Weighted voting games apply to a wide variety of multi-agent settings. They enable the formalization of power indices which quantify the coalitional power of players. We take a novel approach to the study of the power of big vs. small players in these games. We model small (big) players as having single (multiple) votes. The aggregate relative power of big players is measured w.r.t. their votes proportion. For this ratio, we show small constant worst-case bounds for the Shapley-Shubik and the Deegan-Packel indices. In sharp contrast, this ratio is unbounded for the Banzhaf index. As an application, we define a false-name strategic normal form game where each big player may split its votes between false identities, and study its various properties. Together, our results provide foundations for the implications of players’ size, modeled as their ability to split, on their relative power.

Weighted voting games is one of the most important classes of cooperative games. Recently, Zhang and Grossi [53] proposed a variant of this class, called delegative simple games, which is well suited to analyse the relative importance of each voter in liquid democracy elections. Moreover, they defined a power index, called the delagative Banzhaf index to compute the importance of each agent (i.e., both voters and delegators) in a delegation graph based on two key parameters: the total voting weight she has accumulated and the structure of supports she receives from her delegators. We obtain several results related to delegative simple games. We first propose a pseudo-polynomial time algorithm to compute the delegative Banzhaf and Shapley-Shubik values in delegative simple games. We then investigate a bribery problem where the goal is to maximize/minimize the voting power/weight of a given voter in a delegation graph by changing at most a fixed number of delegations. We show that the problems of minimizing/maximizing a voter's power index value are strongly NP-hard. Furthermore, we prove that having a better approximation guarantee than $1-1/e$ to maximize the voting weight of a voter is not possible, unless $P = NP$, then we provide some parameterized complexity results for this problem. Finally, we show that finding a delegation graph with a given number of gurus that maximizes the minimum power index value an agent can have is a computationally hard problem.

Weighted voting games allow for studying the distribution of power between agents in situations of collective decision making. While the conventional version of these games assumes that any agent is always ready to cooperate with all others, recently, more involved models have been proposed, where cooperation is subject to restrictions. Following Myerson [1977], such restrictions are typically represented by a graph that expresses available communication links among agents. In this paper, we study the time complexity of computing two well-known power indices - the Shapley-Shubik index and the Banzhaf index - in the graph-restricted weighted voting games. We show that both are #P-complete and propose a dedicated dynamic-programming algorithm that runs in pseudo-polynomial time for graphs with the bounded treewidth.

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.

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.