On ordinal equivalence of power measures given by regular semivalues☆
Introduction
We will deal here with simple games, each one of which is defined by a finite player set and a monotonic family of winning coalitions. For such games, we consider the complete preorderings (rankings) induced in the set of players by the application of regular semivalues. A semivalue is a cooperative solution concept that can be viewed, when restricted to simple games, as a generalization of the power index notion, and a semivalue is called regular whenever all its weighting coefficients are positive. We refer the reader to Carreras and Freixas (2002), Dubey et al. (1981), Weber, 1979, Weber, 1988 for semivalues in cooperative games, Carreras et al. (2003), Laruelle and Valenciano, 2002, Laruelle and Valenciano, 2003, Laruelle and Valenciano, 2005 for semivalues restricted to simple games, and Carreras and Freixas (1999), Carreras et al. (2003), Freixas and Gambarelli (1997) for regular semivalues. The most distinguished preorderings associated with regular semivalues are those induced by the Shapley–Shubik index (Shapley, 1953, Shapley and Shubik, 1954) and the Penrose–Banzhaf–Coleman index (Banzhaf, 1965, Coleman, 1971, Penrose, 1946), respectively. Both power indices give rise to complete preorderings. For properties and behavior of power indices we refer to Brams (1975), Brams and Affuso, 1976, Brams and Affuso, 1985, Deegan and Packel (1982), Deemen and Rusinowska (2003), Dreyer and Schotter (1980), Felsenthal and Machover, 1995, Felsenthal and Machover, 1998, Fisher and Schotter (1978), Kilgour (1974), Laruelle and Valenciano (2005), and Saari and Sieberg (2000).
From the multiplicity of power measures in simple games it follows that comparisons between these indices should be made. The allocations assigned to a given player in a game by the various indices are so scattered that a fruitful idea consists in basing the comparison on the corresponding rankings instead of the numerical values provided by each index. Tomiyama (1987) has proven that, for every weighted majority game, the Shapley–Shubik and Penrose–Banzhaf–Coleman preorderings coincide. He calls this property the ordinal equivalence of these indices. He also gives examples of simple games where these preorderings differ. Diffo Lambo and Moulen (2002) have extended Tomiyama's result to all linear simple games. They have also highlighted as an open problem the characterization of the class of simple games for which the rankings induced by both indices coincide.
The present paper follows Diffo Lambo and Moulen's work. We introduce here a new class of games: the weakly linear simple games. We do not solve Diffo Lambo and Moulen's problem. However, we have achieved the following two significant steps:
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The rankings induced by both classical indices coincide for weakly linear games, which form a class wider than linear games. This is a generalization of Diffo Lambo and Moulen's (and hence Tomiyama's) result.
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The rankings induced by all regular semivalues (among which there are the Shapley–Shubik and Penrose–Banzhaf–Coleman indices) coincide for a simple game if and only if the game is weakly linear.
The notion of weakly linear game is an intuitive and promising generalization of that of swap robust or linear game. In order to give evidence of the existence of weakly linear but nonlinear simple games, we use the key idea of automorphism group of a game and consider transitive simple games, which supply a source of examples of this kind.
The paper is organized as follows. Basic definitions and preliminary results are included in Section 2. A comparison between different preorderings is given in Section 3, which contains the main results of the paper. Section 4 is devoted to the study of transitive games in order to provide examples of weakly linear games which are not linear. Some conclusions are given in Section 5.
Section snippets
Simple games and preorderings
In the sequel, N = {1,2,…, n} will denote a fixed but otherwise arbitrary finite set of players. Any subset S ⊆ N is a coalition. A cooperative game v (in N, omitted hereafter) is a simple game if (a) v(S) = 0 or 1 for all S, (b) is monotonic, i.e. v(S) ≤ v(T) whenever S ⊂ T, and (c) v(N) = 1. Either the family of winning coalitions or the subfamily of minimal winning coalitions determines the game.
Weakly linear simple games
Definition 3.1 A simple game v is weakly linear whenever the weak desirability relation ≿d is complete.
The reason for this name is that, as we will see, the completeness of the desirability relation ≿D (linearity of game v) implies the completeness of the weak desirability relation ≿d but the converse is not true, so that the second condition is weaker than the former and, therefore, all linear games are also weakly linear. Lemma 3.2 Let v be a simple game. Then: The desirability relation ≿D is a sub-preordering of the
Transitive games
Tomiyama's (1987) and Diffo Lambo and Moulen's (2002) results have been generalized in two senses. First, we have extended the ordinal equivalence of the Shapley–Shubik and Penrose–Banzhaf–Coleman indices (coincidence of ≿φ and ≿β) to all regular semivalues ψ. Second, it has been shown that this general ordinal equivalence holds not only for weighted or, more generally, linear games but also for all weakly linear games. While the first extension is clearly meaningful, we find it convenient to
Conclusions
The paper introduces the concept of weakly linear simple game and provides characterizations of this class of games. It generalizes a result by Diffo and Lambo and Moulen which is, in turn, an extension of the ordinal equivalence of the Shapley–Shubik and Penrose–Banzhaf–Coleman indices for weighted majority games stated by Tomiyama. Tomiyama showed that, for every weighted majority game, the preorderings induced by these indices coincide. Diffo Lambo and Moulen extended this result to all
Acknowledgements
The authors wish to thank two anonymous referees for carefully reading an earlier version of this work and pointing out useful comments that have allowed us to improve the presentation of the paper.
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Research partially supported by Grants SGR 2005-00651 of Generalitat de Catalunya and MTM 2006-06064 of the Education and Science Spanish Ministry and the European Regional Development Fund.