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I dont know how to get started with the following question: How do I show that in a game with a nonempty core the nucleolus always is a core element? I mean, if the core is nonempty its quite obvious there is a core element....

Think it has something to do with the sign of the excesses of core/non-core elements but I can't exactly figure it out.... Guess I cant do much more than that....

Someone a hint/suggestion or an example would be nice?

Thanks!

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If a payoff-vector $x$ is in the core, the excess $v(S)-\sum_{i\in S}x_i$ must be nonpositive for every coalition $S$, for otherwise, the coalition could block the payoff-vector and the absence of the possibility to block characterizes the core.

Now a payoff-vector is in the nucleolus if the largest excess of any coalition is minimized. If the core is nonempty, there must be a payoff-vector at which the excess of every coalition is at most $0$. In particular, a payoff-vector in which the largest excess of any coalition is minimized (an element of the nucleolus), must have an excess of no more than $0$ and therefore be in the core.

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    Oh ofcourse :/ Just didnt link a negative number to a "block" and therefore not in core..... Thanks!2012-12-10