Coca-Cola and PepsiCo repeatedly play a stage game in which they simultaneously set prices for a can of their respective colas. Let p1 0 denote the price set by Coca-Cola andp2 0 the price set by PepsiCo. The payoff of Coca-Cola is v1(p1, p2) = p1(1 p1 + p2), and the payoff of PepsiCo is v2(p1, p2) = p2(1 p2 + p1).

(a) [2 points] What is the best-response p1 = BR1(p2) of Coca-Cola to a price p2 chosen by PepsiCo in the stage game?

(b) [3 points] Find the Nash equilibrium of the stage game.

(c) [4 points] Suppose the two companies play the stage game an infinite number of rounds (T = ) and have a common discount factor (0, 1). Consider the following strategy profile in the infinitely repeated game: the companies start playing (p1, p2) = (3, 3), and keep playing (p1, p2) = (3, 3) as long as neither company deviated from a price of 3; as soon as a company deviates, they switch to playing (p1, p2) = (1, 1) forever. For which values of does this strategy profile constitute a subgame perfect equilibrium?

(d) [1 point] Modify the strategies constructed in part (c) so that the punishment of playing (p1, p2) = (1, 1) following a deviation from (p1, p2) = (3, 3) lasts a single round, and then players revert to playing (p1, p2) = (3, 3). For which values of does the modified strategy profile constitute a subgame perfect equilibrium?