On the ordinal equivalence of the Johnston, Banzhaf and Shapley power indices

European Journal of Operational Research

Kaniovski

2014

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We propose a new power index based on the minimum sum representation (MSR) of a weighted voting game. The MSR offers a redesign of a voting game, such that voting power as measured by the MSR index becomes proportional to voting weight. The MSR index is a coherent measure of power that is ordinally equivalent to the Banzhaf, Shapley-Shubik and Johnston indices. We provide a characterization for a bicameral meet as a weighted game or a complete game, and show that the MSR index is immune to the bicameral meet paradox. We discuss the computation of the MSR index using a linear integer program and the inverse MSR problem of designing a weighted voting game with a given distribution of power.

Section: The Msr As An Index Of Voting Powermentioning

confidence: 98%

“…The Banzhaf, Shapley-Shubik or Johnston indices satisfy strong monotonicity. These three indices are ordinally equivalent in a large class of games that contains all CSVGs (Freixas, Marciniak and Pons 2012).…”

Section: Coherent Power Measuresmentioning

confidence: 99%

The minimum sum representation as an index of voting power

European Journal of Operational Research

Kaniovski

2014

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“…Examples of application of our results naturally apply to political institutions, but also in management enterprisers and even in reliability systems where voters are replaced by device components with three input levels. Examples in these di¤erent contexts can be found in: Levitin [22], Obata and Ishii [23], Alonso-Meijide et al [1], Sueyoshi et al [28] or Freixas et al [16]. The paper is organized as follows.…”

Section: Introductionmentioning

confidence: 99%

Achievable hierarchies in voting games with abstention

European Journal of Operational Research

Tchantcho

Tedjeugang

2014

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: It is well known that he in ‡uence relation orders the voters the same way as the classical Banzhaf and Shapley-Shubik indices do when they are extended to the voting games with abstention (VGA) in the class of complete games. Moreover, all hierarchies for the in ‡uence relation are achievable in the class of complete VGA. The aim of this paper is twofold. Firstly, we show that all hierarchies are achievable in a subclass of weighted VGA, the class of weighted games for which a single weight is assigned to voters. Secondly, we conduct a partial study of achievable hierarchies within the subclass of H-complete games, that is, complete games under stronger versions of in ‡uence relation.

“…Some works are devoted to verify the properties of dominance or local monotonicity (among others) for some power indices and to show failures for some other power indices (see among others, Felsenthal and Machover [8] or Freixas et al [12]). Other works are devoted to study subclasses of games for which a given power index not fulfilling local monotonicity satisfies it for such a subclass of games (see for instance, Holler et al [18] and Holler and Napel [16] for the Public Good Index).…”

Section: Introductionmentioning

confidence: 99%

The Cost of Getting Local Monotonicity

Kurz

2014

SSRN Journal

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Abstract. In [15] Manfred Holler introduced the Public Good index as a proposal to divide a public good among players. In its unnormalized version, i.e., the raw measure, it counts the number of times that a player belongs to a minimal winning coalition. Unlike the Banzhaf index, it does not count the remaining winning coalitions in which the player is crucial. Holler noticed that his index does not satisfy local monotonicity, a fact that can be seen either as a major drawback [9, 221 ff.] or as an advantage [16].In this paper we consider a convex combination of the two indices and require the validity of local monotonicity. We prove that the cost of obtaining it is high, i.e., the achievable new indices satisfying local monotonicity are closer to the Banzhaf index than to the Public Good index. All these achievable new indices are more solidary than the Banzhaf index, which makes them as very suitable candidates to divide a public good.As a generalization we consider convex combinations of either: the Shift index, the Public Good index, and the Banzhaf index, or alternatively: the Shift Deegan-Packel, Deegan-Packel, and Johnston indices.