Core(N, w) (N, w) balanced. Combining the three previous exercises establishes the cycle of equivalences (N, w) balanced δ ¥ 1 (core(N, w) ) (N,

Core(N, w) ‰  ˆ… ‡’ (N, w) balanced.
Combining the three previous exercises establishes the cycle of equivalences
(N, w) balanced ‡’ δ ‰¥ 1 ‡’ (core(N, w) ‰  ˆ…) ‡’ (N, w) balanced and establishes once again the Bondareva-Shapley theorem.
The Bondareva-Shapley theorem can be used both positively to establish that game has a nonempty core and negatively to prove that a game has an empty core. To establish that a game has an empty core, the theorem implies that it is sufficient to find a single-balanced family of coalitions B for which

On the other hand, to show that a game has a nonempty core, we have to show that it is balanced, that is,

for every balanced family of coalitions B. We give an example of each usage.

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