New issues in voting power analysis

2000 ◽  
Vol 9 (2) ◽  
Author(s):  
František Turnovec
Keyword(s):  
Simple Model ◽  
Power Analysis ◽  
Power Indices ◽  
Weighted Voting ◽  
Voting Power ◽  
Voting Game ◽  
Weighted Voting Game

The paper introduces concept of power indices with constraints on voting configuration formation, so called constrained power indices. It is shown in the paper that some of most frequently used power indices based on extremely simple model of weighted voting game can be extended for more sophisticated models of voting situations.

scholarly journals False-Name Manipulation in Weighted Voting Games is Hard for Probabilistic Polynomial Time

2014 ◽  
Vol 50 ◽  
pp. 573-601 ◽  
Author(s):  
A. Rey ◽  
J. Rothe
Keyword(s):  
Lower Bounds ◽  
Polynomial Time ◽  
Power Indices ◽  
Banzhaf Index ◽  
Weighted Voting ◽  
Weighted Voting Games ◽  
Voting Games ◽  
Voting Game ◽  
Probabilistic Polynomial Time ◽  
Weighted Voting Game

False-name manipulation refers to the question of whether a player in a weighted voting game can increase her power by splitting into several players and distributing her weight among these false identities. Relatedly, the beneficial merging problem asks whether a coalition of players can increase their power in a weighted voting game by merging their weights. For the problems of whether merging or splitting players in weighted voting games is beneficial in terms of the Shapley--Shubik and the normalized Banzhaf index, merely NP-hardness lower bounds are known, leaving the question about their exact complexity open. For the Shapley--Shubik and the probabilistic Banzhaf index, we raise these lower bounds to hardness for PP, "probabilistic polynomial time," a class considered to be by far a larger class than NP. For both power indices, we provide matching upper bounds for beneficial merging and, whenever the new players' weights are given, also for beneficial splitting, thus resolving previous conjectures in the affirmative. Relatedly, we consider the beneficial annexation problem, asking whether a single player can increase her power by taking over other players' weights. It is known that annexation is never disadvantageous for the Shapley--Shubik index, and that beneficial annexation is NP-hard for the normalized Banzhaf index. We show that annexation is never disadvantageous for the probabilistic Banzhaf index either, and for both the Shapley--Shubik index and the probabilistic Banzhaf index we show that it is NP-complete to decide whether annexing another player is advantageous. Moreover, we propose a general framework for merging and splitting that can be applied to different classes and representations of games.

scholarly journals An Empirical Comparison of the Performance of Classical Power Indices

2002 ◽  
Vol 50 (1) ◽  
pp. 1-22 ◽  
Author(s):  
Dennis Leech
Keyword(s):  
A Priori ◽  
Power Indices ◽  
Power Structures ◽  
Empirical Comparison ◽  
Shareholder Voting ◽  
Banzhaf Index ◽  
Weighted Voting ◽  
Voting Power ◽  
Relative A Priori ◽  
New Algorithms

Power indices are general measures of the relative a priori voting power of individual members of a voting body. They are useful for both positive and normative analysis of voting bodies particularly those using weighted voting. This paper applies new algorithms for computing the rival Shapley-Shubik and Banzhaf indices for large voting bodies to shareholder voting power in a cross section of British companies. Each company is a separate voting body and there is much variation in ownership between them resulting in different power structures. Because the data are incomplete, both finite and ‘oceanic’ games of shareholder voting are analysed. The indices are appraised, using reasonable criteria, from the literature on corporate control. The results are unfavourable to the Shapley-Shubik index and suggest that the Banzhaf index much better reflects the variations in the power of shareholders between companies as the weights of shareholder blocs vary.

scholarly journals Finding Optimal Solutions for Voting Game Design Problems

2014 ◽  
Vol 50 ◽  
pp. 105-140 ◽  
Author(s):  
B. De Keijzer ◽  
T. B. Klos ◽  
Y. Zhang
Keyword(s):  
Inverse Problem ◽  
Power Distribution ◽  
Game Design ◽  
Distance Measure ◽  
Power Index ◽  
Weighted Voting ◽  
Target Distribution ◽  
Voting Game ◽  
Distribution Of Power ◽  
Weighted Voting Game

In many circumstances where multiple agents need to make a joint decision, voting is used to aggregate the agents' preferences. Each agent's vote carries a weight, and if the sum of the weights of the agents in favor of some outcome is larger than or equal to a given quota, then this outcome is decided upon. The distribution of weights leads to a certain distribution of power. Several `power indices' have been proposed to measure such power. In the so-called inverse problem, we are given a target distribution of power, and are asked to come up with a game in the form of a quota, plus an assignment of weights to the players whose power distribution is as close as possible to the target distribution (according to some specied distance measure). Here we study solution approaches for the larger class of voting game design (VGD) problems, one of which is the inverse problem. In the general VGD problem, the goal is to find a voting game (with a given number of players) that optimizes some function over these games. In the inverse problem, for example, we look for a weighted voting game that minimizes the distance between the distribution of power among the players and a given target distribution of power (according to a given distance measure). Our goal is to find algorithms that solve voting game design problems exactly, and we approach this goal by enumerating all games in the class of games of interest. We first present a doubly exponential algorithm for enumerating the set of simple games. We then improve on this algorithm for the class of weighted voting games and obtain a quadratic exponential (i.e., 2^O(n^2)) algorithm for enumerating them. We show that this improved algorithm runs in output-polynomial time, making it the fastest possible enumeration algorithm up to a polynomial factor. Finally, we propose an exact anytime-algorithm that runs in exponential time for the power index weighted voting game design problem (the `inverse problem'). We implement this algorithm to find a weighted voting game with a normalized Banzhaf power distribution closest to a target power index, and perform experiments to obtain some insights about the set of weighted voting games. We remark that our algorithm is applicable to optimizing any exponential-time computable function, the distance of the normalized Banzhaf index to a target power index is merely taken as an example.

scholarly journals On the Complexity of the Inverse Semivalue Problem for Weighted Voting Games

2019 ◽  
Vol 33 ◽  
pp. 1869-1876 ◽  
Author(s):  
Ilias Diakonikolas ◽  
Chrystalla Pavlou
Keyword(s):  
Inverse Problem ◽  
Power Index ◽  
Power Indices ◽  
Weighted Voting ◽  
Weighted Voting Games ◽  
Voting Games ◽  
Shapley Values ◽  
Voting Game ◽  
Index Problem ◽  
Popular Power

Weighted voting games are a family of cooperative games, typically used to model voting situations where a number of agents (players) vote against or for a proposal. In such games, a proposal is accepted if an appropriately weighted sum of the votes exceeds a prespecified threshold. As the influence of a player over the voting outcome is not in general proportional to her assigned weight, various power indices have been proposed to measure each player’s influence. The inverse power index problem is the problem of designing a weighted voting game that achieves a set of target influences according to a predefined power index. In this work, we study the computational complexity of the inverse problem when the power index belongs to the class of semivalues. We prove that the inverse problem is computationally intractable for a broad family of semivalues, including all regular semivalues. As a special case of our general result, we establish computational hardness of the inverse problem for the Banzhaf indices and the Shapley values, arguably the most popular power indices.

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