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

For each cooperativen-person gamev and eachh∈{1, 2, ⋯,n}, letv h be the average worth of coalitions of sizeh andv h i the average worth of coalitions of sizeh which do not contain playeri∈N. The paper introduces the notion of a proportional average worth game (or PAW-game), i.e., the zero-normalized gamev for which there exist numbersc h ∈ℝ such thatv h −v h i =c h (v n−1−v n −1/i ) for allh∈{2, 3, ⋯,n−1}, andi∈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 non-separable contribution value and the egalitarian non-average contribution value of a PAW-game are collinear. The class of PAW-games contains strictly the class ofk-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.