Limit property of a multi-choice value and the fuzzy value

Abstract

Multi-choice games and fuzzy games, both of which are generalizations of cooperative games, enable us to deal with more than two participation levels. We consider generalized Shapley values for both type of games which are called a multi-choice value and a fuzzy value, respectively. By constructing a fuzzy game from a multi-choice game using a piecewise multilinear function, we can confirm that the multi-choice value converges to the fuzzy value when the number of levels is sufficiently large.