Solutions for Cooperative Fuzzy Games and Their Application in Exchange Economies

Xia Zhang

Research output: PhD ThesisPhD-Thesis - Research and graduation internal

103 Downloads (Pure)

Abstract

This thesis discusses solutions for cooperative games and exchange economies, giving special attention to fuzziness in these models. Starting with solutions for cooperative games with transferable utility (TU-games), we enrich the model by considering fuzzy payoffs in TU-games, and finally consider fuzzy preferences in a model of an exchange economy. In Chapter 2, we deal with the weighted excesses of players in cooperative games which are obtained by summing up all the weighted excesses of all coalitions to which they belong. We first show that lexicographically minimizing the individual weighted excesses of players gives the same minimal weighted excess for every player. Moreover, we show that the associated payoff vector is the corresponding least square value. Second, we show that minimizing the variance of the players' weighted excesses on the preimputation set, again yields the corresponding least square value. Third, we show that these results give rise to lower and upper bounds for the core payoff vectors. Using these bounds, we define the weighted super core as a polyhedron that contains the core, which is one of the main set-valued solutions for both cooperative games as well as exchange economies. It turns out that the least square values can be seen as a center of this weighted super core, giving a third new characterization of the least square values. Finally, these lower and upper bounds for the core inspire us to introduce a new solution for cooperative TU games that has a strong similarity with the Shapley value. In Chapter 3, we introduce a new approach to measure the dissatisfaction for coalitions of players in cooperative transferable utility games. This is done by considering affine (and convex) combinations of the classical excess and the proportional excess. Based on this so-called alpha-excess, we define new solution concepts for cooperative games, such as the alpha-prenucleolus and the alpha-prekernel. The classical prenucleolus and prekernel are special cases when alpha=0. We characterize the alpha-prekernel by strong stability and the alpha-balanced surplus property. Also, we show that the payoff vector generated by the alpha-prenucleolus belongs to the alpha-prekernel. In Chapter 4, we propose a total order relation of fuzzy numbers based on the expected values of fuzzy numbers. We show that three concepts of the indifference fuzzy core, nucleolus and bargaining sets of cooperative games with fuzzy payoffs are well-defined using this total order relation. Moreover, we obtain that the indifference fuzzy bargaining sets coincide with the indifference fuzzy core for convex cooperative games with fuzzy payoffs. Moreover, we characterize the class of superadditive cooperative games with fuzzy payoffs for which these sets coincide. In Chapter 5, we set up a new fuzzy binary relation on consumption sets to evaluate fuzzy preferences. Besides, we prove that there exists a continuous fuzzy order-preserving function on a reference set for a given fuzzy preference relation. Subsequently, we focus on a model of a pure exchange economy with fuzzy preferences. The existence of a fuzzy competitive equilibrium for a pure exchange economy with fuzzy preferences is shown by using a fixed point theorem. Finally, we show that fuzzy competitive equilibria can be characterized as a solution to an associated quasi-variational inequality, giving rise to an equilibrium solution.
Original languageEnglish
QualificationPhD
Awarding Institution
  • Vrije Universiteit Amsterdam
Supervisors/Advisors
  • van den Brink, Rene, Supervisor
  • Sun, Hao, Supervisor, -
  • Estevez Fernandez, Arantza, Co-supervisor
Award date14 Dec 2021
Publication statusPublished - 14 Dec 2021

Fingerprint

Dive into the research topics of 'Solutions for Cooperative Fuzzy Games and Their Application in Exchange Economies'. Together they form a unique fingerprint.

Cite this