Complexity of some aspects of control and manipulation in weighted voting games

An important aspect of mechanism design in social choice protocols and multiagent systems is to discourage insincere behaviour. Manipulative behaviour has received increased attention since the famous Gibbard-Satterthwaite theorem. We examine the computational complexity of manipulation in weighted voting games which are ubiquitous mathematical models used in economics, political science, neuroscience, threshold logic, reliability theory and distributed systems. It is a natural question to check how changes in weighted voting game may affect the overall game. Tolerance and amplitude of a weighted voting game signify the possible variations in a weighted voting game which still keep the game unchanged. We characterize the complexity of computing the tolerance and amplitude of weighted voting games. Tighter bounds and results for the tolerance and amplitude of key weighted voting games are also provided. Moreover, we examine the complexity of manipulation and show limits to how much the Banzhaf index of a player increases or decreases if it splits up into sub-players. It is shown that the limits are similar to the previously examined limits for the Shapley-Shubik index. A pseudo-polynomial algorithm to find the optimal split is also provided.