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

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.

On Weights and Quotas for Weighted Majority Voting Games

In this paper, we analyze the frequency distributions of weights and quotas in weighted majority voting games (WMVG) up to eight players. We also show different procedures that allow us to obtain some minimum or minimum sum representations of WMVG, for any desired number of players, starting from a minimum or minimum sum representation. We also provide closed formulas for the number of WMVG with n players having a minimum representation with quota up to three, and some subclasses of this family of games. Finally, we complement these results with some upper bounds related to weights and quotas.

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

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.

Poisson voting games under proportional rule

We analyze strategic voting under proportional rule and two parties, embedding the basic spatial model into the Poisson framework of population uncertainty. We prove that there exists a unique Nash equilibrium. We show that it is characterized by a cutpoint in the policy space that is always located between the average of the two parties’ positions and the median of the distribution of voters’ types. We also show that, as the expected number of voters goes to infinity, the equilibrium converges to that of the case with deterministic population size.

Simple Voting Games and Cartel Damage Proportioning

Individual contributions by infringing firms to the compensation of cartel victims must reflect their “relative responsibility for the harm caused” according to EU legislation. Several studies have argued that the theoretically best way to operationalize this norm is to apply the Shapley value to an equilibrium model of cartel prices. Because calibrating such a model is demanding, legal practitioners prefer workarounds based on market shares. Relative sales, revenues, and profits however fail to reflect causal links between individual behavior and prices. We develop a pragmatic alternative: use simple voting games to describe which cartel configurations can(not) cause significant price increases in an approximate, dichotomous way; then compute the Shapley-Shubik index. Simulations for a variety of market scenarios document that this captures relative responsibility better than market share heuristics can.

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

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.

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

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.