区间值合作对策的广义区间Shapley值

Generalized Interval Shapley Value for Interval-valued Cooperative Games

研究区间Shapley值通常对区间值合作对策的特征函数有较多约束,本文研究没有这些约束条件的区间值合作对策,以拓展区间Shapley值的适用范围。首先,本文指出广义H-差在减法与加法运算中存在的问题,进而提出了一种改进的广义H-差,称为扩展的广义H-差。然后,基于扩展的广义H-差,定义了区间值合作对策的广义区间Shapley值,并用区间有效性、区间对称性、区间哑元性和区间可加性等四条公理刻画了该广义区间Shapley值。同时,证明了该值的存在性与唯一性,而且得到了该值的一些性质。研究表明,任意的区间值合作对策的广义区间Shapley值都存在。最后,以算例说明该广义区间Shapley值的可行性与实用性。

The research of interval Shapley value for interval-valued cooperative games often assumes that there are some restrictions on characteristic function of this class of games, such as superadditive, convex and size monotonic. To expand the scope of the research, this paper investigates the interval-valued cooperative games without these restrictions. Firstly, we point out several shortcomings of the generalized Hukuhara difference and propose the so-called extended generalized Hukuhara difference. Then, based on extended generalized Hukuhara difference, the generalized interval Shapley value for interval-valued cooperative games is introduced. We char- acterize the generalized interval Shapley value using the axioms of interval efficiency, interval symmetry, interval dummy player and interval additivity. Meanwhile, we prove the existence and uniqueness of the generalized interval Shapley value and discuss some properties of this value. The study shows that an arbitrary interval-valued cooperative game has a unique generalized interval Shapley value. Finally, a numerical example is given to illus- trate the feasibility and practicability of the generalized, interval Shapley value.