Complexity of comparison of influence of players in simple games

Coalitional voting games appear in different forms in multi-agent systems, social choice and threshold logic. In this paper, the complexity of comparison of influence between players in coalitional voting games is characterized. The possible representations of simple games considered are simple games represented by winning coalitions, minimal winning coalitions, weighted voting game or a multiple weighted voting game. The influence of players is gauged from the viewpoint of basic player types, desirability relations and classical power indices such as Shapley-Shubik index, Banzhaf index, Holler index, Deegan-Packel index and Chow parameters. Among other results, it is shown that for a simple game represented by minimal winning coalitions, although it is easy to verify whether a player has zero or one voting power, computing the Banzhaf value of the player is #P-complete. Moreover, it is proved that multiple weighted voting games are the only representations for which it is NP-hard to verify whether the game is linear or not. For a simple game with a set Wm of minimal winning coalitions and n players, a O(n.|W|+nlog(n)) algorithm is presented which returns ‘no’ if the game is non-linear and returns the strict desirability ordering otherwise. The complexity of transforming simple games into compact representations is also examined.