Some Surprising Properties of Power Indices - Donal G. Saari; Katri K. Sieberg (Northwestern University, Kellogg School of Management, Center for Mathematical Studies in Economics and Management Science, 1999)
A troubling aspect about power indices is how the values assigned to players can depend upon the index. As shown, the problem is more severe; different indices can generate radically different rankings; e.g., a 15-player game exists with over a trillion different strict power index rankings of the players. It is shown that certain indices always share the same ranking, but this assertion depends on the number of players; e.g., the Shapley and Banzhaf rankings agree with three players, but with more players they can even have opposite rankings. It is also shown how changes in certain assumptions affect the outcomes. This includes demonstrating how the rankings change when players drop out of a game.
Keywords: 330