Prove that the core of this game is empty

Question

A game (N, v) is simple if for every coalition S is a proper subset of N, either v(S) = 0 or v(S) = 1, In a simple game, a player, i, is said to be a veto player, if v(N \ {i}) = 0.

Suppose (N,v) is a simple game with v(N) = 1, and suppose there are no veto players in (N,v). Prove that the core of this game is empty.

Can someone show me how to do this question?

Answer 1

We have $v\left(N\setminus\{i\}\right)=1$ for all $i\in N$, and thus

$$ \sum_{i\in N}v\left(N\setminus\{i\}\right)=|N| $$

but

$$ \sum_{i\in N}\sum_{j\in N\setminus\{i\}}x_j=(|N|-1)\sum_{j\in N}x_j=|N|-1 $$

for any allocation $x$, contradicting coalitional rationality.