Bondareva-Shapley theorem 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.

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.