Problem Set 3, MIEG, Winter Term, 2015
Osborne: 468.1, 468.2,
1. Consider a infinitely repeated ‘offer counteroffer model’. Players have discounted utilities.
(i) Is the following strategy profile Nash equilibrium?
Player 1: (proposer) Offer (12,12) in period 1. Offer (1,0) in any other period.
Player 1: (responder) Accept any offer (x,1−x) if and only if x= 1 Player 2: (proposer) Offer (0,1) in every period.
Player 2: (responder) Accept any offer (x,1−x) if and only if x≤ 12 Is it subgame perfect equilibrium? If not show a profitable deviation.
(ii) Is the following strategy profile Nash equilibrium?
Player 1: (proposer) Offer (0,1) in every period Player 1: (responder) Accept any offer (x,1−x) Player 2: (proposer) Offer (0,1) in every period.
Player 2: (responder) Accept any offer (x,1−x) if and only if x= 0 Is it subgame perfect equilibrium? If not show a profitable deviation.
2.Without using the ‘uniqueness of SPNE proof’, find subgame perfect equi- librium of infinitely repeated ‘offer counteroffer model’ when (i) δ1 = 0, δ2 >0, (ii) δ2 = 0, δ1 >0.
3.Find subgame perfect equilibrium of infinitely repeated ‘offer counteroffer model’ when player 1 proposes in two consecutive periods (date 1,2,4,5,7,8, . . .) while player 2 proposes in every third periods (date 3,6,9, . . .).
4.Find subgame perfect equilibrium of infinitely repeated ‘offer counteroffer model’ where each player i incurs a cost ci > 0 for every period in which agreement is not reached. Suppose that player 1 makes the first offer.
1
a) Show that if c1 < c2 then the game has unique SPNE, where player 1 keeps the entire surplus.
b) Show that if c1 =c2 =c <1 then the game has many SPNE. What is the minimum surplus that player 1 can keep in equilibrium?
5. Consider the following two player bargaining game. At each time period t, one of the players is randomly selected as the proposer, Player 1 is se- lected with probability 13, Player 2 is selected with probability 13 and with the remaining probability no player is selected, in that case game ends with zero payoff to each player. If a player is selected as a proposer, he proposes a division (x,1−x) wherex∈[0,1] and the other player can accepts or rejects.
If the proposal is accepted by the other player, the game ends with payoffs (δt−1x, δt−1(1−x)) (to the proposer, to the other player). Otherwise, they proceed to t+ 1. Each player gets 0 if no proposal is ever accepted. (i) Find the SPNE of this game. (ii) What if, when no player is selected instead of ending the game, nothing happens that day and the game proceed to next period t+ 1.
6.The players in the following game are Alpha, who is a DSE passout looking for a job, and Bank Beta. He has also received a wage offer ‘r’ from Gamma, but we do not consider Gamma as a player. Alpha and Beta are negotiating over wage rate. They use alternating offer bargaining, Alpha making offers at odd dates t = 1,3,5, . . . and Beta offering at even dates t= 2,4,6. . . When Alpha makes an offer of wage w, Beta either accepts the offer, by hiring Alpha at wage w and ending the bargaining process, or rejects the offer and the negotiation continues. When Beta makes an offer w, then Alpha can choose one of the following three actions
- accepts the offer and starts working for Beta for wage w, ending the game - rejects the offer w and takes the Gamma’s offer r and starts working for Gamma and ending the game
2
- rejects the offerwand continue the negotiation process by making a counter offer
If the game continues to date t ≤ ∞, then the game ends with zero payoffs for both players. If Alpha takes Gamma’s offer at t < t, then the payoff of Alpha is rδt−1 and the payoff of Beta is 0, where δ ∈(0,1). If Alpha starts working for Beta at t < tfor wagew, then Alpha’s payoff iswδt−1 and Beta’s payoff is (π−w)δt−1, where r < π. (Note he cannot work for both Beta and Gamma)
a) Compute the SPNE for the case t = 4. (There are three rounds of bar- gaining.)
b) Take t = ∞ and also assume π2 < r < π. Conjecture a SPNE and check that the conjectured strategy profile is indeed an SPNE.
7. Mr X has lost his left arm due to complications in a surgery. He is suing the Doctor. The court date is set at date 2n+ 1. It is known that if they go to court, the judge will order the Doctor to pay J >0 to X.
They can negotiate for a settlement before the court date. At each date t ∈ {1,3, . . . ,2n−1}, if they have not yet settled, X offers a settlement st, and the Doctor decides whether to accept or reject it. If he accepts, the game ends with the Doctor paying st to X, otherwise game continues.
At dates t∈ {2,4, . . . ,2n}, the Doctor offers a settlement st, and X decides whether to accept the offer which leads to the ending of the game with Doctor paying st to X, or to reject it and negotiation continues.
X pays his lawyer a share of the money he gets from the Doctor as his fees.
He pays (1−α)st if they settle at date t (before court date), (1−β)J if they go to the court, where 0< β < α <1. The Doctor pays his lawyercfor each day they negotiate and an extra C if they go to the court.
Each party tries to maximize the expected amount of money it has at the end of the game.
3
a) Apply backward induction to find the SPNE of this game.
b) Suppose now that with probability φ the Judge may become sick on the court date and a Substitute Judge decide the case in the court. The Substi- tute Judge is sympathetic to the doctors and will dismiss the case. In that case, the Doctor does not pay anything to X. How would your answer to part (a) change?
8. Consider an economy with two goods and two agents. Agents 1 has an endowment of 0.6 units of good X and 0.6 units of good Y. Whereas agent 2 has an endowment of 0.4 units of both the goods. Utility functions are as follows, u1(x, y) =x+ 2y and u2(x, y) = 2x+y. Agents are bargaining over the total social endowment of 1 unit of good X and 1 unit of good Y. Find the Nash bargaining outcome.
9. A subcontractor can sell 1 unit of intermediate input (say car air condi- tioner) either to a car manufacturer or to the open market. Manufacturer gets a value 1 from the intermediate input, while it fetches price p∗ < 1 in the open market. At period 1, subcontractor chooses a technology which determines her cost of production. Cost of production for manufacturer (cm) and the open market (c) are different and depend on the technology choice.
Technology choice trade-off is given by cm = f(c), where cm is a decreasing function of c (that is, f0(.) < 0 and f00(.) > 0). At period 2, manufacturer and subcontractor bargain over a price, which is determined by the Nash bargaining rule.
a) Which technology will be chosen in equilibrium?
b) Which technology is socially optimal?
c) Why the equilibrium choice is socially sub-optimal?
4