Microarray Data Analysis via Weighted Indices and Weighted Majority Games

A recent paper ([12]) introduces the notion of microarray game, a new class of cooperative games, with the aim of ranking genes potentially responsible of genetic diseases (especially tumors). A microarray game is for short an average of unanimity games, where each game corresponds to a patient: the underlying assumption is that the set of the abnormally expressed genes is, as a whole, responsible for the disease in each patient, and thus is globally the (minimal) "winning" coalition of the corresponding game. The Shapley index is then (axiomatized on the class of the microarray games and) used to rank the genes. Subsequent papers ([8], [9]) deal with the same issue, by using different indices. Here we propose a definition of extended microarray game, which allows using weighted power indices; this is very useful to better rank the genes: by using only unanimity games, the enormous amount of players in each game results in considering too many of them as symmetric players, and this does not allow to significantly differentiate them. Moreover, the extended game allows us also playing a (average of) weighted majority game(s) played by the genes in the first positions according to the ranking obtained by the previous method, since it provides a natural way to attach weights to the players. We then apply the machinery to some real data concerning tumoral diseases, and we observe that our ranking highlights the role of some genes already considered in the literature as important in the onset of the disease.