ABSTRACT

The Shapley value belongs to the family of semivalues: it is the unique efficient semivalue, and so the only one that can be profitably used as solution for the whole class of the TU games. On restricted classes, for instance on the class of simple games, when values are used to measure the power of players, also the other semivalues have interesting features. In this chapter, a first section is dedicated to the characterization of the family of semivalues looking at the form they assume on unanimity games. Then we use semivalues in two different contexts. First, we consider the so-called microarray games, games defined with the aim of detecting genes, which are crucial in the onset of specific diseases. Applying semivalues to these games provides a ranking of the genes. Our results, compared with other rankings from the medical literature, show an interesting uniformity between the different approaches. The last section is dedicated to the use of semivalues in a problem of Social Choice, to consider the problem of extending a ranking of objects to a ''compatible'' ranking on the family of the subsets of the objects. While the main extensions considered in the literature do not allow interactions among the objects, we use semivalues to describe situations where there are some interactions.