through efficient Nash equilibrium.An axiomatic discussion of the Owen value allocation function is made in terms of linearity,collective rationality,and hierarchical strength.(3)Stability of Nash equilibrium set based on Shapley value and Owen value.In view of the lack of stability research of the biform game Nash equilibrium,the stability of n-person strategic-form game Nash equilibrium is extended to the biform game by using nonlinear analysis and stability research method,and the generic stability and essential component of Nash equilibrium set based on Shapley value and Owen value are obtained.Based on the utility function,rationality function,and rationality requirement,the generic stability of the bounded rationality model of the biform game is studied,and the equivalent relation between the essential bounded rationality Nash equilibria and the structural stability of the biform game is obtained.2.Characteristic function and cooperative solution of n-person strategic-coalition form game.Based on the coalition function,the characteristic function is defined and the cooperative solution is given by the cooperative game approach.Include:(1)Cooperative solution based on the characteristic function of minmax representation.To obtain cooperative solutions for n-person strategic-coalition form games,based on coalition function,the minmax characteristic function of n-person strategic-form game(i.e.normal-form game)is extended to n-person strategic-coalition form game,and the cooperative game model in the form of characteristic function is established and the cooperative solution is given.The stability of the core is given by the nucleolus.(2)Cooperative solution based on the characteristic function of maxmax defenseequilibrium representation.The main purpose is to address the problem that there are multiple Nash equilibria between complementary coalitions and-,and the characteristic function value cannot be obtained from the defense-equilibrium representation,the characteristic function value of coalition is obtained by considering the “maxmax” method in which the complementary coalitions and- successively maximize their utilities on the Nash equilibrium set,so as to build a cooperative game model in characteristic function form to give the cooperative solution.(3)Cooperative solution based on Owen value.For the coalition structure where the players are restricted by the precedence constraint,a cooperative game model in the characteristic function form is developed,and the restricted Owen value under the precedence constraint is used as the cooperative solution.In order to address the problem of the lack of individual rationality of the core under the precedence constrained coalition structure,we discuss the internal and external stability of the precedence coalitions based on the utility gap.3.Nash equilibrium of an n-person strategic-form game under the biform game analysis(the n-person game is a special form of n-person strategic-coalition form game).For an n-person strategic-form game with transferable utility(side payment),it is transformed into a biform game by introducing a transferable utility vector and a bottom-line value vector.The refining problem of Nash equilibria in the n-person noncooperative games is discussed by using biform game analysis.Include:(1)Nash equilibrium for the n-person strategic-form game with transferable utility.The transfer utility vector is defined on each strategy profile,the transfer utility game based on the bottom-line value on each strategy profile is considered as the cooperative game stage of the biform game analysis,and the utility function after the transferring utility forms an n-person noncooperative game whose Nash equilibrium is the solution of the biform game analysis.Based on the transfer utility vector and the bottom-line value vector,we discuss the satisfaction stability of players under biform game analysis by defining the satisfaction oscillation value and the satisfaction stability.(2)Nucleolus-rationality Nash equilibrium of the n-person noncooperative game.For the multiplicity of Nash equilibria,the biform game analysis method is applied.In the first stage,the players choose strategies noncooperatively to generate a set of Nash equilibria.In the second stage,based on the fair principle of the nucleolus and selection method,nucleolus-rationality Nash equilibrium is selected from the Nash equilibrium set as the solution of n-person noncooperative game.Based on the utility dominance principle and the fairness dominance principle,the nucleolus-rationality Nash equilibrium is stable.