Computation of several power indices by generating functions

Abstract In this paper we propose methods to compute the Deegan-Packel, the Public Good, and the Shift power indices by generating functions for the particular case of weighted voting games. Furthermore, we define a new power index which combines the ideas of the Shift and the Deegan-Packel power indices and also propose a method to compute it with generating functions. We conclude by some comments about the complexity to compute these power indices.

[1]  Haris Aziz,et al.  Algorithmic and complexity aspects of simple coalitional games , 2009 .

[2]  Ron Johnston,et al.  On the Measurement of Power: Some Reactions to Laver , 1978 .

[3]  José María Alonso-Meijide,et al.  Generating Functions for Coalitional Power Indices: An Application to the IMF , 2005, Ann. Oper. Res..

[4]  Stefan Bolus,et al.  Power indices of simple games and vector-weighted majority games by means of binary decision diagrams , 2011, Eur. J. Oper. Res..

[5]  J. R. Isbell,et al.  A class of simple games , 1958 .

[6]  Takehiro Inohara,et al.  New interpretation of the core of simple games in terms of voters' permission , 2000, Appl. Math. Comput..

[7]  Albert Nijenhuis,et al.  Combinatorial Algorithms for Computers and Calculators , 1978 .

[8]  Gregory Levitin,et al.  Reliability optimization for weighted voting system , 2001, Reliab. Eng. Syst. Saf..

[9]  H. Wilf generatingfunctionology: Third Edition , 1990 .

[10]  William F. Lucas,et al.  Measuring Power in Weighted Voting Systems , 1983 .

[11]  William S. Zwicker,et al.  Simple games - desirability relations, trading, pseudoweightings , 1999 .

[12]  Jesús Mario Bilbao,et al.  Weighted multiple majority games with unions: Generating functions and applications to the European Union , 2009, Eur. J. Oper. Res..

[13]  Katsuhisa Ohno,et al.  The performance evaluation of a multi-stage JIT production system with stochastic demand and production capacities , 2011, Eur. J. Oper. Res..

[14]  Gregory Levitin,et al.  Asymmetric weighted voting systems , 2002, Reliab. Eng. Syst. Saf..

[15]  Enrico Zio,et al.  A clustering procedure for reducing the number of representative solutions in the Pareto Front of multiobjective optimization problems , 2011, Eur. J. Oper. Res..

[16]  Martin Shubik,et al.  A Method for Evaluating the Distribution of Power in a Committee System , 1954, American Political Science Review.

[17]  Gregory Levitin Weighted voting systems: reliability versus rapidity , 2005, Reliab. Eng. Syst. Saf..

[18]  Takehiro Inohara,et al.  Methods for comparison of coalition influence on games in characteristic function form and their interrelationships , 2010, Appl. Math. Comput..

[19]  David G. Cantor,et al.  On The Ambiguity Problem of Backus Systems , 1962, JACM.

[20]  Forming Coalitions and Measuring Voting Power , 1982 .

[21]  Igor Ushakov,et al.  The method of generalized generating sequences , 2000, Eur. J. Oper. Res..

[22]  José María Alonso-Meijide,et al.  Computing power indices: Multilinear extensions and new characterizations , 2008, Eur. J. Oper. Res..

[23]  Josep Freixas,et al.  A new power index based on minimal winning coalitions without any surplus , 2010, Decis. Support Syst..

[24]  Gregory Levitin,et al.  Uneven allocation of elements in linear multi-state sliding window system , 2005, Eur. J. Oper. Res..

[25]  Dennis Leech,et al.  Computation of Power Indices , 2002 .

[26]  Gregory Levitin,et al.  Optimal allocation of multi-state elements in linear consecutively connected systems with vulnerable nodes , 2003, Eur. J. Oper. Res..

[27]  Jesús A. De Loera,et al.  Pareto Optima of Multicriteria Integer Linear Programs , 2009, INFORMS J. Comput..

[28]  Philip Wolfe,et al.  Contributions to the theory of games , 1953 .

[29]  S. Brams,et al.  Power and size: A new paradox , 1976 .

[30]  Gregory Levitin,et al.  Evaluating correct classification probability for weighted voting classifiers with plurality voting , 2002, Eur. J. Oper. Res..

[31]  G. Owen Multilinear extensions and the banzhaf value , 1975 .

[32]  G. Owen Multilinear Extensions of Games , 1972 .

[33]  J. M. Bilbao,et al.  Generating functions for computing power indices efficiently , 2000 .

[34]  J. Deegan,et al.  A new index of power for simplen-person games , 1978 .

[35]  Philippe Flajolet,et al.  An introduction to the analysis of algorithms , 1995 .

[36]  Keitarou Ishikawa,et al.  Fundamentals of simple games from a viewpoint of blockability relations , 2009, Appl. Math. Comput..

[37]  L. S. Shapley,et al.  17. A Value for n-Person Games , 1953 .

[38]  Gregory Levitin,et al.  Threshold optimization for weighted voting classifiers , 2003 .