Consistency à la Hart and Mas-Colell of 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 values for TU-games, this paper establishes a similar axiomatization for almost every value that is supposed to be efficient, linear, and symmetric (like the Shapley value). For that purpose, we introduce a general type of reduced game that takes into account the (probabilities of) two events that a removed player joins or does not join a proposed coalition in the reduced game. Similar to Hart and Mas-Colell's reduced game (in which the player joins the coalition with probability one), the general model of the reduced game involves the value itself. According to this unified approach, almost every efficient, linear, and symmetric value is consistent with respect to an appropriately chosen reduced game. The relevant reduced game varies whenever the efficient, linear, and symmetric value varies, nevertheless we present an operational criterion how to determine the appropriate reduced game (by solving an associated system of linear equations in a recursive manner). The second result states that the resulting consistency property, together with some kind of standardness for two-person games, fully characterize the given value. This paper extends the main result from a 1998 article of Driessen, Radzik and Wanink on the consistency (or reduced game) property for values that are supposed to have a weighted potential representation (since the consistency theory developed avoids the concept called a weighted potential function). Finally, the consistency theory is illustrated in the context of several known values, among which the least square values (including the Shapley value)