MBA 710 Economic Analysis
December 14, 2004 Professor Malamud ______
Your Name
Final Examination
Question I There’s a dominant outcome (strategy profile) in here somewhere.
a) Trace your steps as you find it. (5 points)
COLUMN CHOICESA / B / C / D
R C
H
O O
I
W C
E
S / a / 1,9 / 2,9 / 2,8 / 7,3
b / 5,3 / 4,4 / 1,1 / 6,2
c / 0,8 / 1,0 / 1,9 / 2,9
d / 2,2 / 3,5 / 3,3 / 1,0
1. Row a dominates row c. (Row c will never be chosen). Now you continue …
2. Column B now dominates Column D. (Column D will never be chosen).
3. Row d now dominates Row a. (Row a will never be chosen).
4. Column B now dominates both Columns A and C. Only Column B will be chosen.
5. Row can now choose between payoffs of 4 (Row b) or 3 (Row d). Row b will be chosen.
b) Is the “dominant outcome” a Nash equilibrium? Why or why not? (5 points)
Row b Column B is a Nash equilibrium. Once there, neither player wants to move from there.
Question II
a) Construct a payoff matrix for a two-player game of chicken (you name the strategies and supply the numbers, 10 points).
This is just one of many payoff matrices that correspond to chicken:
COLUMN CHOICESBlink / Dare
C
R H
O
O I
C
W E
S / Blink / 0,0 / -5,20
Row loses face
Dare / 20,-5
Column loses face / -100,-100
Mutual destruction
b) How should you play this game? (10 points)
You’d really rather not play this game. But if you must and if you can move first, tie-your-hands, e.g., move your troops to your opponent’s border so it’s clear that you’ll have to follow through (Dare). Your opponent can then either Blink (and lose 5) or Dare (and lose everything). If your opponent moves first and ties his hands, you’d better Blink if you’re rational. If the game is repetitive, you don’t want to be stuck at the He Dares – You Blink Nash equilibrium. So next time make your move first (it’s clear that alternating between the two Nash equilibria beats Blink – Blink and certainly beats Dare – Dare). Maybe you can use a lunar calendar or a Yalta agreement to coordinate who Dares and who Blinks when and where, as the people of East Berlin (’53), Budapest (’56), and Prague (’68) learned to their regret. [Osama bin Laden may have had something like this in mind in his recent video offering to leave us alone if we stay out of “Arab lands.”]
If you’re forced to play this game of chicken and choose simultaneously, both you and your opponent should Blink with probability 95/115 = 82.6% and Dare with probability 20/115 = 17.4%. When each plays with these “optimal” probabilities, giving no advantage to the other, each has an expected loss of 100/115 = 0.870 with each play of the game, which gets back to why you really don’t want to play chicken.
Question III
Crazy Edie and Nutty Nellie are competing in the college textbook market. Based on their profit margins and the quantities demanded at different combinations of the prices they charge for a typical textbook, they can expect the profit outcomes shown in the table below (all numbers are in $000). If, for example, Edie charged $100 for a typical textbook while Nellie charged $60, Edie’s profit would be $24,000 for the semester and Nellie’s profit would be $40,000. There are, of course, a great variety of textbooks but the demands for all of them are similar. Thus, if Edie randomly priced one-third of her textbooks at $100, one-third at $80, and one-third at $60 while Nellie did the same, both Edie and Nellie’s prices for a particular book would be $100 one-ninth of the time, Edie’s price would be $100 while Nellie’s would be $80 one-ninth of the time, etc., and each would earn a profit of $41,556. They can, of course, do better than that.
