Computing power indices for weighted voting games via dynamic programming (original) (raw)
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The Shapley value and Banzhaf index are two well known indices for measuring the power a player has in a voting game. However, the problem of computing these indices is computationally hard. To overcome this problem, we analyze approximation methods for computing these indices. Although these methods have polynomial time complexity, finding an approximate Shapley value using them is easier than finding an approximate Banzhaf index. We also find the absolute error for the methods and show that this error for the Shapley value is lower than that for the Banzhaf index.
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Power indices methodology of weighted majority game is widely used to measure a priori voting power of members of a committee. In this paper we present a computer implementation of the main power indices: Shapley-Shubik, Banzhaf, Johnston, Deegan-Packel and the Holler-Packel power indices. This computer implementation allows comparing the different indices. The system was developed for multiplataform: Linux and Windows. We compute the power indices for two examples and analyze its results.
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A voting situation is given by a set of voters and the rules of legislation that determine minimal requirements for a group of voters to pass a motion. A priori measures of voting power, such as the Shapley-Shubik index and the Banzhaf value, show the influence of the individual players. We used to calculate them by looking at marginal contributions in a simple game consisting of winning and losing coalitions derived from the rules of the legislation. We introduce a new way to calculate these measures directly from the set of minimal winning coalitions. This new approach logically appealing as it writes measures as functions of the rules of the legislation. For certain classes of games that arise naturally in applications the logical shortcut drastically simplifies calculations. The technique generalises directly to all semivalues.
Coalitional Power Indices Applied to Voting Systems
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We describe voting mechanisms to study voting systems. The classical power indices applied to simple games just consider parties, players or voters. Here, we also consider games with a priori unions, i.e., coalitions among parties, players or voters. We measure the power of each party, player or voter when there are coalitions among them. In particular, we study real situations of voting systems using extended Shapley-Shubik and Banzhaf indices, the so-called coalitional power indices. We also introduce a dynamic programming to compute them.
Approximating power indices: theoretical and empirical analysis
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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 power to affect the outcome of the game. Prior research on weighted voting games has explored power 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 power indices such as the Banzhaf power index and the Shapley-Shubik power 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 power indices. We also provide empirical results regarding our method, and show that it typically achieves much better accuracy and confidence than those required.
Computational implementation of indices of power
2000
D ep artam ento de M atem áticas-IM A SL (In stitu to d e Matem áticas Aplicad a S an Lu is) e-m ail: jovied o@ uns l.edu .ar (***) D ep artam ento de M atem áticas-IM A SL (Institu to d e Matem áticas Aplicad a San Lu is)-C O NIC ET
Computing the Banzhaf power index in network flow games
Proceedings of the 6th international joint conference on Autonomous agents and multiagent systems - AAMAS '07, 2007
Preference aggregation is used in a variety of multiagent applications, and as a result, voting theory has become an important topic in multiagent system research. However, power indices (which reflect how much "real power" a voter has in a weighted voting system) have received relatively little attention, although they have long been studied in political science and economics. The Banzhaf power index is one of the most popular; it is also well-defined for any simple coalitional game.
2008
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 power to affect the outcome of the game. Prior research on weighted voting games has explored power 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.