Ordinal notions of submodularity

We consider several ordinal formulations of submodularity, defined for arbitrary binary relations on lattices. Two of these formulations are essentially due to David Kreps (A Representation Theorem for “Preference for Flexibility”, Econometrica, 1979) and one is a weakening of a notion due to Paul Milgrom and Chris Shannon (Monotone Comparative Statics, Econometrica, 1994). We show that any reflexive binary relation satisfying either of Kreps’s definitions also satisfies Milgrom and Shannon’s definition, and that any transitive and monotonic binary relation satisfying the Milgrom and Shannon condition satisfies both of Kreps’s conditions.