Given two side-payment gamesv andw, both defined for the same finite player-setN, the following three welfare criteria are characterized in terms of the datav andw: (A) For everyy ∃ C(w) there existsx ∃ C(v) such thaty≤x; (A′) For everyx∃C(v) there existsyeC(w) such thaty≤x; and (B) There existyeC(w) andxeC(v) such thaty≤x. (HereC(v) denotes the core ofv.) Given two non-side-payment gamesv andw, sufficient conditions for the criteria (A′) and (B) are established, by observing that an ordinal convex game has a large core.