Annexations and Merging in Weighted Voting Games - The Extent of Susceptibility of Power Indices

This paper discusses weighted voting games and two methods of manipulating those games, called annexation and merging. These manipulations allow either an agent, called an annexer to take over the voting weights of some other agents in the game, 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 manipulations in weighted voting games of the following prominent power indices: Shapley-Shubik, Banzhaf, and DeeganPackel indices. We found that for unanimity weighted voting games of n agents and for the three indices: the manipulability, (i.e., the extent of susceptibility to manipulation) via annexation of any one index does not dominate that of other indices, and the upper bound on the extent to which an annexer may gain while annexing other agents is at most n times the power of the agent in the original game. Experiments on non unanimity weighted voting games suggest that the three indices are highly susceptible to manipulation via annexation while they are less susceptible to manipulation via merging. In both annexation and merging, the ShapleyShubik index is the most susceptible to manipulation among the indices.