Several Interval-Valued Solutions of Interval-Valued Cooperative Games and Simplified Methods

Abstract

The aim of this chapter is to develop direct and effective simplified methods for computing interval-valued cooperative games. In this chapter, we propose several commonly used and important concepts of interval-valued solutions such as the interval-valued equal division value, the interval-valued equal surplus division value, the interval-valued Shapley value, the interval-valued egalitarian Shapley value, the interval-valued discounted Shapley value, the interval-valued solidarity value, and the interval-valued generalized solidarity value as well as the interval-valued Banzhaf value. Through adding some conditions such as the size monotonicity, we prove that the aforementioned corresponding solutions of cooperative games are continuous, monotonic, and non-decreasing functions of coalitions’ values. Hereby, the aforementioned interval-valued solutions of interval-valued cooperative games can be directly and explicitly obtained by determining their lower and upper bounds, respectively. Moreover, we discuss these interval-valued solutions’ important properties. Thus, we may overcome the issues of the Moore’s interval subtraction. The feasibility and applicability of the methods proposed in this chapter are illustrated with real numerical examples.