Theory of Computing Systems

, Volume 63, Issue 1, pp 150–174 | Cite as

Analyzing Power in Weighted Voting Games with Super-Increasing Weights

  • Yuval FilmusEmail author
  • Joel Oren
  • Yair Zick
  • Yoram Bachrach
Part of the following topical collections:
  1. Special Issue on Algorithmic Game Theory (SAGT 2016)


Weighted voting games (WVGs) are a class of cooperative games that capture settings of group decision making in various domains, such as parliaments or committees. Earlier work has revealed that the effective decision making power, or influence of agents in WVGs is not necessarily proportional to their weight. This gave rise to measures of influence for WVGs. However, recent work in the algorithmic game theory community have shown that computing agent voting power is computationally intractable. In an effort to characterize WVG instances for which polynomial-time computation of voting power is possible, several classes of WVGs have been proposed and analyzed in the literature. One of the most prominent of these are super increasing weight sequences. Recent papers show that when agent weights are super-increasing, it is possible to compute the agents’ voting power (as measured by the Shapley value) in polynomial-time. We provide the first set of explicit closed-form formulas for the Shapley value for super-increasing sequences. We bound the effects of changes to the quota, and relate the behavior of voting power to a novel function. This set of results constitutes a complete characterization of the Shapley value in weighted voting games, and answers a number of open questions presented in previous work.


Weighted voting games Shapley values 



Significant portions of the research presented in this work were done while Filmus and Oren were affiliated with the University of Toronto; Zick was affiliated with Nanyang Technological University and then with Carnegie Mellon University; Bachrach was affiliated with Microsoft Research, Cambridge. A preliminary version of this work was presented in SAGT 2016; the authors express their gratitude to the anonymous SAGT reviewers for their useful suggestions.


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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Technion — Israel Institute of TechnologyHaifaIsrael
  2. 2.Yahoo! ResearchHaifaIsrael
  3. 3.National University of SingaporeSingaporeSingapore
  4. 4.DeepMindLondonUK

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