Manipulation of Weighted Voting Games and the Effect of Quota

Abstract

The Shapley-Shubik, Banzhaf, and Deegan-Packel indices are three prominent power indices for measuring voters’ power in weighted voting games. We consider two methods of manipulating weighted voting games, called annexation and merging. These manipulations allow either an agent, called an annexer to take over the voting weights of some other agents, or the coming together of some agents to form a bloc of manipulators to have more power over the outcomes of the games. We evaluate the extent of susceptibility to these forms of manipulation and the effect of the quota of a game on these manipulation for the three indices. Experiments on weighted voting games suggest that the three indices are highly susceptible to annexation while they are less susceptible to merging. In both annexation and merging, the Shapley-Shubik index is the most susceptible to manipulation among the indices. Further experiments on the effect of quotas of weighted voting games suggest the existence of an inverse relationship between the susceptibility of the indices to manipulation and the quotas for both annexation and merging. Thus, weighted voting games with large quota values closer to the total weight of agents in the games may be less vulnerable to annexation and merging than those with corresponding smaller quota values.