Solving Weighted Voting Game Design Problems Optimally: Representations, Synthesis, and Enumeration

61 Pages Posted: 2 May 2012  

Bart de Keijzer

affiliation not provided to SSRN

Tomas Klos

Delft University of Technology

Yingqian Zhang

Erasmus School of Economics

Date Written: May 1, 2012

Abstract

We study the inverse power index problem for weighted voting games: the problem of finding a weighted voting game in which the power of the players is as close as possible to a certain target distribution. Our goal is to find algorithms that solve this problem exactly. Thereto, we study various subclasses of simple games, and their associated representation methods. We survey algorithms and impossibility results for the synthesis problem, i.e., converting a representation of a simple game into another representation. We contribute to the synthesis problem by showing that it is impossible to compute in polynomial time the list of ceiling coalitions of a game from its list of roof coalitions, and vice versa. Then, we proceed by studying the problem of enumerating the set of weighted voting games. We present first a naive algorithm for this, running in doubly exponential time. Using our knowledge of the synthesis problem, we then improve on this naive algorithm, and we obtain an enumeration algorithm that runs in quadratic exponential time. Moreover, we show that this algorithm runs in output-polynomial time, making it the best possible enumeration algorithm up to a polynomial factor. Finally, we propose an exact anytime algorithm for the inverse power index problem that runs in exponential time. By the genericity of our approach, our algorithm can be used to find a weighted voting game that optimizes any exponential time computable function. We implement our algorithm for the case of the normalized Banzhaf index, and we perform experiments in order to study performance and error convergence.

Keywords: algorithms, inverse power index problem, synthesis problem, weighted voting games

JEL Classification: C6, C7

Suggested Citation

de Keijzer, Bart and Klos, Tomas and Zhang, Yingqian, Solving Weighted Voting Game Design Problems Optimally: Representations, Synthesis, and Enumeration (May 1, 2012). ERIM Report Series Reference No. ERS-2012-006-LIS. Available at SSRN: https://ssrn.com/abstract=2049715

Bart De Keijzer

affiliation not provided to SSRN

Tomas Klos

Delft University of Technology ( email )

Stevinweg 1
Stevinweg 1
Delft, 2628 CN
Netherlands

HOME PAGE: http://www.st.ewi.tudelft.nl/~tomas/

Yingqian Zhang

Erasmus School of Economics ( email )

Burgemeester Oudlaan 50
3000 DR Rotterdam, Zuid-Holland 3062PA
Netherlands

HOME PAGE: http://people.few.eur.nl/yqzhang/

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