Egalitarianism in convex fuzzy games
Introduction
The concept of egalitarianism, mainly based on Lorenz domination, has generated several core-related solution concepts on the set of cooperative crisp games with transferable utility (cooperative TU-games): the constrained egalitarian solution (Dutta and Ray, 1989), the Lorenz solution (Hougaard et al., 2001), the Lorenz stable set and the egalitarian core (Arin and Inarra, 2001). The class of convex crisp games is the only standard class of cooperative TU-games for which the constrained egalitarian solution exists and, moreover, it belongs to the core and Lorenz dominates every other core allocation. It turns out that all the other egalitarian solutions mentioned above coincide for convex crisp games with the constrained egalitarian solution. On this class of cooperative TU-games, alternative axiomatic characterizations of the constrained egalitarian solution are provided by Dutta (1990), Hokari (2000), Klijn et al. (2000). This solution for a convex crisp game can be obtained using the algorithm proposed by Dutta and Ray (1989) or the formula suggested by Hokari (2000).
Another solution concept related to the norm of equity is the equal division core proposed by Selten (1972). He introduces it in order to explain outcomes in experimental cooperative games and notes that in 76% of 207 experimental games, the outcomes have a “strong tendency to be in the equal division core”. Axiomatic characterizations of this solution concept on two classes of cooperative TU-games are provided by Bhattacharya (2002).
The main purpose of this paper is to introduce on one hand the egalitarian solution in the context of convex fuzzy games as proposed by Branzei et al., 2002, Branzei et al., 2003, and on the other hand the equal division core for arbitrary fuzzy games.
Cooperative fuzzy games have proved to be suitable for modelling cooperative behavior of agents in economic situations Billot, 1995, Nishizaki and Sakawa, 2001 and political situations Butnariu, 1978, Lebret and Ziad, 2001 in which some agents do not fully participate in a coalition but only to a certain extent. For example, in a class of production games, partial participation in a coalition means to offer a part of the resources while full participation means to offer all the resources. A coalition including players who participate partially can be treated in the context of cooperative game theory as a so-called fuzzy coalition, introduced by Aubin, 1974, Aubin, 1981.
The theory of cooperative fuzzy games started with the cited work of Aubin where the notions of a fuzzy game and the core of a fuzzy game are introduced. In the meantime, many solution concepts have been developed (cf. Branzei et al., 2003, Branzei et al., 2002, Butnariu, 1978, Molina and Tejada, 2002, Nishizaki and Sakawa, 2001, Sakawa and Nishizaki, 1994, Tsurumi et al., 2001).
The outline of the paper is as follows. Sections 2 and 3 provide the necessary notions and facts for cooperative crisp and fuzzy games, respectively. Section 4 introduces an egalitarian solution for convex fuzzy games by adjusting the classical Dutta–Ray algorithm for convex crisp games. It starts by pointing out that computing the egalitarian solution for a supermodular fuzzy game is not a priori easy. It is proved that adding coordinate-wise convexity to supermodularity guarantees the existence of a maximal fuzzy coalition corresponding to a crisp coalition, at each step of the adjusted Dutta–Ray algorithm. It turns out that the introduced egalitarian solution lies in the core of the convex fuzzy game and coincides with the Dutta–Ray egalitarian solution of the corresponding crisp game. In Section 5, the equal division core of an arbitrary fuzzy game is introduced and it is shown that for any convex fuzzy game, the egalitarian solution is an allocation in the equal division core of the game, and the equal division core of a convex fuzzy game coincides with the equal division core of the corresponding crisp game. This leads straightforwardly to an axiomatic characterization of the egalitarian solution for convex fuzzy games. Section 6 concludes with some final remarks.
Section snippets
Cooperative crisp games
A cooperative crisp game 〈N, w〉 consists of a finite set of players N, N={1,2, …, n} and a map w: 2N→ with w(∅)=0. For S∈2N, w(S) is called the worth of coalition S and it is interpreted as the amount of money (utility) the coalition can obtain, when the players in S work together. The class of crisp games with player set N is denoted by GN.
A game 〈N, w〉∈GN is called convex if for each S, T∈2N
The core of a game 〈N, w〉∈GN is the convex set
Cooperative fuzzy games
Given the set N={1, 2, …, n} of players, a fuzzy coalition is a vector s∈[0, 1]N. The i-th coordinate si of s is called the participation level of player i in the fuzzy coalition s. Instead of [0, 1]N, we will also write for the set of fuzzy coalitions. A crisp coalition S∈2N corresponds in a canonical way to the fuzzy coalition es, where is the vector with (es)i=1 if i∈S, and (es)i=0 if i∈N\S. The fuzzy coalition es corresponds to the situation where the players in S fully cooperate
An egalitarian solution for convex fuzzy games
In this section, we are interested in introducing an egalitarian solution only for convex fuzzy games. We do this in a constructive way by adjusting the classical Dutta–Ray algorithm for a convex crisp game.
As mentioned in Section 2, at each step of the Dutta–Ray algorithm for convex crisp games, a largest element exists. Note that for the crisp case, supermodularity of the characteristic function is equivalent to convexity of the corresponding game.
Although the cores of a convex fuzzy game and
The equal division core for convex fuzzy games
Given a cooperative fuzzy game 〈N, v〉, we define the equal division core EDC (N, v) as the set
So x∈EDC(N, v) can be seen as a distribution of the value of the grand coalition eN, where for each fuzzy coalition s, there is a player i with a positive participation level for which the pay off xi is at least as good as the equal division share α (s, v) of v (s) in s.
Some interesting facts w.r.t. the equal division core for convex fuzzy games are
Final remarks
In this paper, we introduce the equal division core for fuzzy games and the egalitarian solution for convex fuzzy games. With the aid of the key result in Lemma 7, we prove the coincidence of the egalitarian solution and the equal division core for a convex fuzzy game with the corresponding solution concepts for its related crisp game. This implies that an easy method comes available to calculate the egalitarian solution of a convex fuzzy game just by considering the corresponding crisp game,
Acknowledgements
The authors gratefully thank Flip Klijn, Herve Moulin, and the anonymous referees of this journal for their valuable comments on the previous version of the paper. The work on this paper started while the authors were research fellows at ZiF (Bielefeld) for the project “Procedural Approaches to Conflict Resolution”, 2002. We thank our hosts for their hospitality. The work of Dinko Dimitrov was partially supported by a Marie Curie Research Fellowship of the European Community programme
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