Let (S, d) ∈ F be a bargaining game. (a) Prove that there exists a unique efficient…

Let (S, d) ∈ F be a bargaining game.

(a) Prove that there exists a unique efficient alternative in S minimizing the absolute value |(x1 − d1) − (x2 − d2)|. Denote this alternative by x∗.

Let Y be the collection of efficient alternatives y in S satisfying the property that the sum of their coordinates y1 + y2 is maximal.

(b) Show that the Nash solution N (S, d) is on the efficient boundary between x∗ and the point in Y that is closest to x. In particular, if x∗ ∈ Y then x∗ = N (S, d).