Download Problem set #2 - due: January 31

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?
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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.
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