scholarly journals A Monotonic Weighted Banzhaf Value for Voting Games

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1343
Author(s):  
Conrado M. Manuel ◽  
Daniel Martín
Keyword(s):  
Total Power ◽  
Banzhaf Value ◽  
Banzhaf Index ◽  
Voting Games ◽  
Multilinear Extension ◽  
Power Property

The aim of this paper is to extend the classical Banzhaf index of power to voting games in which players have weights representing different cooperation or bargaining abilities. The obtained value does not satisfy the classical total power property, which is justified by the imperfect cooperation. Nevertheless, it is monotonous in the weights. We also obtain three different characterizations of the value. Then we relate it to the Owen multilinear extension.

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 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 A heuristic approximation method for the Banzhaf index for voting games

2012 ◽  
Vol 8 (3) ◽  
pp. 257-274 ◽  
Author(s):  
Shaheen Fatima ◽  
Michael Wooldridge ◽  
Nicholas R. Jennings
Keyword(s):  
Approximation Method ◽  
Banzhaf Index ◽  
Voting Games

scholarly journals The Multilinear Extension and the Symmetric Coalition Banzhaf Value

2005 ◽  
Vol 59 (2) ◽  
pp. 111-126 ◽  
Author(s):  
J. M. Alonso-Meijide ◽  
F. Carreras ◽  
M. G. Fiestras-Janeiro
Keyword(s):  
Banzhaf Value ◽  
Multilinear Extension

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 Structural Control in Weighted Voting Games

2018 ◽  
Vol 18 (2) ◽  
Author(s):  
Anja Rey ◽  
Jörg Rothe
Keyword(s):  
Computational Complexity ◽  
Cooperative Games ◽  
Structural Control ◽  
Structural Changes ◽  
Transferable Utility ◽  
Original Game ◽  
Banzhaf Index ◽  
Weighted Voting ◽  
Weighted Voting Games ◽  
Voting Games

Abstract Inspired by the study of control scenarios in elections and complementing manipulation and bribery settings in cooperative games with transferable utility, we introduce the notion of structural control in weighted voting games. We model two types of influence, adding players to and deleting players from a game, with goals such as increasing a given player’s Shapley–Shubik or probabilistic Penrose–Banzhaf index in relation to the original game. We study the computational complexity of the problems of whether such structural changes can achieve the desired effect.

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 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 False-Name Manipulations in Weighted Voting Games

2011 ◽  
Vol 40 ◽  
pp. 57-93 ◽  
Author(s):  
H. Aziz ◽  
Y. Bachrach ◽  
E. Elkind ◽  
M. Paterson
Keyword(s):  
Decision Making ◽  
Computational Complexity ◽  
Lower Bounds ◽  
Experimental Evaluation ◽  
Randomized Algorithms ◽  
Power Index ◽  
Upper And Lower Bounds ◽  
Banzhaf Index ◽  
Weighted Voting ◽  
Voting Games

Weighted voting is a classic model of cooperation among agents in decision-making domains. In such games, each player has a weight, and a coalition of players wins the game if its total weight meets or exceeds a given quota. A player's power in such games is usually not directly proportional to his weight, and is measured by a power index, the most prominent among which are the Shapley-Shubik index and the Banzhaf index.In this paper, we investigate by how much a player can change his power, as measured by the Shapley-Shubik index or the Banzhaf index, by means of a false-name manipulation, i.e., splitting his weight among two or more identities. For both indices, we provide upper and lower bounds on the effect of weight-splitting. We then show that checking whether a beneficial split exists is NP-hard, and discuss efficient algorithms for restricted cases of this problem, as well as randomized algorithms for the general case. We also provide an experimental evaluation of these algorithms. Finally, we examine related forms of manipulative behavior, such as annexation, where a player subsumes other players, or merging, where several players unite into one. We characterize the computational complexity of such manipulations and provide limits on their effects. For the Banzhaf index, we describe a new paradox, which we term the Annexation Non-monotonicity Paradox.

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