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AEC 512 Winter 2007 Michael Raith Problem set #2 - due: January 31 1. Three-firm Stackelberg: Three oligopolists operate in a market with inverse demand given by P (Q) = a − Q, where Q = q1 + q2 + q3 , and qi is the quantity produced by firm i. Each firm has a constant marginal cost of c, and no fixed cost. The firms choose their quantities sequentially as follows: (1) Firm 1 chooses q1 > 0; (2) Firms 2 and 3 observe q1 and then simultaneously choose q2 and q3 , respectively. (a) What is the subgame perfect equilibrium of this game? (b) Find a Nash equilibrium that is not subgame-perfect. Show that your suggested strategy profile is indeed a Nash equilibrium, and argue briefly why it is not subgame-perfect (other than noting that it’s different from the profile of (a)). 2. Pirates have captured a ship with exactly one million gold dollars. There exists a fixed rank order among the 318 pirates on the ship, where no. 1, the captain, ranks highest. Very soon, the pirates fight about about how to divide the money, until the captain comes up with a plan: “We should proceed like this: Number 1, that’s me, proposes an allocation of the money to each of us 318 men. Then we all vote. If this plan is not rejected by a strict majority, the plan shall be carried out (that is, also if there is a tie). If, however, the plan is rejected, number 1 is out of the game and is not allowed to vote any more. Then number 2 proposes an allocation, and we proceed in the same way: If it is not rejected by strict majority of the remaining players, it shall be accepted; otherwise, number 2 is out, number 3 proposes an allocation, and so on. Does this sound fair?” The pirates cheer and praise the democratic attitude of their captain, not knowing that in his student days, he took a course in game theory... (a) Find the unique subgame-perfect equilibrium outcome of this game. The gold dollars are not divisible, i.e. have to be allocated in integer numbers. Also, assume that each pirate rejects any plan that allocates a payoff of zero to him. (b) Give an example of a Nash equilibrium that gives each pirate at least $2. Why is such an equilibrium not subgame-perfect? 1 3. Sequential bargaining: (a) Consider the following model of (infinite-horizon) sequential bargaining between a seller and a buyer over the price of an indivisible good. The cost of selling the good to the seller is zero, its value to the buyer is v. In each period, the player to make an offer is randomly determined. With probability q, the seller (player 1) is chosen to make an offer, and with probability 1−q, the buyer (player 2) is chosen. The player chosen to make an offer proposes a price p. If the other player accepts, trade occurs and the players’ payoffs are (p, v − p), and the game ends; otherwise the game continues into the next period. The players are risk-neutral and have the same discount factor δ. Determine the subgame-perfect equilibrium, and the players’ expected equilibrium payoffs, of this game. (b) Now suppose there are two identical buyers. Consider the game in which in every period, the seller is first randomly matched with one of the buyers, then one of the matched parties is randomly chosen to make an offer; suppose q = 1/2. The responder either accepts the offer, in which case trade occurs and the game ends, or rejects the offer, in which case play moves to the next period, in which the seller is again randomly matched with one of the two buyers. Determine the subgame-perfect equilibrium and the associated payoffs of this game. What happens as δ approaches 1? 4. Three-way bargaining: Huey, Dewey and Louie have a dollar to split. Huey gets to offer first, and offers shares d and l to Dewey and Louie, keeping h for himself (so h + d + l = 1). If both accept, the game is over and the dollar is divided accordingly. If either Dewey or Louie rejects the offer, however, they come back the next day and start again, this time Dewey making the offer to Huey and Louie, and if this is rejected, on the third day Louie gets to make the offer. If this is rejected, they come back on the fourth day with Huey again making the offer, and so on forever or until an offer is accepted. All three players have the same (daily) discount factor of δ. Determine the unique symmetric, stationary (i.e. players make the same offers every time they move) SPE of this game. Note: It is possible to compute the equilibrium based on symmetry considerations alone. Don’t do that, though; instead determine the equilibrium using subgame-perfection arguments similar to those normally used in sequential bargaining games. 5. Gibbons, problem 2.11. 2