Example of deferred acceptance procedure:-
Find the matchings produced by the deferred acceptance procedure both with proposals by X's and with proposals by Y's for the preferences given in Figure. Verify the results in the previous paragraph. (Argue that the only matchings in the core are the two generated by the procedures.)
In summary, every two-sided one-to-one matching game has a nonempty core, which contains the matching generated by each deferred acceptance procedure. The matching generated by the procedure is the best one in the core for the side making proposals, and the worst one in the core for the other side.