The lattice structure of the S-Lorenz core

For any TU game and any ranking of players, the set of all preimputations compat- ible with the ranking, equipped with the Lorenz order, is a bounded join semi-lattice. Furthermore the set admits as sublattice the S-Lorenz core intersected with the region compatible with the ranking. This result uncovers a new property about the structure of the S-Lorenz core. As immediate corollaries we obtain complementary results to the findings of Dutta and Ray, Games Econ. Behav., 3(4) p. 403-422 (1991), by showing that any S-constrained egalitarian allocation is the (unique) Lorenz greatest element of the S-Lorenz core on the rank-preserving region the allocation belongs to. Besides, our results suggest that the comparison between W- and S-constrained egalitarian allocations is more puzzling than what is usually admitted in the literature.