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

For each cooperative n-person game nu and each h is an element of{1, 2,..., n}, let nu(h) be the average worth of coalitions of size h and nu(h)(i) the average worth of coalitions of size h which do not contain player i is an element of N. The paper introduces the notion of a proportional average worth game (or PAW-game), i.e., the zero-normalized game nu for which there exist numbers c(h) is an element of R such that nu(h) - nu(h)(i) = c(h) (nu(n-1) - nu(n-1)(i)) for all h is an element of {2, 3,..., n - 1}, and i is an element of 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.