To construct a suitable space for the application of a separation argument, consider the set of points

where eS is characteristic vector of the coalition S (example 3.19) and w(S) is its worth. Let A be the conic hull of A0, that is,

Let B be the interval

Clearly, A and B are convex and nonempty.
We assume that the game is balanced and construct a payoff in the core.
1. Show that A and B are disjoint if the game is balanced.
2. Consequently there exists a hyperplane that separates A and B. That is, there exists a nonzero vector (z, z0) ˆˆ „œn × „œ such that

for all y ˆˆ A and all ε > 0. Show that
a. (eˆ…, 0) ˆˆ A implies that c = 0.
b. (eN, w(N) ˆˆ A implies that z0 3. Show that (36) implies that the payoff vector z satisfies the inequalities

Therefore z belongs to the core.

A" = {(es, w(S)) : S N), Asles, w Z, 20 (36) 12 w(S) for every S g N and e ,z n(N)