The lattice of embedded subsets

In cooperative game theory, games in partition function form are real-valued function on the set of the so-called embedded coalitions, that is, pairs (S,π)(S,π) where SS is a subset (coalition) of the set NN of players, and ππ is a partition of NN containing SS. Despite the fact that many studies have been devoted to such games, surprisingly nobody clearly defined a structure (i.e., an order) on embedded coalitions, resulting in scattered and divergent works, lacking unification and proper analysis. The aim of the paper is to fill this gap, thus to study the structure of embedded coalitions (called here embedded subsets), and the properties of games in partition function form.