Date: March 28, 2012
Instructor: Yosuke YASUDA
1. Extensive Form (16 points)
For each of the game trees (a) and (b) below, answer the following questions: (1) How many information sets are there?
(2) How many subgames (including the entire game) are there? (3) Derive the subgame perfect Nash equilibrium.
2. Duopoly Game (20 points)
Consider a duopoly game in which two firms, denoted by Firm 1 and Firm 2, simultaneously and independently select their own prices, p1 and p2, respectively. The firms’ products are differentiated. After the prices are set, consumers demand A − p1+ p2
2 units of the firm 1’s good and A − p2+ p1
2 units of the firm 2’s good. Assume that the firms have identical (and constant) marginal costs c(< A), and the payoff for each firm is equal to the firm’s profit, denoted by π1 and π2.
(1) Write the payoff functions π1 and π2 (as a function of p1 and p2). (2) Derive the best response function for each player.
(3) Find the pure-strategy Nash equilibrium of this game.
(4) Derive the prices (p1, p2) that maximize joint-profit, i.e., π1+ π2.
(5) Suppose that this game is played finitely many times, say T (≥ 2) times. De- rive the subgame perfect Nash equilibrium of such a finitely repeated game. Assume that payoff of each player is sum of each period payoff.
(6) Now suppose that the game is played infinitely many times: payoff of each player is discounted sum of each period payoff with some discount factor δ ∈ (0, 1). Assume specifically that A = 16, c = 8. Then, derive the condition under which the trigger strategy sustains the joint-profit maximizing prices you derived in (3) (as a subgame perfect Nash equilibrium).
3. Auction (14 points)
Suppose that a seller auctions one object to two buyers, = 1, 2. The buyers submit bids simultaneously, and the buyer with higher bid receives the object. The loser pays nothing while the winner pays the average of the two bids b+ b
2 where b is the winner’s bid, b′ is the loser’s bid. Assume that the valuation of the object for each buyer is private and it is independently and uniformly distributed between 0 and 1.
(1) Suppose that buyer 2 takes a linear strategy, b2 = αv2. Then, derive the probability such that buyer 1 wins (as a function of b1).
(2) Solve a Bayesian Nash equilibrium. You can assume (without proof) that equilibrium bidding strategy is symmetric and linear: b1 = αv1, and b2 = αv2.