van den Brink, J.R., Levinsky, R. and Zeleny, M. (2015). On proper Shapley values for monotone TU-games International Journal of Game Theory, 44(2):449--471.

The Shapley value of a cooperative transferable utility game distributes the dividend of each coalition in the game equally among its members. Given exogenous weights for all players, the corresponding weighted Shapley value distributes the dividends proportionally to their weights. A proper Shapley value, introduced in Vorob{\textquoteright}ev and Liapounov (Game Theory and Applications, vol IV. Nova Science, New York, pp 155–159, 1998), assigns weights to players such that the corresponding weighted Shapley value of each player is equal to her weight. In this contribution we investigate these proper Shapley values in the context of monotone games. We prove their existence for all monotone transferable utility games and discuss other properties of this solution.