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.
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