PIM821 – Problem Set 3
April-June 2012
Mike Conlin
SOLUTIONS
1. Below is an editorial entitled “Why FTC Opposes The Staples Merger” which appeared in the Wall
Street Journal.
For simplicity, assume Staples and Office Depot have no fixed costs. Assume that the marginal cost,
average variable cost and average total cost for staplers are constant at $0.50. The market demand for
staplers in Washington D.C. is depicted on the graph below.
6.5
6
5.5
5
4.5
4
3.5
3
2.5
2
1.5
1
MC
0.5
0
0
20
40
60
80
100
120
140
160
180
200
220
240
MR
Suppose Staples and Office Depot in Washington D.C. are each charging the same price for staplers and
each making profits of $90. Also assume that they are the only two stores that sell office supplies in
Washington D.C. How much will the price of a stapler increase if the merger occurs? EXPLAIN.
If both Staples and Office Depot are making profits of $90, then each must be charging a price of $1.50
and selling 90 staplers each. You can determine this price by just trial and error.
6.5
6
5.5
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
Firm 1’s Profits
Firm 2’s Profits
MC=AVC=ATC
0
20
40
60
80
100
120
140
160
180
200
220
240
[You can figure this out by doing the math - taking the demand of P=6-.025*Q, plugging it into the
equation representing the firms making a joint profit of $180 (P*Q-.5*Q=180), solving for Q and the
plugging this Q into P=6-.025*Q,. I do not expect you to do this but I received some questions about it
after class.]
If this is the case, then each firm’s profits are 1.50(90)-.5(90)=90. (It could also be the case that each is
charging a price of $5 and each selling 20 staplers. That would result in each making a profit of 90; but
it is very unlikely that Staples and Office Depot would collude on a price greater than the monopoly price
of 3.25.)
If a merger occurs, there would be a monopoly in the office supplies industry in Washington D.C.
Therefore, one may expect that the price set by this monopolist to be 3.25 (see above graph). Therefore,
the price increases from $1.50 before the merger to $3.25 after the merger.
If the merger of Staples and Office Depot did occur, how is it likely to affect Bob’s Office Supply, an
office supply retail store in Arlington, Virginia? Be precise in your explanation. (Arlington is a town just
outside of Washington, D.C.)
If a merger occurs and the price of a stapler in DC increases, it would increase the demand for staplers
at Bob’s Office Supply store (because staplers at Staples and Office Depot are substitutes to staplers at
Bob’s Office Supply store). This would result in Bob setting a higher price and obtaining greater profits.
2. Software Developers Inc. and AMC Applications Inc. are two companies developing statistical
software programs. These companies must simultaneously decide whether to develop software that
operates on the Unix operating system, Microsoft’s Windows operating system or an objectorientated operating system. The table below indicates Software Developers Inc. and AMC
Applications Inc. profits depending on the operating system they develop their statistical software for.
AMC Applications Inc.
Software
Developers
Inc.
Unix
Windows
ObjectOrientated
Unix
40 , 50
30 , 40
30 , 30
Windows
45 , 40
80 , 90
60 , 20
Object-Orientated
25 , 60
60 , 75
70 , 80
a)
Does Software Developers Inc. or AMC Applications Inc. have a dominant strategy? Explain.
No, the best strategy of Software Developers Inc. depends on the strategy of AMC Applications Inc. and
the best strategy of AMC Applications Inc. depends on the strategy of Software Developers Inc..
b)
Identify all pure strategy Nash Equilibria.
The first strategy listed identifies Software Developers Inc. strategy and the second is AMC Applications
Inc. strategy.
Two Nash Equilibria:
(Windows , Windows)
(Object-Orientated , Object-Orientated)
For the (Windows , Windows) Nash Equilibrium, Software Developers Inc. cannot increase their payoff of
80 by changing their strategy of choosing Windows given that AMC Applications Inc. chooses Windows.
In addition, AMC Applications Inc. cannot increase their payoff of 90 by changing their strategy of
choosing Windows given that Software Developers Inc. chooses Windows. Therefore, (Windows ,
Windows) is a Nash Equilibrium. The same logic can be used for (Object-Orientated , ObjectOrientated).
c)
Can you explain why profits may be greater for both firms when they develop software for the
same operating system?
