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By definition, an imputation is a vector $\alpha \in \Re^n$ such that

(1) $~ \alpha_i \ge v(i)~~~\forall i \in N $
(2) $ \sum_{i \in N} \alpha_i = v(N) $

where N is the coalition with all the players. Besides we can prove that an imputation $\alpha$ belongs to the core iff

(3) $ \sum_{i \in S} \alpha_i \ge v(S) ~~~ \forall S \ne N $

Now, the demostration of the Bondareva-Shapley theorem starts by noticing that the core is not empty if the result of the optimization problem

$\min_{\alpha} \sum_{i \in N} \alpha_i$
subject to (3)

is less or equal than $v(N)$.

My question is: if, for every imputation, (2) must hold, how can the result of the optimization problem be different than $v(N)$?

Sorry if the question is trivial but I'm studying these things for the first time.

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up vote 1 down vote accepted

Let $N=\{A,B\}$, $v(N)=3$, $v(A)=v(B)=1$. Now $$\min_{(\alpha_A,\alpha_B)} \alpha_A+\alpha_B$$ subject to $\alpha_A\geq v(A)$ and $\alpha_B\geq v(B)$ gives you $2<3=v(N)$.

Note that the minimization problem is over all vectors, not just imputations.

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