Efficient Algorithm for Designing Weighted Voting Games

Aziz, Haris; Paterson, Mike; Leech, Dennis

doi:10.1109/inmic.2007.4557718

Cited by 28 publications

(37 citation statements)

References 23 publications

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“…Aziz, Paterson and Leech [1] discuss an iterative algorithm which seems to work well in practice, but there is a fundamental problem in analyzing it fully. There are no results about how well one can approximate a given vector.…”

Section: Introductionmentioning

confidence: 99%

The inverse Banzhaf problem

Alon

Edelman

2009

Soc Choice Welf

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Section: Introductionmentioning

confidence: 99%

The inverse Banzhaf problem

Alon

Edelman

2009

Soc Choice Welf

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“…• We provide numerical computation for large and structured games with up to 100 players in the weighted voting games [1,4] and up to 75 players in the coalitional skill games [2]. This is a significant improvement compared to the literature where numerical results are shown for computing the nucleoli of games with at most 20 players.…”

Section: Algorithms # Lps and Their Sizesmentioning

confidence: 99%

“…Suppose so far we have generated the set of linearly independent tight constraints {z (1) , z (2) , ..., z (s) }. In order to restrict z not to be in the linear combination of {z (1) , z (2) , ..., z (s) }, we modify the constraint generation problem to:…”

Section: Producing a Representative Setmentioning

confidence: 99%

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Finding the nucleoli of large cooperative games

Nguyen

Thomas

2016

European Journal of Operational Research

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Highlights• We propose a new nested-LPs-based approach for finding the nucleolus• We resolve subtle issues in handling multiple optimal solutions in each LP• The approach involves fewer and smaller LPs compared to existing methods• The approach proves to work well in several large combinatorial games AbstractThe nucleolus is one of the most important solution concepts in cooperative game theory as a result of its attractive properties -it always exists (if the imputation is non-empty), is unique, and is always in the core (if the core is non-empty). However, computing the nucleolus is very challenging because it involves the lexicographical minimization of an exponentially large number of excess values. We present a method for computing the nucleoli of large games, including some structured games with more than 50 players, using nested linear programs (LP). Although different variations of the nested LP formulation have been documented in the literature, they have not been used for large games because of the large size and number of LPs involved. In addition, subtle issues such as how to deal with multiple optimal solutions and with tight constraint sets need to be resolved in each LP in order to formulate and solve the subsequent ones. Unfortunately, this technical issue has been largely overlooked in the literature. We treat these issues rigorously and provide a new nested LP formulation that is smaller in terms of the number of large LPs and their sizes. We provide numerical tests for several games, including the general flow games, the coalitional skill games and the weighted voting games, with up to 100 players.

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“…Leech (2003) and Aziz, Paterson and Leech (2007) proposed two fixed-point iteration methods for obtaining the inverse solution for the Banzhaf index. For a general method based on linear programs, see Kurz (2012b).…”

Section: The Msr As An Index Of Voting Powermentioning

confidence: 99%

The minimum sum representation as an index of voting power

Freixas

Kaniovski

2014

European Journal of Operational Research

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We propose a new power index based on the minimum sum representation (MSR) of a weighted voting game. The MSR offers a redesign of a voting game, such that voting power as measured by the MSR index becomes proportional to voting weight. The MSR index is a coherent measure of power that is ordinally equivalent to the Banzhaf, Shapley-Shubik and Johnston indices. We provide a characterization for a bicameral meet as a weighted game or a complete game, and show that the MSR index is immune to the bicameral meet paradox. We discuss the computation of the MSR index using a linear integer program and the inverse MSR problem of designing a weighted voting game with a given distribution of power.

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Efficient Algorithm for Designing Weighted Voting Games

Cited by 28 publications

References 23 publications

The inverse Banzhaf problem

The inverse Banzhaf problem

Finding the nucleoli of large cooperative games

The minimum sum representation as an index of voting power

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