There are likely to be benefits associated with having a standardized operating system. For example,
using a standardized operating system may increase the overall demand for software (which is very
likely).
I did not ask you to identify mixed strategy Nash Equilibria because it is beyond the scope of the
class. However, I thought you might be interested. (If you are not interested, just ignore.  )
First note that in any mixed strategy Nash Equilibria Software Developers Inc. would play Unix with
probability zero because no matter what AMC Applications Inc. strategy is, Software Developers Inc.
obtains a higher payoff from choosing Windows. Therefore, Software Developers Inc. will never choose
Unix as part of any Nash Equilibria. If this is the case, AMC Applications Inc. best response to Software
Developers Inc.’s strategy cannot be to choose Unix because AMC Applications Inc.’s payoff from
choosing Windows is higher than from choosing Unix given that Software Developers Inc. does not
choose Unix. Therefore, define pS as the probability Software Developers Inc chooses Windows and (1ps) as the probability Software Developers Inc chooses Object-Orientated. Define pA as the probability
AMC Applications Inc chooses Windows and (1-pA) as the probability AMC Applications Inc chooses
Object-Orientated. Have each player randomize to make the other player indifferent between choosing
Windows and Object-Orientated and then solve for ps and pA.
3.
Here is a story many of you have heard.
There are two friends taking MBA814 this semester. Both had done pretty well on all of the
homeworks and the midterm, so that going into the final they had a solid 4.0. They were so
confident the weekend before the final that they decided to go to a party in Chicago. The party
was so good that they overslept all day Sunday, and got back too late to study for the final that
was scheduled for Monday morning. Rather than take the final unprepared, they went to Prof.
Conlin with a sob story. They said they had gone to Chicago and had planned to come back in
plenty of time to study for the final but had had a flat tire on the way back. Because they did
not have a spare, they had spent most of the night looking for help. Now they were really tired,
so could they please have a makeup final the next day? Prof. Conlin thought it over and agreed.
The two studied all of Monday evening and came well prepared on Tuesday morning. Prof.
Conlin placed them in separate rooms and handed the test to each. The first question on the
first page, worth 10 points, was very easy. Each of them wrote a good answer, and greatly
relieved, turned the page. It had just one question, worth 90 points. It was: “Which tire?”
Suppose that each student’s “payoff” is 100 (because they receive an 4.0 in the class) if they answer the
second question the same and each student’s “payoff” is 30 (because they receive a 2.0 in the class) if
they answer the second question differently.
a) Depict the above situation as a normal form game.
Front-Left
Student 1
Student 2
FrontBack-Left
Right
30,30
30,30
Back-Right
Front-Left
100,100
30,30
Front-Right
30,30
100,100
30,30
30,30
Back-Left
30,30
30,30
100,100
30,30
Back-Right
30,30
30,30
30,30
100,100
b) Does either student have a dominant strategy? If so, please identify the dominant strategy.
Neither student has a dominant strategy. The best strategy for Student 1 depends on the strategy of
Student 2 and the best strategy for Student 2 depends on the strategy of Student 1.
c) Identify all pure strategy Nash Equilibria.
There are 4 pure strategy Nash Equilibria. For both students to select Front-Left, for both students to
select Front- Right, for both students to select Back-Left, or for both students to select Back-Right.
Do you really believe that one of these 4 will likely happen and which one do you expect is more
likely? It probably depends on the student’s past experiences with flat tires.
d) Identify one mixed strategy Nash Equilibrium.
One mixed strategy Nash Equilibrium is for Student 1 to say Front-Left, Front-Right, Back-Left and BackRight each with probability .25 and for Student 2 to say Front-Left, Front-Right, Back-Left and BackRight each with probability .25. Student 1’s expected payoff no matter what she says is
.25*100+.75*30=47.5. Student 2’s expected payoff no matter what she says is .25*100+.75*30=47.5.
Therefore, a best response for each student is to randomize over the four different strategies.
Another mixed strategy Nash Equilibrium is for Student 1 to say Front-Left and Front-Right each with
probability .5 and for Student 2 to say Front-Left and Front-Right each with probability .5. Student 1’s
expected payoff if she says Front-Left or Front-Right is .5*100+.5*30=65 while her expected payoff from
saying Back-Left or Back-Right is 30 given Student 2’s strategy . A similar story holds for Student 2.
There are obviously many mixed strategy Nash Equilibria.
