NP-completeness of some problems concerning voting games (original) (raw)
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An approximation method for power indices for voting games
2011
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.
Annals of Operations Research, 2013
This paper is a twofold contribution. First, it contributes to the problem of enumerating some classes of simple games and in particular provides the number of weighted games with minimum and the number of weighted games for the dual class as well. Second, we focus on the special case of bipartite complete games with minimum, and we compare and rank these games according to the behavior of some efficient power indices of players of type 1 (or of type 2). The main result of this second part establishes all allowable rankings of these games when the Shapley-Shubik power index is used on players of type 1.
Coalitional Power Indices Applied to Voting Systems
Proceedings of the 9th International Conference on Operations Research and Enterprise Systems, 2020
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.
Computing power indices for weighted voting games via dynamic programming
Operations Research and Decisions, 2021
We study the efficient computation of power indices for weighted voting games using the paradigm of dynamic programming. We survey the state-of-the-art algorithms for computing the Banzhaf and Shapley-Shubik indices and point out how these approaches carry over to related power indices. Within a unified framework, we present new efficient algorithms for the Public Good index and a recently proposed power index based on minimal winning coalitions of smallest size, as well as a very first method for computing Johnston indices for weighted voting games efficiently. We introduce a software package providing fast C++ implementations of all the power indices mentioned in this article, discuss computing times, as well as storage requirements.
On the dimensionality of voting games
2008
In a yes/no voting game, a set of voters must determine whether to accept or reject a given alternative. Weighted voting games are a well-studied subclass of yes/no voting games, in which each voter has a weight, and an alternative is accepted if the total weight of its supporters exceeds a certain threshold. Weighted voting games are naturally extended to k-vector weighted voting games, which are intersections of k different weighted voting games: a coalition wins if it wins in every component game. The dimensionality, k, of a kvector weighted voting game can be understood as a measure of the complexity of the game. In this paper, we analyse the dimensionality of such games from the point of view of complexity theory. We consider the problems of equivalence, (checking whether two given voting games have the same set of winning coalitions), and minimality, (checking whether a given k-vector voting game can be simplified by deleting one of the component games, or, more generally, is equivalent to a k -weighted voting game with k < k). We show that these problems are computationally hard, even if k = 1 or all weights are 0 or 1. However, we provide efficient algorithms for cases where both k is small and the weights are polynomially bounded. We also study the notion of monotonicity in voting games, and show that monotone yes/no voting games are essentially as hard to represent and work with as general games.
Power indices expressed in terms of minimal winning coalitions
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.
Complexity of some aspects of control and manipulation in weighted voting games}
2009
An important aspect of mechanism design in social choice protocols and multiagent systems is to discourage insincere behaviour. Manipulative behaviour has received increased attention since the famous Gibbard-Satterthwaite theorem. We examine the computational complexity of manipulation in weighted voting games which are ubiquitous mathematical models used in economics, political science, neuroscience, threshold logic, reliability theory and distributed systems. It is a natural question to check how changes in weighted voting game may affect the overall game. Tolerance and amplitude of a weighted voting game signify the possible variations in a weighted voting game which still keep the game unchanged. We characterize the complexity of computing the tolerance and amplitude of weighted voting games. Tighter bounds and results for the tolerance and amplitude of key weighted voting games are also provided. Moreover, we examine the complexity of manipulation and show limits to how much the Banzhaf index of a player increases or decreases if it splits up into sub-players. It is shown that the limits are similar to the previously examined limits for the Shapley-Shubik index. A pseudo-polynomial algorithm to find the optimal split is also provided.
Compound voting and the Banzhaf index
Games and Economic Behavior, 2005
We present three axioms for a power index defined on the domain of simple (voting) games. Positivity requires that no voter has negative power, and at least one has positive power. Transfer requires that, when winning coalitions are enhanced in a game, the change in voting power depends only on the change in the game, i.e., on the set of new winning coalitions. The most crucial axiom is composition: the value of a player in a compound voting game is the product of his power in the relevant first-tier game and the power of his delegate in the second-tier game. We prove that these three axioms categorically determine the Banzhaf index.
On the Chacteristic Numbers of Voting Games
International Game Theory Review, 2006
This paper deals with the non-emptiness of the stability set for any proper voting game. We present an upper bound on the number of alternatives which guarantees the non emptiness of this solution concept. We show that this bound is greater than or equal to the one given by Le Breton and Salles [6] for quota games.
Mathematical Social Sciences, 2000
In a weighted majority game each player has a positive integer weight and there is a positive integer quota. A coalition of players is winning (losing) if the sum of the weights of its members exceeds (does not exceed) the quota. A player is pivotal for a coalition if her omission changes it from a winning to a losing one. Most game theoretic measures of the power of a player involve the computation of the number of coalitions for which that player is pivotal. Prasad and Kelly [Prasad, K., Kelly, J.S., 1990. NP-completeness of some problems concerning voting games. International Journal of Game Theory 19, 1-9] prove that the problem of determining whether or not there exists a coalition for which a given player is pivotal is NP-complete. They also prove that counting the number of coalitions for which a given player is pivotal is [P-complete. In the present paper we exhibit classes of weighted majority games for which these problems are easy.