Using the Multilinear Extension to Study Some Probabilistic Power Indices

Josep Freixas () and Montserrat Pons ()
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Josep Freixas: Universitat Politècnica de Catalunya (Barcelona–Tech)
Montserrat Pons: Universitat Politècnica de Catalunya (Barcelona–Tech)

Group Decision and Negotiation, 2017, vol. 26, issue 3, No 1, 437-452

Abstract: Abstract We consider binary voting systems modeled by a simple game, in which voters vote independently of each other, and the probability distribution over coalitions is known. The Owen’s multilinear extension of the simple game is used to improve the use and the computation of three indices defined in this model: the decisiveness index, which is an extension of the Banzhaf index, the success index, which is an extension of the Rae index, and the luckiness index. This approach leads us to prove new properties and inter-relations between these indices. In particular it is proved that the ordinal equivalence between success and decisiveness indices is achieved in any game if and only if the probability distribution is anonymous. In the anonymous case, the egalitarianism of the three indices is compared, and it is also proved that, for these distributions, decisiveness and success indices respect the strength of the seats, whereas luckiness reverses this order.

Keywords: Game theory; Voting; Simple games; Power indices; Multilinear extension; 91A12; 91A80 (search for similar items in EconPapers)
JEL-codes: C71 D72 (search for similar items in EconPapers)
Date: 2017
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Citations: View citations in EconPapers (3)

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DOI: 10.1007/s10726-016-9514-6

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