Shubik Index Research Articles

Using equations from power indices to analyze figure skating teams

Power indices were originally developed to measure voting power. However, Saari and Sieberg (Games Econ Behav 36:241–263, 2001) and Saari (Chaotic elections, American Mathematical Society, Providence, 2001a) have suggested that the equations from power indices could potentially be used in some sports contexts as a way of evaluating athletes. This article explores this idea in the context of figure skating. The International Skating Union developed team events in figure skating for the 2014 Winter Olympic Games in Sochi, Russia and for other major competitions. In this article, we show how the Shapley–Shubik and Banzhaf indices can be used to analyze contributions of athletes to their countries’ teams in figure skating team events. We illustrate this approach by analyzing the results from the 2014 Winter Olympic Games figure skating team event. We also discuss some ways in which the numbers assigned by the equations from power indices can be used in the figure skating context.

False-Name Manipulation in Weighted Voting Games is Hard for Probabilistic Polynomial Time

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.

Approximating power indices: theoretical and empirical analysis

Many multiagent domains where cooperation among agents is crucial to achieving a common goal can be modeled as coalitional games. However, in many of these domains, agents are unequal in their to affect the outcome of the game. Prior research on weighted voting games has explored indices, which reflect how much real power a voter has. Although primarily used for voting games, these indices can be applied to any simple coalitional game. Computing these indices is known to be computationally hard in various domains, so one must sometimes resort to approximate methods for calculating them. We suggest and analyze randomized methods to approximate indices such as the Banzhaf index and the Shapley---Shubik index. Our approximation algorithms do not depend on a specific representation of the game, so they can be used in any simple coalitional game. Our methods are based on testing the game's value for several sample coalitions. We show that no approximation algorithm can do much better for general coalitional games, by providing lower bounds for both deterministic and randomized algorithms for calculating indices. We also provide empirical results regarding our method, and show that it typically achieves much better accuracy and confidence than those required.

Power analysis of the Nice Treaty on the future of European integration

Power analysis of changes in voting weights and rules in the Nice Treaty is done by applying methods that use Shapley–Shubik and Banzhaf indices. Significant decreases in voting power of small countries make widening of integration more acceptable to incumbent members due to small size of the applicants. Relative increases in the conciliatory power of smaller members, and relative increases in the independent power of bigger members make smaller members compromise more, and improve the position of large members for further deepening of the integration. Lastly, the fairness analysis reveals a more federalist EU in the way votes are distributed.