The Family of Ideal Values for Cooperative Games

In view of the nature of pursuing profit, a selfish coefficient function is employed to describe the degrees of selfishness of players in different coalitions, which is the desired rate of return to the worth of coalitions. This function brings in the concept of individual expected reward to every player. Built on different selfish coefficient functions, the family of ideal values can be obtained by minimizing deviations from the individual expected rewards. Then, we show the relationships between the family of ideal values and two other classical families of values: the procedural values and the least square values. For any selfish coefficient function, the corresponding ideal value is characterized by efficiency, linearity, an equal-expectation player property and a nullifying player punishment property, and also interpreted by a dynamic process. As two dual cases in the family of ideal values, the center of gravity of imputation set value and the equal allocation of nonseparable costs value are raised from new axiomatic angles.