(Solution) - Core N w N w balanced Combining the three -(2025 Original AI-Free Solution)

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