The proportional value for positive cooperative games
By generalizing in a natural way the idea of proportional division of surplus for two person games, a new solution concept for TU cooperative games is introduced and called the proportional value. Its domain is restricted to strictly positive games but, as is noticed, this solution applies as well to strictly negative games. A motivating example is given in Section 1. In Section 2 it is first shown that a value admits a (multiplicative) potential if, and only if, it preserves ratios -- a proportionally balanced contribution property. Next two axiomatic characterizations are provided for the proportional value: the first one, in terms of efficiency and ratio preservation; the second, by means of proportionality for two person games and consistency with regard to a classical notion of reduced game. For computation purposes, two recursive formulae concerning the potential are supplied in Section 3. Finally, several further properties of the proportional value are shown in Section 4: homogeneity, monotonicity, the dummy player property, anonymity and the inessential game property. Perhaps it should have been also mentioned that the proportional value fails to satisfy additivity, as this may cause some trouble when trying to evaluate, e.g., a discount problem in terms of both cost and saving share. Of course, the paper will be of interest to mathematicians (cooperative game theorists) and economists.
scientific article; zbMATH DE number 1488615