A new family of regular semivalues and applications

Abstract

We define a new family of values for cooperative games, including as a particular case the Shapley value. They are defined on the collection of the unanimity games, then extended by linearity. Our most relevant result shows that the family of the weighting coefficients characterizing the values so defined is an open curve on the simplex of the regular semivalues. We give an explicit formula for the values when the parameter characterizing the family is a natural number and we offer an algorithm to calculate them in weighted majority games, slightly extending previous results (see Bilbao et al., TOP, 8:191–213, 2000). The paper ends with two applications. The first one is classical, and serves to see how the indices behave with respect to the Shapley and Banzhaf values in the case of the EU parliament and in the UN Security Council. The second one is much more recent: it deals with the microarray games, introduced in Moretti et al. (TOP, 15:256–280, 2007), which are average of unanimity games. The idea is to rank genes taken from DNA of patients affected by a specific disease, with the aim of singling out a group of genes potentially responsible of the disease. In this last case we consider some microarray data available on the net and concerning some specific diseases and we show that several genes mentioned in the medical literature as potentially responsible for the onset of the disease are present in the first places according to our rankings.