Nash equilibrium.

A strategy for player 1 is a value for x₁ from the set X. Similarly, a strategy
for player 2 is a value for x₂ from the set X. Player 1’s payoff is V₁(x₁, x₂) =
5 + x₁ - 2x₂ and player 2’s payoff is V₂(x₁, x₂) = 5 + x₂ - 2x₁.
a. Assume that X is the interval of real numbers from 1 to 4 (including 1
and 4). (Note that this is much more than integers and includes such numbers as 2.648 and 1.00037). Derive all Nash equilibria.
b. Now assume that the game is played infinitely often and a player’s payoff is the present value of his stream of single-period payoffs, where d
is the discount factor.
(i) Assume that X is composed of only two values: 2 and 3; thus, a
player can choose 2 or 3, but no other value. Consider the following
symmetric strategy profile: In period 1, a player chooses the value 2. In period t(≥2), a player chooses the value 2. In period a player chooses the value 2 if both players chose 2 in all previous periods; otherwise, she chooses the value 3. Derive conditions which ensure that this is a subgame perfect Nash equilibruim.
(ii) Return to assuming that X is the interval of numbers from 1 to 4, so that any number between 1 and 4 (including 1 and 4) can be selected by a player. Consider the following symmetric strategy profile: In period 1, a player chooses y. In period t(≥2),  a player chooses y if both players chose y in all previous periods; otherwise, he chooses z. y and z come from the set X, and furthermore, suppose 1 ≤ y ≤ z ≤ 4. Derive conditions on y, z, and δ whereby this is a subgame perfect Nash equilibrium.