Let (N, w) be a TP-coalitional game with a compact set of outcomes X.
For every outcome x ∈ X, let d (x) be a list of coalitional deficits arranged in decreasing order (example 1.49). Let di (x) denote the ith element of d (x)
1. Show that X1 = {x ∈ X : d1(x)Ud1…y† for every y A Xg is nonempty
and compact.
2. For k ˆ 2; 3; . . . ; 2n, define Xk = {x ∈ Xk-1}: dk (x) :dk(y) for every y ∈ Xk-1}. Show that Xk is nonempty and compact.
3. Show that Nu = X2n , which is nonempty and compact.
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