NUTTY NELLIE’S PRICE$100 / $80 / $60
C
R
A P
Z
Y R
E I
D
I C
E
’ E
S / $100 / 60,60 / 54,64 / 24,40
$80 / 64,54 / 52,52 / 20,36
$60 / 40,24 / 36,20 / 30,30
Question III, continued
$100 / $80 / $60
C
R
A P
Z
Y R
E I
D
I C
E
’ E
S / $100 / 60,60 / 54,64 / 24,40
$80 / 64,54 / 52,52 / 20,36
$60 / 40,24 / 36,20 / 30,30
a) Does either competitor have a dominant strategy? In the absence of a dominant strategy, what price or mix of prices should Nellie set when she and Edie set prices simultaneously? (You needn’t come up with a precise answer; just describe Nellie’s thought process in reaching an answer, always with Edie’s responses in mind. HINT: Neither Nellie nor Edie would set any textbook’s price at $60, but you should explain why.) (10 points)
This game has three Nash equilibria: Row 80 Column 100 with payoffs (64,54); Row 100 Column 80 with payoffs (54,64); and Row 60 Column 60 with payoffs (30,30). The only time Edie would want to charge $60 is when Nellie is charging $60 and vice versa. Edie knows she’d rather not charge $60 and it’s safe that Nellie would rather not either: if Nellie assumes Edie is not charging $60, the only reason she would have for charging $60 is for spite … and spite doesn’t pay.
Just concentrating on the $100 and $80 strategies, it’s clear that some mix of these strategies is called for by both Edie and Nellie. The worst that can happen to either is a profit of $54,000 when she charges $100 and $52,000 when she charges $80. Charging $100 promises the best of the worst payoffs, but when both charge $100, each wants to switch to the $80 strategy. If they both switch to $80 at the same time, however, they both want to switch back to $100. Each would like to lock into the $80 strategy first, forcing the other to charge $100, but they must act simultaneously. Given these numbers, each should price 1/3 of all books at $100 and 2/3 at $80. [Edie sets these proportions so Nellie has no clear advantage whatever strategy she chooses … and vice versa, i.e., 60P100 +54P80 = 64 P100 + 52 P80.] Each can then expect to earn a profit of $56,000.
Question III, continued
$100 / $80 / $60
C
R
A P
Z
Y R
E I
D
I C
E
’ E
S / $100 / 60,60 / 54,64 / 24,40
$80 / 64,54 / 52,52 / 20,36
$60 / 40,24 / 36,20 / 30,30
c) Nellie sees Edie advertising, “I will not be undersold! I guarantee the lowest price on each and every textbook or your money back.” In view of this information i) what price should Nellie charge for a typical textbook? ii) what price can she expect Edie to set? iii) what profit will each of the competitors earn? and iv) is this outcome a Nash equilibrium? Explain why or why not. (10 points)
If Nellie charges $100, Edie will charge $80 (64>60) and Nellie earns $54,000; if Nellie charges $80, Edie will charge $80 (she can’t charge more than $80) and Nelly earns $52,000. Since $54,000 > $52,000, Nellie will charge $100. Nellie will earn $54,000 and Edie will earn $64,000. This is a Nash equilibrium since, with Edie charging $80, Nellie will not want to switch her price from $100 … and with Nellie charging $100, Edie will not want to switch her price either.
Question III, continued
$100 / $80 / $60
C
R
A P
Z
Y R
E I
D
I C
E
’ E
S / $100 / 60,60 / 0,64 / 0,40
$80 / 20,36 / 52,52 / 0,36
$60 / 40-,0- / 36-,0- / 30,30
c) Nellie now advertises the following promise to her customers: “If you find a book selling at a lower price than mine, I will refund you double the difference. If I charge $100 for a book, for example, and Edie charges $80 for the same book, I will refund you $40. All you will really pay for the book is $60.” In view of Nellie’s advertisement, i) what price should Edie charge for a typical book? ii) what price should Nellie set in response? iii) what profit will each of the competitors earn? and iv) is this outcome a Nash equilibrium? Explain why or why not. (10 points)
Nellie’s promise, combined with Edie’s promise made earlier, changes the payoffs in this game. All strategy profiles (outcomes) where Edie charges more than Nellie are disastrous for Edie: she would have to refund all her sales and would earn nothing. Also, if Nellie charged $100 and Edie charged $80, Nellie would really wind up charging only $60; Edie’s profit would then be $20,000 while Nellie’s would be $36,000. Nellie’s $100 strategy is now dominated by her $80 strategy. Edie will therefore charge $80 as well (she could hurt Nellie real bad if she charged $60, but she would hurt herself real bad as well: Nellie’s effective price would then be $40 ($80 – 2x(80-60)) and Edie would lose most of her sales). Since Nellie had better not charge more than Edie does … she’d have to refund double the difference …, Nellie too will charge $80. Both Edie and Nellie earn $52,000. Given Edie and Nellie’s commitments, neither wants to move from what is now a Nash equilibrium … Edie can’t move to a higher price and it would be extremely costly for Nellie to raise her price (since that would effectively lower it).
