Abstract: Most of the existing studies on interval-valued cooperative games in which the values of coalitions S are expressed with intervals υ(S)=[υ_L(S),υ_R(S)]are based on the interval arithmetic (e.g., interval subtraction) and ranking functions of intervals and hereby are some extensions of the classic Shapley value. The main purpose of this paper is to develop an effective method for solving n-person interval-valued cooperative games based on the least square method. Firstly, according to the concept of the distance between intervals and the least square method, an optimization mathematical model is constructed through considering that players in coalitions try to guarantee their payoffs' sums being as close to the coalitions' values as possible. Through solving the constructed optimization mathematical model, all players' interval-valued payoffs x_i=[x_Li, x_Ri] (i=1,2,…,n) can be obtained, which can be determined by the analytical formula[X_L,X_R]=[A^-1 B_L,A^-1 B_R], where B_L=(υ_L(S),υ_L(S),…,υ_L(S))^T,B_R=(υ_L(S),υ_L(S),…,υ_L(S))^T,A^-1=(1/2^n-2)(a'_ij)_n×n, and a'_ij=-/(n+1)(i≠j or n/(n+1) if i=j. Then, the auxiliary optimization mathematical model is extended so that it satisfies some conditions x(N)=υ(N) and hereby all players' interval-valued payoffs x'_i=[x'_Li, x'_Ri] (i=1,2,…,n) are solved, which can be determined by the analytical formula[X'_L,X'_R]=[X'_L+(υ_L(N)-x_Li)e/n,X_R+(υ_R(N)-x_Ri)e/n]. Finally, a numerical example of the dispatch coalition problem is used to conduct the validation and comparison analysis, which has shown that the proposed models and method are of the validity, the applicability, and the superiority. The models and method proposed in this paper can effectively avoid the magnification of uncertainty resulted from the subtraction of intervals and provide a new theoretical angle and suitable tool for solving interval-valued cooperative games.

Key words: interval-valued cooperative game, least square method, loss function, dispatch coalition, mathematical programming