Extensions of Hart and Mas-Colell's consistency to efficient, linear, and symmetric values for TU-games

By Hart and Mas-Colell’s axiomatization, it is known that the Shapley value for TU-games is fully characterized by its 1-standardness for two-person games and its consistency property with respect to a particular reduced game. In the framework of TU-games, this paper establishes a similar axiomatization (with reference to some kind of consistency and standardness for two-person games) for values that verify efficiency, linearity, and symmetry. The fundamental idea in this unified consistency approach involves the introduction of a new type of reduced game. The construction of this game takes into account, besides the value itself, the probabilities of two events that a removed player joins or does not join a proposed coalition. Although the reduced game varies whenever the efficient, linear, and symmetric value varies, an operational criterion is presented to determine the appropriate reduced game by solving an associated system of linear equations recursively. Finally, the impact of the unified consistency approach is illustrated in the context of several known values, in particular the least square values and the Shapley value.