Game theory

Coca-Cola and PepsiCo repeatedly play a stage game in which they simultaneously set prices for a can of their respective colas. Let pl 2 0 denote the price set by Coca-Cola and P2 2 0 the price set by PepsiCo. The payoff of CocaCola is 01(p1, p2) = p1(1 331 + 392), and the payoff of PepsiCo is 112(p1, p2) = 392(1 p2 +311). (a) [2 points] What is the best-response p1 = BR1(p2) of Coca-Cola to a price 102 chosen by PepsiCo in the stage game? (b) {3 points] Find the Nash equilibrium of the stage game. (0) [4 points] Suppose the two companies play the stage game an innite number of rounds (T = 00) and have a common discount factor 5 E (0, 1). Consider the following strategy prole in the innitely 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 (1);, p2) = (1, 1) forever. For which values of 5 does this strategy prole constitute a subgame perfect equilibrium? (d) [1 point] Modify the strategies constructed in part (c) so that the punishment of playing (121,102) = (1, 1) following a deviation from (m, p2) = (3,3) lasts a single round, and then players revert to playing (pl, 192) = (3, 3). For which values of :5 does the modied strategy prole constitute a subgame perfect equilibrium