Abstract
In this chapter we review basic concepts about fuzzy sets. We introduce the concept of fuzzy coalition and also of game with fuzzy coalitions. Although our study is interrelated to these games we only need some superficial notes about them. Two generalizations of the Shapley value for games with fuzzy coalitions are defined. Given a crisp cooperative game there are several ways to extend it to a game with fuzzy coalitions. We analyze three very known. The multilinear extension is a probabilistic version using a fuzzy coalition as a set of probability distributions for the players and then calculating an expected worth for each coalition. The proportional extension and the Choquet one use the fuzzy coalition as participation levels for the players in two different ways depending on the players’ behavior.
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