NP-completeness of some problems concerning voting games (original) (raw)

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.

Easy weighted majority 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.

Heuristic and exact solutions to the inverse power index problem for small voting bodies

Annals of Operations Research, 2014

Power indices are mappings that quantify the influence of the members of a voting body on collective decisions a priori. Their nonlinearity and discontinuity makes it difficult to compute inverse images, i.e., to determine a voting system which induces a power distribution as close as possible to a desired one. This paper considers approximations and exact solutions to this inverse problem for the Penrose-Banzhaf index, which are obtained by enumeration and integer linear programming techniques. They are compared to the results of three simple solution heuristics. The heuristics perform well in absolute terms but can be improved upon very considerably in relative terms. The findings complement known asymptotic results for large voting bodies and may improve termination criteria for local search algorithms.

Compound Voting and the Banzhaf Power Index

RePEc: Research Papers in Economics, 2003

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

Lecture Notes in Computer Science, 2014

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. Analogously to this splitting problem, the beneficial merging problem asks whether a coalition of players can increase their power in a weighted voting game by merging their weights. Aziz et al. [ABEP11] analyze the problem of whether merging or splitting players in weighted voting games is beneficial in terms of the Shapley-Shubik and the normalized Banzhaf index, and so do Rey and Rothe [RR10] for the probabilistic Banzhaf index. All these results provide merely NP-hardness lower bounds for these problems, 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," and provide matching upper bounds for beneficial merging and, whenever the number of false identities is fixed, also for beneficial splitting, thus resolving previous conjectures in the affirmative. It follows from our results that beneficial merging and splitting for these two power indices cannot be solved in NP, unless the polynomial hierarchy collapses, which is considered highly unlikely.

A power analysis of linear games with consensus

Mathematical Social Sciences, 2004

Linear simple games with consensus are considered. These games are obtained by intersecting a linear simple game (a game where the desirability order for players is total) and a symmetric (k-outof-n) game. We investigate the behavior of the Shapley -Shubik power index when passing from a linear game with consensus level q 1 to the same game with consensus level q 2 >q 1 . We also introduce a ''range notion'', which intuitively represents the egalitarianism of the Shapley -Shubik index, obtain an upper bound on this measure, and characterize when it is achieved. Finally, the theory developed here is illustrated with several real-world voting systems. D