An application of the representations of symmetric groups to characterizing solutions of games in partition function form

Our concern in this paper is to present a different perspective from the more ‘traditional’ approaches to studying solutions of games in partition function form. We provide a decomposition of the space of such games under the action of the symmetric group, for the cases with three and four players. In particular, we identify all the irreducible subspaces that are relevant to the study of linear symmetric solutions. We then use such a decomposition to derive a characterization of the class of linear and symmetric solutions, as well as of the class of linear, symmetric and efficient solutions.