(Solution) - To construct a suitable space for the application of a -(2025 Original AI-Free Solution)

Paper Details

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) ? ?n × ? such that
for all y ? A and all ? > 0. Show that
a. (e?, 0) ? A implies that c = 0.
b. (eN, w(N) ? A implies that z0 < 0. Without loss of generality, we can normalize so that z0 = -1.
3. Show that (36) implies that the payoff vector z satisfies the inequalities
Therefore z belongs to the core.