In this exercise, we show that given Theorem does not hold without the condition that the core is nonempty. Let (N; v) be a two-player coalitional game with payoff function
v(1) = v(2) = 0, v(1, 2) = 1,
and let B = {{1},{2}}.
(a) Show that the core relative to the coalitional structure B is empty.
(b) Find an imputation in X(B; v) that is in the bargaining set M(N; v;B).
Theorem:- If the core of the coalitional game (N; v) for a coalitional structure B is nonempty, then it is the set of all imputations in X(B; v) at which no player has an objection against another player in the same coalition in B.