d) What, in Kreps’ words, has Nellie done to improve her outcome? (5 points)
By tying her hands, Nellie has actually worsened her outcome. The law of unintended consequences applies here. Nellie should have promised to match any price charged elsewhere, perhaps with a sweetener of a $5 gift to whoever can prove a book’s price is lower at Edie’s bookstore than at hers. It would then be a no-brainer that Edie would charge $100 for all books, Nellie would match the $100 price, and both would earn $60,000.
Question IV
Ace Camera and Zenith Optics are the sole suppliers of a specialty camera used in scientific research. Ace has the larger market share. Each knows a way to improve the camera using a more expensive production technology. Doing so would increase sales but, because of customer price resistance, might or might not increase each firm’s annual profits depending on what the other firm does. Possible outcomes are summarized in the table below, with all profits in $000.
ZENITH OPTICSImproved Camera / Same old, same old
C
A A
M
C E
R
E A / Improved
Camera / 10,2 / 15,0
Same old, same old / 3,3 / 12,5
a) What prediction would a game theorist make about the outcome of this competitive situation when Ace and Zenith choose their strategies simultaneously? Explain why. (10 points)
For Ace, Improve dominates Same Old. Recognizing this, Zenith will also choose to Improve; the respective payoffs are (10,2).
Question IV, continued
Improved Camera / Same old, same old
C
A A
M
C E
R
E A / Improved
Camera / 10,2 / 15,0
Same old, same old / 3,3 / 12,5
b) Suppose there is a 12% chance a new firm enters the market with a comparable (same old, same old) camera in any year that neither Ace nor Zenith offers an improved model. It would then be a whole new game. In addition, suppose both Ace and Zenith apply a 10% discount rate to profits earned in future years. Does either Ace or Zenith have an incentive to defect from a cooperative (same old, same old) strategy in this repetitive, year after year game, knowing its competitor will introduce an improved camera the year after it introduces one? Show your calculations[1]. (10 points)
Suppose both firms have settled at the cooperative, Same Old strategy and are earning $12,000 and $5,000, respectively. Looking into the future as far as the eye can see …
Ace, if it continues to cooperate (and assuming that Zenith continues to cooperate as well), sees an income stream of $12,000 each year into the future discounted at a 10% rate but with only a 88% chance that the game will continue each succeeding year. The expected present value of this income stream is
EPV(coop) = 12 + .88 x (12/1.1) + .882 x (12/1.12) + … = 12 x (1 + .8 + . 82 + …) = 5 x 12 = 60
If Ace defects, i.e., introduces an improved camera, it can be certain Zenith will follow next year, in which case there would be no threat of future entry by a third firm. And Ace would realize a $15,000 profit immediately. Ace’s expected income stream discounted to a present value would then be
EMV(defect) = 15 + 10/1.1 + 10/1.12 + … = 15 + (10/1.1) x (1 + 1/1.12 + …) = 15 + 100 = 115
Ace should therefore defect and introduce an improved camera immediately. Knowing this, Zenith should also introduce an improved camera immediately. The two firms wind up “cooperating” in keeping a third firm out of the market.
Question IV, continued
Improved Camera / Same old, same old
C
A A
M
C E
R
E A / Improved
Camera / 10,2 / 15,0
Same old, same old / 3,3 / 12,5
c) Suppose Ace Camera could somehow commit to its action before Zenith chooses what to do. (A commitment to introduce an improved camera is easy to make: Ace simply has to announce it. A commitment not to introduce an improved model is more difficult but possible: Ace can visibly undertake a substantial investment in facilities to produce the old model.) This changes the strategic form game of part (a) above into an extended form game. Sketch and solve this extended form game where Ace moves first. Show the payoffs at each branch of the game and highlight the path that each player would take at each decision point. Circle or state the outcome you expect. (10 points)
If both Ace and Zenith consider the expected present values of the outcomes, Ace has an overwhelming incentive to defect (EPV of $110 when, as is logical, Zenith also defects) and thus keep the game going for sure … and given that Ace defects at first, Zenith should also defect.