Complementary cooperation, minimal winning coalitions, and power indices

We introduce a new simple game, which is referred to as the complementary weighted multiple majority game (C-WMMG for short). C-WMMG models a basic cooperation rule, the complementary cooperation rule, and can be taken as a sister model of the famous weighted majority game (WMG for short). In C-WMMG, each player is characterized by a nonnegative vector with a fixed dimension, and players in the same coalition cooperate by producing a characteristic vector for this coalition (each dimension of this vector equals the maximum of the corresponding dimensions of its members). The value of a coalition is 1 if and only if the sum of its characteristic vector is larger than that of its complementary coalition, in which case the coalition is called winning. Otherwise, the coalitional value is 0. In this paper, we concentrate on the two dimensional C-WMMG. An interesting property of this case is that there are at most n+1 minimal winning coalitions (MWCs for short), and they can be enumerated in time O(nlogn), where n is the number of players. This property guarantees that the two dimensional C-WMMG is more handleable than WMG. In particular, we prove that the main power indices, i.e. the Shapley-Shubik index, the Penrose-Banzhaf index, the Holler-Packel index, and the Deegan-Packel index, are all polynomially computable. To make a comparison with WMG, we know that it may have exponentially many MWCs, and none of the four power indices is polynomially computable (unless P=NP). Still for the two dimensional case, we show that local monotonicity holds for all of the four power indices. In WMG, this property is possessed by the Shapley-Shubik index and the Penrose-Banzhaf index, but not by the Holler-Packel index or the Deegan-Packel index. Since our model fits very well the cooperation and competition in team sports, we hope that it can be potentially applied in measuring the values of players in team sports, say help people give more objective ranking of NBA players and select MVPs, and consequently bring new insights into contest theory and the more general field of sports economics. It may also provide some interesting enlightenments into the design of non-additive voting mechanisms. Last but not least, the threshold version of C-WMMG is a generalization of WMG, and natural variants of it are closely related with the famous airport game and the stable marriage/roommates problem.