A method to compare influence of coalitions on group decision other than desirability relation

Abstract

This paper proposes a method to compare influence of coalitions on group decision, called blockability relation. This relation is defined both on the set of all coalitions and on the set of all feasible coalitions. Examples show that this relation is definitely different from the desirability relation, which have been dealt with in the literature. In particular, an example shows that the desirability relation is not always transitive, and a proposition verifies that the blockability relation is always transitive. In order to clarify the simple games on which these relations are defined and to investigate interrelationships among these simple games, three types of operators, called L-intersection, L-restriction and extension, respectively, on simple games are defined, where L is a family of all feasible coalitions. It is verified that the extension of the L-intersection of a simple game coincides with the restriction of the simple game.

Introduction

Desirability relations [2], [6], [8] are in the most important methods in the literature to compare influence of coalitions in a group decision making situation. In the framework of simple games [4], [5], [6], Einy [2] discussed the comparability of a desirability relation and noted that the desirability relation is comparable when the simple game is symmetric [3], [9]. Taylor and Zwicker [6] comprehensively treated desirability relations also in the simple game framework. Yamazaki et al. [8] dealt with the comparability of a desirability relation in the framework of voting committees [3] with the consideration of the permission of voters [7]. Yamazaki et al. [8] also proposed a method called a hopefulness relation mainly for comparing winning coalitions, while the desirability relations are, in effect, for comparing non-winning coalitions.

This paper concentrates on proposing new methods to compare, in effect, non-winning coalitions in the framework of simple games. The methods are called the blockability relations and are defined both on the set of all coalitions [7] and on the set of all feasible coalitions [1]. To present precise definitions of blockability relations and to give examples which verifies that the blockability relations are definitely different from the desirability relations are the main purposes of this paper. The difference of these two relations comes from that of their treatment of the influence of coalitions on group decision: being a winning coalition is regarded as the influence in the desirability relations, whereas the influence is expressed as the capability of a coalition to block other coalitions to be winning in the blockability relations. In fact, the desirability relations compare coalitions with respect to how close the coalitions are to be winning coalitions, that is, how close the coalitions are to have enough power to completely control the decision of the situation. On the other hand, the blockability relations compare coalitions in terms of how much the coalitions can make winning coalitions non-winning, that is, how much the coalitions can make other coalitions not have enough power to completely control the decision of the situation. It turns out that they are different from each other. Moreover, an example shows that the desirability relation is not always transitive, and a proposition verifies that the blockability relation is always transitive. This implies that the blockability relation is more appropriate than the desirability relation for the purpose of comparison of influence of coalitions on group decision making.

When we consider methods to compare coalitions in a simple game, it is quite important to clearly treat winning coalitions in the simple game. In fact, there are at least two ways on treating winning coalitions in a simple game with consideration on the feasibility of forming coalitions. A winning coalition has to be feasible in one way [1]. In the other way, it is enough for a coalition being winning to include a feasible winning coalition and it is not necessary that the coalition itself is feasible [7]. This paper gives a framework that makes us possible to distinguish these ways of treating winning coalitions. Actually, the purposes of this paper include to propose three types of operators, called L-intersection, L-restriction and extension, respectively, on simple games, where L denotes the family of all feasible coalitions, in order to clarify the simple games and the winning coalitions on which the blockability relation and the desirability relation are defined. Together with the family of minimal winning coalitions, some interrelationships among these operators are verified. In particular, it is shown that the extension of the L-intersection of a simple game coincides with the simple game if and only if all minimal winning coalitions are feasible. It is also verified that the extension of the L-intersection of a simple game coincides with the L-restriction of the simple game.

The structure of this paper is as follows: the framework of simple games is presented in the following section. There, the three operators on simple games together with the family of minimal winning coalitions are defined, and some propositions on the interrelationships among these operators are verified. In Section 3, the definitions of desirability relations and the newly proposed blockability relations are provided. The examples that show that these two relations are different from each other and analysis on the transitivity of these relations are also given in this section. Section 4 is devoted for concluding remarks.