Ichiro Nishizaki

A lexicographical solution concept in an n -person cooperative fuzzy game

Abstract In this paper, a new solution concept in an n -person cooperative fuzzy game with sidepayment is proposed. We first define an excess with respect to a player in a fuzzy game with sidepayment. Usually, the concept of an excess is defined for each coalition. In contrast however, we define an excess of a player by summing up all the excess of a coalition to which he belongs. By considering a payoff vector which minimizes the excess of a player in the lexicographical order, we proposed a new solution concept in a fuzzy game as well as in a nonfuzzy game. Then, we further consider the problem to extend a nonfuzzy game into a fuzzy game. Such an extension can be described by the mappings from a characteristic function in a nonfuzzy game into a characteristic function in a fuzzy game. An illustrative numerical example of the proposed solution concept to some extended fuzzy games is also presented.

Max-min solutions for fuzzy multiobjective matrix games

Abstract In this paper we consider two-person zero-sum games with fuzzy multiple payoff matrices. We assume that each player has a fuzzy goal for each of the payoffs. A degree of attainment of the fuzzy goal is defined and the max-min strategy with respect to the degree of attainment of the fuzzy goal is examined. If all of the membership functions both for the fuzzy payoffs and for the fuzzy goals are linear, the formulated mathematical programming problem which yields the max-min strategy can be reduced to the linear programming problem by making use of Sakawas method, the variable transformations, and the relaxation procedure.

Equilibrium solutions for multiobjective bimatrix games incorporating fuzzy goals

Equilibrium solutions in terms of the degree of attainment of a fuzzy goal for games in fuzzy and multiobjective environments are examined. We introduce a fuzzy goal for a payoff in order to incorporate ambiguity of human judgments and assume that a player tries to maximize his degree of attainment of the fuzzy goal. A fuzzy goal for a payoff and the equilibrium solution with respect to the degree of attainment of a fuzzy goal are defined. Two basic methods, one by weighting coefficients and the other by a minimum component, are employed to aggregate multiple fuzzy goals. When the membership functions are linear, computational methods for the equilibrium solutions are developed. It is shown that the equilibrium solutions are equal to the optimal solutions of mathematical programming problems in both cases. The relations between the equilibrium solutions for multiobjective bimatrix games incorporating fuzzy goals and the Pareto-optimal equilibrium solutions are considered.

Interactive support for fuzzy trade-off evaluation in group decision making

Abstract In this paper, we are concerned with constructing interactive support systems for group decision making using fuzzy utility evaluation. In particular, fuzzy indifference experiments for articulating diversified value trade-offs in collective choice are discussed in the multiobjective environments. The fuzzy indifference curve is derived along with the fuzzy scaling constants for single-attribute utility functions. The intelligent decision support systems (IDSS) are used for supporting this process. With this device, some notions for heuristics construction of the fuzzy multiattribute utility function are suggested.

The Nucleolus in Multiobjective n-person Cooperative Games

In this paper, multiobjective n -person cooperative games are defined, and the nucleolus is considered in such games. First, a multiobjective game is reduced to single-objective games by using the scalarizing methods of multiobjective programming, i.e., the weighting coefficients method and the weighted minimax method. Second, the nucleolus is defined directly in a multiobjective n-person cooperative game. Computational methods for deriving the nucleolus are shown when excess functions are defined as distance from the ideal point or a Pareto optimal point.

N-Person Cooperative Games with Multiple Scenarios

In this paper, solution concepts for an n-person cooperative game with multiple scenarios are considered. A characteristic function of the n -person cooperative game with multiple scenarios associates a subset of the set of all players with its real vector value. We consider the nucleolus, which is one of solution concepts based on the lexicographical framework in an n -person cooperative game, in the game with multiple scenarios. To define extended nucleoli, three aggregation methods using weighting coefficients, a minimum component, and constraints are employed. The computational methods for the three extended nucleoli are developed by repeatedly solving linear programming problems. Finally, a numerical example illustrates the proposed methods.

Multiobjective n-person cooperative games

In this chapter, we investigate multiobjective n-person cooperative games. First of all, as a topic related to the multiobjective games, we consider n- person cooperative games under fuzziness, uncertainty or risk. To introduce fuzziness, uncertainty or risk into a cooperative game, we define a mapping which associates a coalition with a fuzzy set or a probability distribution, instead of the characteristic function. A representation of cooperative games in which a coalition value is represented as a random variable has already been studied by Charnes and Granot [29, 30] . The cooperative game under uncertainty is formally represented as a cooperative game with vector-valued characteristic function. Such games were also investigated by Bergstresser and Yu [11] in the study on multiobjective games. They called such games multiobjective cooperative games but we think it would be appropriate to call them cooperative games with multiple scenarios or cooperative games under uncertainty. Bergstresser and Yu mainly considered the core defined by the domination structures and referred to a couple of solution concepts which yield a unique solution such as the nucleolus in n-person cooperative games. Sakawa and Nishizaki considered the nucleolus in n-person cooperative games with multiple scenarios [121].

Lexicographical Solutions in N-Person Cooperative Games with Multiple Scenarios

In this paper, lexicographical solution concepts in an n-person cooperative game with multiple scenarios are considered. A characteristic function of the n-person cooperative game with multiple scenarios associates a subset of the set of all the players with its real vector value. Fuzziness, uncertainty or risk is incorporated in the cooperative games, and especially, we examine the cooperative games in which the values of the coalitions have discrete distributions. In such games, the nucleolus and its related lexicographical solutions are defined and the computational methods for obtaining the solutions are developed.

A Cooperarive Fuzzy Games for International Conflict Solving

This paper concerns an effective formation of an international concord for international conflict solving under the fuzzy decision environments. For treating this problem, an n-person cooperative fuzzy game in characteristic function form is constructed, where the characteristic function is assessed as the fuzzy number embodying diversified evaluation. The nucleolus as the solution concept of the game is derived also in fuzzy terms by solving a fuzzy linear programming problem which comes to formulate a parametric programming problem

A Solution Concept in Multiobjective Matrix Games with Fuzzy Payoffs and Fuzzy Goals

In this paper we consider multiobjective matrix games with fuzzy payoffs and fuzzy goals. Games considered here and conventional matrix games differ by the following three points. First, we employ representing entries of a payoff matrix as fuzzy numbers for expressing ambiguity and imprecision in information which is utilized in modeling of games because such information is not always accurate. Second, multiple payoffs are considered in games because most of decision making problems under conflict possess multiple objectives such as cost, time and productivity. Third, we assume that each player has a fuzzy goal for each objective in order to incorporate ambiguity of human judgment.