Enumeration and exact design of weighted voting games

In many multiagent settings, situations arise in which agents must collectively make decisions while not every agent is supposed to have an equal amount of influence in the outcome of such a decision. Weighted voting games are often used to deal with these situations. The amount of influence that an agent has in a weighted voting game can be measured by means of various power indices. This paper studies the problem of finding a weighted voting game in which the distribution of the influence among the agents is as close as possible to a given target value. We propose a method to exactly solve this problem. This method relies on a new efficient procedure for enumerating weighted voting games of a fixed number of agents. The enumeration algorithm we propose works by exploiting the properties of a specific partial order over the class of weighted voting games. The algorithm enumerates weighted voting games of a fixed number of agents in time exponential in the number of agents, and polynomial in the number of games output. As a consequence we obtain an exact anytime algorithm for designing weighted voting games.