In the exercise prove the given theorem:

Theorem A set E ⊆ R² is the set of Nash equilibrium payoffs in a two-player game in strategic form if and only if E is the union of a finite number of rectangles of the form [a, b] × [c, d] (the rectangles are not necessarily disjoint from each other, and we do not rule out the possibility that in some of them a = b and/or c = d).

For every distribution x over a finite set S, the support of x, which is denoted supp(x), is the set of all elements of S that have positive probability under x:

supp(x) := {s ∈ S : x(s) > 0}.

(a) Let (x₁, y₁) and (x₂, y₂) be two equilibria of a two-player strategic-form game with payoffs (a, c) and (b, d) satisfying supp(x₁) = supp(x₂) and supp(y₁) = supp(y₂). Prove that for every 0 ≤ α, β ≤ 1 the strategy vector (αx1 + (1 - α)x₂, βy₁ + (1 - β)y₂) is a Nash equilibrium with the same support, and with payoff (αa + (1 - α)c, βb + (1 - β)d).

(b) Deduce that for any subset S" I of Player I's pure strategies, and any subset S" II of Player II's pure strategies, the set of Nash equilibria payoffs yielded by strategy vectors (x, y) satisfying supp(x) = S" I and supp(y) = S" II is a rectangle.

(c) Since the number of possible supports is finite, deduce that the set of equilibrium payoffs of every two-player game in strategic form is a union of a finite number of rectangles of the form [a, b] × [c, d].

(d) In this part, we will prove the converse of Theorem 5.56. Let K be a positive integer, and let (ak, bk, ck, dk) K k=1 be positive numbers satisfying ak ≤ bk and ck ≤ dk for all k. Define the set A = K k=1([ak, bk] × [ck, dk]), which is the union of a finite number of rectangles (if ak = bk and/or ck = dk, the rectangle is degenerate). Prove that the set of equilibrium payoffs in the following game in strategic form in which each player has 2K actions is A.