Dynamic Programming for Computing Power Indices for Weighted Voting Games with Precoalitions

Games ◽  
2021 ◽  
Vol 13 (1) ◽  
pp. 6
Author(s):  
Jochen Staudacher ◽  
Felix Wagner ◽  
Jan Filipp
Keyword(s):  
Dynamic Programming ◽  
Power Indices ◽  
Efficient Computation ◽  
Banzhaf Index ◽  
Weighted Voting ◽  
Computing Power ◽  
Weighted Voting Games ◽  
Voting Games ◽  
Large Numbers ◽  
New Algorithms

We study the efficient computation of power indices for weighted voting games with precoalitions amongst subsets of players (reflecting, e.g., ideological proximity) using the paradigm of dynamic programming. Starting from the state-of-the-art algorithms for computing the Banzhaf and Shapley–Shubik indices for weighted voting games, we present a framework for fast algorithms for the three most common power indices with precoalitions, i.e., the Owen index, the Banzhaf–Owen index and the symmetric coalitional Banzhaf index, and point out why our new algorithms are applicable for large numbers of players. We discuss implementations of our algorithms for the three power indices with precoalitions in C++ and review computing times, as well as storage requirements.

scholarly journals Dynamic Programming for Computing Power Indices for Weighted Voting Games with Precoalitions

2021 ◽  
Author(s):  
Jochen Staudacher ◽  
Felix Wagner ◽  
Jan Filipp
Keyword(s):  
Dynamic Programming ◽  
Power Indices ◽  
Efficient Computation ◽  
Banzhaf Index ◽  
Weighted Voting ◽  
Computing Power ◽  
Weighted Voting Games ◽  
Voting Games ◽  
Large Numbers ◽  
New Algorithms

We study the efficient computation of power indices for weighted voting games with precoalitions amongst subsets of players (reflecting, e.g., ideological proximity) using the paradigm of dynamic programming. Starting from the state-of-the-art algorithms for computing the Banzhaf and Shapley-Shubik indices for weighted voting games we present a framework for fast algorithms for the three most common power indices with precoalitions, i.e. the Owen index, the Banzhaf-Owen index and the Symmetric Coalitional Banzhaf index, and point out why our new algorithms are applicable for large numbers of players. We discuss implementations of our algorithms for the three power indices with precoalitions in C++ and review computing times as well as storage requirements.

scholarly journals Computing power indices for weighted voting games via dynamic programming

2021 ◽  
Vol 31 (2) ◽  
Author(s):  
Jochen Staudacher ◽  
László Á. Kóczy ◽  
Izabella Stach ◽  
Jan Filipp ◽  
Marcus Kramer ◽  
...  
Keyword(s):  
Dynamic Programming ◽  
Power Indices ◽  
Efficient Computation ◽  
Unified Framework ◽  
Weighted Voting ◽  
Weighted Voting Games ◽  
The Public ◽  
Voting Games ◽  
Carry Over ◽  
Minimal Winning Coalitions

We study the efficient computation of power indices for weighted voting games using the paradigm of dynamic programming. We survey the state-of-the-art algorithms for computing the Banzhaf and Shapley-Shubik indices and point out how these approaches carry over to related power indices. Within a unified framework, we present new efficient algorithms for the Public Good index and a recently proposed power index based on minimal winning coalitions of smallest size, as well as a very first method for computing Johnston indices for weighted voting games efficiently. We introduce a software package providing fast C++ implementations of all the power indices mentioned in this article, discuss computing times, as well as storage requirements.

scholarly journals Worst-case Bounds on Power vs. Proportion in Weighted Voting Games with Application to False-name Manipulation

2021 ◽  
Author(s):  
Yotam Gafni ◽  
Ron Lavi ◽  
Moshe Tennenholtz
Keyword(s):  
Power Indices ◽  
Relative Power ◽  
Banzhaf Index ◽  
Weighted Voting ◽  
Worst Case ◽  
Weighted Voting Games ◽  
Voting Games ◽  
Small Constant ◽  
Novel Approach ◽  
Multi Agent

Weighted voting games are applicable to a wide variety of multi-agent settings. They enable the formalization of power indices which quantify the coalitional power of players. We take a novel approach to the study of the power of big vs.~small players in these games. We model small (big) players as having single (multiple) votes. The aggregate relative power of big players is measured w.r.t.~their votes proportion. For this ratio, we show small constant worst-case bounds for the Shapley-Shubik and the Deegan-Packel indices. In sharp contrast, this ratio is unbounded for the Banzhaf index. As an application, we define a false-name strategic normal form game where each big player may split its votes between false identities, and study its various properties. Together our results provide foundations for the implications of players' size, modeled as their ability to split, on their relative power.

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 Worst-case Bounds on Power vs. Proportion in Weighted Voting Games with an Application to False-name Manipulation

2021 ◽  
Vol 72 ◽  
pp. 99-135
Author(s):  
Yotam Gafni ◽  
Ron Lavi ◽  
Moshe Tennenholtz
Keyword(s):  
Power Indices ◽  
Relative Power ◽  
Banzhaf Index ◽  
Weighted Voting ◽  
Worst Case ◽  
Weighted Voting Games ◽  
Voting Games ◽  
Small Constant ◽  
Novel Approach ◽  
Multi Agent

Weighted voting games apply to a wide variety of multi-agent settings. They enable the formalization of power indices which quantify the coalitional power of players. We take a novel approach to the study of the power of big vs. small players in these games. We model small (big) players as having single (multiple) votes. The aggregate relative power of big players is measured w.r.t. their votes proportion. For this ratio, we show small constant worst-case bounds for the Shapley-Shubik and the Deegan-Packel indices. In sharp contrast, this ratio is unbounded for the Banzhaf index. As an application, we define a false-name strategic normal form game where each big player may split its votes between false identities, and study its various properties. Together, our results provide foundations for the implications of players’ size, modeled as their ability to split, on their relative power.

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 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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