Collinearity between the Shapley value and the egalitarian division rules for cooperative games

For each cooperative n-person game v and each hε{1, 2 . . . . . n}, let ν_h be the average worth of coalitions of size h and νⁱ_h the average worth of coalitions of size h which do not contain player iε N. The paper introduces the notion of a proportional average worth game (or PAW-game), i.e., the zero-normalized game v for which there exist numbers c_hεℝ such that ν_h- ν_hⁱ=c_h (ν_n-1-v _n-1ⁱ) for all hε{2, 3 . . . . , n-1}, and iε N. The notion of average worth is used to prove a formula for the Shapley value of a PAW-game. It is shown that the Shapley value, the value representing the center of the imputation set, the egalitarian nonseparable contribution value and the egalitarian non-average contribution value of a PAW-game are collinear. The class of PAW-games contains strictly the class of k-coalitional games possessing the collinearity property discussed by Driessen and Funaki (1991). Finally, it is illustrated that the unanimity games and the landlord games are PAW-games.