An Axiomatic Approach to Probabilistic Efficient Values for Cooperative Games

he paper provides two characterizations of probabilistic values satisfying three classical axioms (linearity, dummy player and any of three “symmetry” axioms) together with a new probabilistic-efficiency axiom. This new axiom requires that players of a game allocate the total amount of their subjectively expected gains composed of the players’ subjective beliefs (arising from the underlying family of probability distributions) on receiving their marginal contributions in the game. In the axiomatic framework, the resulting type of a value suggests that players divide the total amount of their subjectively expected gains according to a new family of probability distributions. For both the Banzhaf and Shapley value it holds that the new family of probability distributions is equal to the underlying family of probability distributions.