Computer Science – Computer Science and Game Theory
Scientific paper
2009-04-14
Computer Science
Computer Science and Game Theory
37 pages
Scientific paper
We introduce a new simple game, which is referred to as the complementary weighted multiple majority game (C-WMMG for short). It models a basic cooperation rule, which we named the complementary cooperation, and can be taken as a sister model of the famous weighted majority game (WMG for short). We concentrate on the two dimensional case in this paper. An interesting property of this case is that there are at most $n+1$ minimal winning coalitions (MWC for short), and they can be enumerated in time $O(n\log n)$, 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. We hope that our model may be of interest to the fields of contest theory and sports economics, and has the possibility to be applied in measuring the values of players and benchwarmers in team sports.
Cao Zhigang
Yang Xiaoguang
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