4. Mr. Clemens and Mr. McNamee are partners in a drug company (called Hall of Fame Results Inc.)
that produces B-12 vitamin supplements (wink-wink). Mr. Clemens is going on vacation the second
week of January and Mr. McNamee is going on vacation the first week of January. At the end of the
second week in January, Hall of Fame Results Inc. is going to introduce a new “supplement”
targeting high school and college athletes. The likely success of the new supplement will depend on
how hard Mr. Clemens and Mr. McNamee work prior to its introduction. While their payoffs
increase with the likelihood of success (holding the amount they work constant), their payoffs
decrease with the amount they work (holding the likelihood of success constant). Mr. Clemens and
Mr. McNamee will not change their vacation plans but must decide whether to work hard or not work
hard during the week they are in the office. Because Mr. Clemens is in the office the first week of
January and Mr. McNamee is in the office the second week of January, Mr. McNamee observes
whether Mr. The extensive form of the game is depicted below.
Mr. Clemens
Work Hard
Don’t Work Hard
Mr. McNamee
Work Hard
Clemens 100
McNamee 80
Mr. McNamee
Don’t
Work Hard
70
90
Work Hard
80
65
Don’t
Work Hard
60
50
What is (are) the Subgame Perfect Equilibrium (ia) of the above game?
The green lines above are obtained by backward induction and represent the following Subgame Perfect
Equilibrium.
(Clemens strategy is Don’t Work Hard ; McNamee strategy is Don’t Work Hard if Clemens selects Work
Hard and Work Hard if Clemens selects Don’t Work Hard)
5.
Lafollete Kitchen and Design is providing Stacy Dickert-Conlin a quote (i.e., specify a price) for a
kitchen renovation. For simplicity, assume Lafollete can either quote a high price of $30,000 or a
low price of $20,000. After Lafollete quotes a price, Stacy Dickert-Conlin decides whether to have
Lafollete renovate the kitchen or not to have the kitchen renovated. (Assume there is no negotiating
over the quoted price.) If Stacy Dickert-Conlin decides to have Lafollete renovate the kitchen,
Lafollete then decides whether to do a high quality job or a low quality job. Note that Lafollete
decides on the quality of the job after Stacy makes her decision. The maximum Stacy DickertConlin is willing to pay for a high quality job is $35,000 and the maximum she is willing to pay for
a low quality job is $29,000. It costs Lafollete $18,000 to do a high quality job and $17,000 to do a
low quality job.
a) Depict the extensive form of the “game” described above (i.e., the game tree).
Lafollete
High Price of $30k
Low Price of $20k
Stacy Dickert Conlin
Don’t Renovate
Stacy Dickert Conlin
Renovate
Don’t Renovate
Renovate
Lafollete
Lafollete’s Payoff:
Stacy Dickert Conlin’s Payoff:
0
0
High Quality
Quality
Lafollete’s Payoff:
Stacy Dickert Conlin’s Payoff:
30k-18k=12k
35k-30k=5k
Lafollete
0
0
Low Quality
30k-17k=13k
29k-30k=-1k
High Quality
20k-18k=2k
35k-20k=15k
Low
20k-17k=3k
29k-20k=9k
b) What is the Subgame Perfect Equilibrium of the game? BE PRECISE AND MAKE SURE YOU
IDENTIFY A STRATEGY FOR EACH PLAYER.
The red lines above represent the Subgame Perfect Equilibrium. This equilibrium is:
Lafollete bids a price of $20k and selects a low quality job if Stacy Dickert-Conlin selects to Renovate
no matter what Lafollete initially bids;
Stacy Dickert-Conlin chooses not to renovate if Lafollete bids $30k and chooses to renovate if Lafollete
bids $20k.
This equilibrium results in a payoff for Lafollete of $3k and a payoff for Stacy Dickert-Conlin of
$9k.
6.
I should have stated in the problem with Lou Anna Simon’s fixed costs are also $10.
9
8
7
6
5
4
3
2
1
0
MR
MR
If Magic has the only Chester’s, his profits are 6*40-2*40-10-FFMJ = 150- FFMJ.
If Lou Anna Simon has the only Chester’s, her profits are 5*30-2*30-10-FFMJ = 80- FFMJ.
The extensive form below represents the game.
