An important aspect of mechanism design in social choice protocols and multiagent systems is to discourage insincere and manipulative behaviour. We examine the computational complexity of false-name manipulation in weighted voting games which are an important class of coalitional voting games. Weighted voting games have received increased interest in the multiagent community due to their compact representation and ability to model coalitional formation scenarios. Bachrach and Elkind in their AAMAS 2008 paper examined divide and conquer false-name manipulation in weighted voting games from the point of view of Shapley-Shubik index. We analyse the corresponding case of the Banzhaf index and check how much the Banzhaf index of a player increases or decreases if it splits up into sub-players. A pseudo-polynomial algorithm to find the optimal split is also provided. Bachrach and Elkind also mentioned manipulation via merging as an open problem. In the paper, we examine the cases where a player annexes other players or merges with them to increase their Banzhaf index or Shapley-Shubik index payoff. We characterize the computational complexity of such manipulations and provide limits to the manipulation. The annexation non-monotonicity paradox is also discovered in the case of the Banzhaf index. The results give insight into coalition formation and manipulation.
Weighted voting games date back at least to John von Neumann and Oskar Morgenstern who developed their theory in their monumental book @cite_35 . WVGs and voting power indices have been analyzed extensively in the game theory literature for instance in @cite_20 @cite_39 . They have been applied to various economic and political bodies such as the EU Council of Ministers and the IMF @cite_26 . Power indices such as the Banzhaf index and the Shapley-Shubik index originated in such a setting in order to gauge the decision making ability of players. These indices have now been utilized in different domains such as networks @cite_16 . Simple games and weighted voting games are known by different names in other literatures and communities. There is considerable work on similar models in threshold logic @cite_17 .
@cite_35 @cite_26 @cite_39 @cite_16 @cite_20 @cite_17
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Weighted voting is a classic model of cooperation among agents in decision-making domains. In such games, each player has a weight, and a coalition of players wins the game if its total weight meets or exceeds a given quota. A player's power in such games is usually not directly proportional to his weight, and is measured by a power index, the most prominent among which are the Shapley-Shubik index and the Banzhaf index. In this paper, we investigate by how much a player can change his power, as measured by the Shapley-Shubik index or the Banzhaf index, by means of a false-name manipulation, i.e., splitting his weight among two or more identities. For both indices, we provide upper and lower bounds on the effect of weight-splitting. We then show that checking whether a beneficial split exists is NP-hard, and discuss efficient algorithms for restricted cases of this problem, as well as randomized algorithms for the general case. We also provide an experimental evaluation of these algorithms. Finally, we examine related forms of manipulative behavior, such as annexation, where a player subsumes other players, or merging, where several players unite into one. We characterize the computational complexity of such manipulations and provide limits on their effects. For the Banzhaf index, we describe a new paradox, which we term the Annexation Non-monotonicity Paradox. Weighted voting games are frequently used in decision making. Each voter has a weight and a proposal is accepted if the weight sum of the supporting voters exceeds a quota. One line of research is the efficient computation of so-called power indices measuring the influence of a voter. We treat the inverse problem: Given an influence vector and a power index, determine a weighted voting game such that the distribution of influence among the voters is as close as possible to the given target value. We present exact algorithms and computational results for the Shapley-Shubik and the (normalized) Banzhaf power index. We study the complexity of the following problem: Given two weighted voting games G? and G?? that each contain a player p, in which of these games is p's power index value higher? We study this problem with respect to both the Shapley-Shubik power index [16] and the Banzhaf power index [3,6]. Our main result is that for both of these power indices the problem is complete for probabilistic polynomial time (i.e., is PP-complete). We apply our results to partially resolve some recently proposed problems regarding the complexity of weighted voting games. We also show that, unlike the Banzhaf power index, the Shapley-Shubik power index is not #P-parsimonious-complete. This finding sets a hard limit on the possible strengthenings of a result of Deng and Papadimitriou [5], who showed that the Shapley-Shubik power index is #P-metric-complete. Weighted voting games provide a simple model of decision-making in human societies and multi-agent systems. Such games are described by a set of players, a list of [email protected]? weights, and a quota; a coalition of the players is said to be winning if the total weight of its members meets or exceeds the quota. The power of a player in a weighted voting game is traditionally identified with her Shapley-Shubik index or her Banzhaf index, two classic power measures that reflect the [email protected]?s marginal contribution under different coalition formation scenarios. In this paper, we investigate by how much one can change a [email protected]?s power, as measured by these indices, by modifying the quota. We give tight bounds on the changes in the individual [email protected]?s power that can result from a change in quota. We then describe an efficient algorithm for determining whether there is a value of the quota that makes a given player a dummy, i.e., reduces her power (as measured by both indices) to 0. We also study how the choice of quota can affect the relative power of the players. Finally, we investigate scenarios where [email protected]?s choice in setting the quota is constrained. We show that optimally choosing between two values of the quota is complete for the complexity class PP, which is believed to be significantly more powerful than NP. On the other hand, we empirically demonstrate that even small changes in quota can have a significant effect on a [email protected]?s power.