Interval Shapley Value for Cooperative Games with Interval Payoffs

In this paper,we make a study of the Shapley value with characteristic functions from the viewpoint that the payoff of each coalition are often only imprecisely or ambiguously known to the players as an interval number.Axioms of Shapley value which was given by Shapley in 1953 have been extended for the interval Shapley value.The explicit and exclusive interval Shapley value has been put forward,which has been applied to profit allocation scheme among partners in supply-chain.Because of the interval payoffs,the results of imputation in this paper are also interval numbers.Moreover,it is proven that any crisp Shapley value that corresponds to a real number belonging to interval range remains with the bound of interval Shapley.