Semivalues as power indices

Carreras, Francesc; Freixas, Josep; Puente, María Albina

doi:10.1016/s0377-2217(02)00453-8

Cited by 30 publications

(32 citation statements)

References 26 publications

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“…The natural generalization to semivalues has been argued by Laruelle and Valenciano [36], Carreras and Freixas [17], and Carreras, Freixas and Puente [20]. By considering here binomial semivalues, we look at the Banzhaf value in perspective, as will be shown by the results of our analysis.…”

Section: A Remark and Two Examplesmentioning

confidence: 68%

“…Carreras, Freixas and Puente [20] gave an axiomatic characterization and a combinatorial description in terms of weighting coefficients for (the restrictions of) semivalues as power indices, which parallel the corresponding ones for semivalues on general cooperative games.…”

Section: A Remark and Two Examplesmentioning

confidence: 99%

“…Laruelle and Valenciano [36] and Carreras, Freixas and Puente [20] investigated semivalues as power indices, that is, by restricting them to simple games. Finally, Carreras and Freixas [17] suggested several applications of semivalues based on their versatility.…”

Section: A Remark and Two Examplesmentioning

confidence: 99%

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Symmetric Coalitional Binomial Semivalues

Carreras

Puente

2011

Group Decis Negot

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We introduce here a family of mixed coalitional values. They extend the binomial semivalues to games endowed with a coalition structure, satisfy the property of symmetry in the quotient game and the quotient game property, generalize the symmetric coalitional Banzhaf value introduced by Alonso and Fiestras and link and merge the Shapley value and the binomial semivalues. A computational procedure in terms of the multilinear extension of the original game is also provided and an application to political science is sketched.

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Section: A Remark and Two Examplesmentioning

confidence: 68%

Section: A Remark and Two Examplesmentioning

confidence: 99%

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Symmetric Coalitional Binomial Semivalues

Carreras

Puente

2011

Group Decis Negot

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“…A good survey on semi-indices is Carreras et al (2003). Let us recall that the Shapley-Shubik index and the Banzhaf absolute index are semi-indices.…”

Section: Deegan-packel Indexδmentioning

confidence: 99%

Egalitarian property for power indices

Freixas

Marciniak

2011

Soc Choice Welf

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In this study, we introduce and examine the Egalitarian property for some power indices on the class of simple games. This property means that after intersecting a game with a symmetric or anonymous game the difference between the values of two comparable players does not increase. We prove that the Shapley-Shubik index, the absolute Banzhaf index, and the Johnston score satisfy this property. We also give counterexamples for Holler, Deegan-Packel, normalized Banzhaf and Johnston indices. We prove that the Egalitarian property is a stronger condition for efficient power indices than the Lorentz domination.

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“…Such axiomatic approaches have been used to create new indices, as well as to champion one power index over others. Generalized power indices, such as semivalues (Carreras, Freixas, and Puente, 2003, Laruelle and Valenciano, 2003a, and Saari and Sieberg, 2001, measure power in broader classes of cooperative games, often following the same axiomatic development. * Department of Mathematical Sciences, Montclair State University, Montclair, NJ 07013 USA.…”

Section: Introductionmentioning

confidence: 99%

The Geometry behind Paradoxes of Voting Power

Jones¹

2009

Mathematics Magazine

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Despite the many useful applications of power indices, the literature on power indices is raft with counterintuitive results or paradoxes, as well as real-life institutions that exhibit these behaviors. This has led to a cataloging of sorts where new and different paradoxes are calculated and then shown to exist in nature. Even though the paradoxes sound different from one another with names like the paradox of redistribution, the donor and transfer paradoxes, the paradox of quarreling members, the paradox of a new member, and the paradox of large size, they can be classified by the underlying geometric properties that induce the counterintuitive results. Perhaps surprisingly, analyzing the geometry behind the paradoxes for three voters is sufficient to understand the geometry behind the paradoxes. Voting power induces a partition on games where two games are in the same part if each player i has the same power in each game. The paradoxes are a result of three geometric ideas and how they interact with the partition: a point passing a hyperplane thereby changing parts, moving hyperplanes that change the size or number of parts in a partition, and changing the dimension of the space by adding or subtracting a voter.Key words : Voting Power, Paradoxes, Geometry Power indices are used to measure the a priori distribution of power among voters under a given voting rule. Many of these power indices uniquely satisfy different sets of axioms, including the most commonly used indices by Penrose (1946), Shapley and Shubik (1954) and Banzhaf (1965), as well as others. Such axiomatic approaches have been used to create new indices, as well as to champion one power index over others. Generalized power indices, such as semivalues (Carreras, Freixas, and Puente, 2003, Laruelle and Valenciano, 2003a, and Saari and Sieberg, 2001, measure power in broader classes of cooperative games, often following the same axiomatic development. * Department of Mathematical Sciences, Montclair State University, Montclair, NJ 07013 USA. jonesm@mail.montclair.edu 193The Geometry Behind Paradoxes of Voting Power As a tool, power indices have been used to examine weighted voting in institutions including the International Monetary Fund (Dreyer andSchotter, 1980 andLeech, 2002c), the Electoral College (Mann and Shapley, 1964), the European Union Council of Ministers (Johnston, 1995 andLeech, 2002b), and the Israeli Knesset (Laruelle, 2001). Not only have power indices been used to analyze existing institutions, but they have been part of the debate about the design of new institutions. For example, Turnovec (1996) and Widgren (1994) use power indices to model the effects of institutional reforms on, and the introduction of new members into, the European Union. Because power indices rarely agree on the measure of power for a voter, let alone on the ranking of the power of voters (cf. Saari and Sieberg, 2001), the selection of a power index is paramount. Although a productive way to generate power indices, the axiomatic approach has not bee...

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Semivalues as power indices

Cited by 30 publications

References 26 publications

Symmetric Coalitional Binomial Semivalues

Symmetric Coalitional Binomial Semivalues

Egalitarian property for power indices

The Geometry behind Paradoxes of Voting Power

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