Abstract

http://ssrn.com/abstract=2049715
 
 

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


Bart De Keijzer


affiliation not provided to SSRN

Tomas Klos


Delft University of Technology

Yingqian Zhang


Erasmus School of Economics

May 1, 2012

ERIM Report Series Reference No. ERS-2012-006-LIS

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.

Number of Pages in PDF File: 61

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

JEL Classification: C6, C7

working papers series


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Date posted: May 2, 2012  

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: http://ssrn.com/abstract=2049715

Contact Information

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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