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Generating Functions of Weighted Voting Games, MacMahon’s Partition Analysis, and Clifford Algebras

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  • Antônio Francisco Neto

    (Departamento de Engenharia de Produção, Administração e Economia, Escola de Minas, Universidade Federal de Ouro Preto, CEP 35.400-000, MG, Brazil)

Abstract

MacMahon’s Partition Analysis (MPA) is a combinatorial tool used in partition analysis to describe the solutions of a linear diophantine system. We show that MPA is useful in the context of weighted voting games. We introduce a new generalized generating function that gives, as special cases, extensions of the generating functions of the Banzhaf, Shapley-Shubik, Banzhaf-Owen, symmetric coalitional Banzhaf, and Owen power indices. Our extensions involve any coalition formation related to a linear diophantine system and multiple voting games. In addition, we show that a combination of ideas from MPA and Clifford algebras is useful in constructing generating functions for coalition configuration power indices. Finally, a brief account on how to design voting systems via MPA is advanced. More precisely, we obtain new generating functions that give, for fixed coalitions, all the distribution of weights of the players of the voting game such that a given player swings or not.

Suggested Citation

  • Antônio Francisco Neto, 2019. "Generating Functions of Weighted Voting Games, MacMahon’s Partition Analysis, and Clifford Algebras," Mathematics of Operations Research, INFORMS, vol. 44(1), pages 74-101, February.
    • Handle: RePEc:inm:ormoor:v:44:y:2019:i:1:p:74-101
    DOI: 10.1287/moor.2017.0915
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