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.

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.

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.

A Monotonic Weighted Banzhaf Value for Voting Games

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.

Reliability Analysis of a Home-scale Microgrid Based on a Threshold System

The reliability of a microgrid power system is an important aspect to analyze so as to ascertain that the system can provide electricity reliably over a specified period of time. This paper analyzes a home-scale model of a microgrid system by using the threshold system model (inadvertently labeled as the weighted k-out-of-n:G system model), which is a system whose success is treated as a threshold switching function. To analyze the reliability of the system, we first proved that its success is a coherent threshold function, and then identified possible (non-unique) values for its weights and threshold. Two methods are employed for this. The first method is called the unity-gap method and the second is called the fair-power method. In the unity-gap method, we utilize certain dominations and symmetries to reduce the number of pertinent inequalities (turned into equations) to be solved. In the fair-power method, the Banzhaf index is calculated to express the weight of each component as its relative power or importance. Finally, a recursive algorithm for computing system reliability is presented. The threshold success function is verified to be shellable, and the non-uniqueness of the set of weights and thresholds is demonstrated to be of no detrimental consequence, as different correct sets of weights and threshold produce equivalent expressions of system reliability.

An Ordinal Banzhaf Index for Social Ranking

We introduce a new method to rank single elements given an order over their sets. For this purpose, we extend the game theoretic notion of marginal contribution and of Banzhaf index to our ordinal framework. Furthermore, we characterize the resulting ordinal Banzhaf solution by means of a set of properties inspired from those used to axiomatically characterize another solution from the literature: the ceteris paribus majority. Finally, we show that the computational procedure for these two social ranking solutions boils down to a weighted combination of comparisons over the same subsets of elements.