Chester’s
FFMJ
MJ
Accept
Reject
Chester’s
Chester’s
FFLAS
FFLAS
LAS
Accept
Chester:
MJ:
LAS:
FFMJ+FFLAS
70- FFMJ
60- FFLAS
LAS
Reject
FFMJ
150- FFMJ
0
Accept
FFLAS
0
80- FFLAS
Reject
0
0
0
80
70
60
40
30
MC
0
100
90
80
70
60
50
40
30
20
10
0
MC
D
20
D
10
10
9
8
7
6
5
4
3
2
1
0
If Lou Anna Simon has only Chester’s in East Lansing
50
If Magic Johnson has only Chester’s in East Lansing
Using backward induction to determine Subgame Perfect Equilibrium.
If Magic Johnson accepted Chester’s offer, Lou Anna Simon would accept FFLAS if FFLAS<60.
If Magic Johnson rejected Chester’s offer, Lou Anna Simon would accept FFLAS if FFLAS<80.
Chester should offer Lou Anna Simon a fixed fee of FFLAS=60 if Magic accepted and a fixed fee of
FFLAS=80 if Magic Johnson rejected the offer of FFMJ.
Magic Johnson should accept FFMJ<70 (realizing that if he accepts, Chester’s will then offer a fixed
fee of 60 to Lou Anna Simon which she will accept).
Chester’s should offer a fixed fee of FFMJ =70 to Magic Johnson which he accepts. .
In the end, Chester’s offers FFMJ =70. Magic Johnson accepts the offer. Chester’s offers FFLAS =60
and Lou Anna Simon accepts this offer.
PART B)
There is a benefit from being able to commit to Magic Johnson. By committing not to offer a
franchisee to Lou Anna Simon, Magic Johnson would then be willing to accept FFMJ<150 and
Chester’s would offer FFMJ=150 while offering this commitment to Magic Johnson. Chester’s
payoff would be $20 more than when they did not commit (part a).
7. Robert Shapiro and Alan Dershowitz are currently lawyers who each have their own practice. Robert
Shapiro makes $8.25 million ($8.25M) a year at his own practice while Alan Dershowitz makes
$8.25M a year at his own practice. They each work 60 hours a week at their own practice. They are
thinking about entering a law partnership. If they establish a partnership, Robert Shapiro believes he
can generate $8M if he works 60 hours a week and believes he can generate $6M if he works 40
hours a week. If they establish a partnership, Alan Dershowitz believes he can generate $9M if he
works 60 hours a week and believes he can generate $6M if he works 40 hours a week. For each of
them, the opportunity cost of working the additional 20 hours a week (from 40 hours to 60 hours) is
$1.5M per year. Suppose Robert Shapiro and Alan Dershowitz simultaneously choose the number of
hours to work. Also assume that if they do enter a partnership, they evenly split the revenues
generated. Use backward induction to determine the most likely outcome to this game.
Identify the Subgame Perfect Equilibrium and provide a detailed explanation as to why this is the
Subgame Perfect Equilibrium.
If the partnership is formed, the normal form game in terms of Robert Shapiro and Alan
Dershowitz choosing 40 hours or 60 hours is depicted below.
Alan Dershowitz
40 hours
60 hours
40 hours
(6+6)/2 , (6+6)/2
(6+9)/2 , (6+9)/2-1.5
60 hours
(8+6)/2-1.5 , (8+6)/2
(8+9)/2-1.5 , (8+9)/2-1.5
Robert Shapiro
SAME AS
Alan Dershowitz
40 hours
60 hours
40 hours
6,6
7.5 , 6
60 hours
5.5 , 7
7,7
Robert Shapiro
Conditional on entering the partnership, the Nash equilibria of this game is for both Robert
Shapiro and Alan Dershowitz to choose 40 hours or for Robert Shapiro to work 40 hours and Alan
Dershowitz to work 60 hours. Robert Shapiro’s payoff in his own practice is 8.25-1.5=6.75 and Alan
Dershowitz’s payoff in his own practice is 8.25-1.5=6.75. No matter what Nash equilibrium in the
partnership game is likely to occur, a subgame perfect equilibrium is for Alan Dershowitz to not
enter the partnership. This is an example of what is called the “free rider problem.” Sometimes,
this free rider problem results in no partnership forming (depending on the payoffs) even when
there is a net benefit associated with the partnership. This is the case in the above example.