ECON/TIM/CMPS 166a
Final Exam Fall 2014
Friedman & Musacchio
Problem Points Score
1 20
2
3
4
5
Total
28
17
15
20
100
1.
Barbara and Carl are dining in a French restaurant. Alice knows French and
Bob doesn’t, so Alice will order the wine from the waiter. Barbara prefers
Loire Valley wine, whereas Carl prefers Rhone Valley wine. Barbara moves first, telling the waiter either (L)oire Valley wine or (R)hone Valley wine. Carl then sees what wine arrives at the table. If it is Rhone valley wine, he is happy and the game ends. If it is Loire Valley wine, Carl chooses whether to
(A)cquiesce , or (P)rotest. Protesting embarrasses both Barbara and Carl, and is the least desirable outcome for both. Alice is happiest if she gets to have
Loire Valley wine and Carl acquiesces. The game tree and numerical payoffs are indicated below: [20 points] a.
Convert the above extensive form game tree into a strategic form matrix game. [5 points]
Solution:
A P
L
R
2,1
1,2
-1,-1
1,2
1

ECON/TIM/CMPS 166a
Final Exam Fall 2014
Friedman & Musacchio b.
Identify all pure strategy Nash equilibria in the matrix game you found in part a. [5 points]
Solution:
Both (L, A) and (R, P) are Nash equilibria. c.
Looking at the original extensive form game. Identify the Subgame
Perfect Equilibrium. [5 points]
Solution:
Only (L, A) is a SPE. d.
What does all this analysis say about the credibility of Carl threatening to protest if Barbara orders Loire Valley wine? [5 points]
Carl’s threat is not credible since once the Loire Valley wine arrives at the table, his best action going forward is to acquiesce. The analysis of SPE captures this fact.
2

ECON/TIM/CMPS 166a
Final Exam Fall 2014
Friedman & Musacchio
2.
You are the CEO of Untidy airlines. Your airline competes with one other airline, called Epsilon Air on the popular New Amsterdam to Los Diablos route. Let q u
and q e
denote the number of seats per day Untidy and Epsilon offer respectively on the route. Suppose that neither airline offers a better product than the other, so neither can charge a price premium. Instead, the average fare on the route depends on the total number of seats per day as
F = $1400 – Q where Q= q u
+q e is the total seats per day on the route. [28 points] a.
Suppose it costs each airline $500 per seat offered on the route, and that every seat offered gets bought at the fare F. Thus Untidy’s daily operating profit is

u
= (1400 – q u
– q e
) q u
– 500 q u
.
Similarly, Epsilon’s profit is

e
= (1400 – q u
– q e
) q e
– 500 q e
.
If both firms simultaneously choose the quantities of seats to offer on the route, what is the Nash Equilibrium number of seats the firms offer? What is each firm’s daily operating profit in equilibrium? [8
points]
Solution:
Both firms have concave payoff functions, so we can find best response of each by taking the derivative and equating it to zero.
Differentiating, Untidy’s payoff function and setting it to zero yields q
* u
=
900
q e .
2
Epsilon’s best response function is identical. The two best responses intersect at q u
= 300 and q e
= 300. The average fare is $800, thus each firm’s profit per seat is
$300, and a profit of $90,000 per day.
3

ECON/TIM/CMPS 166a
Final Exam Fall 2014
Friedman & Musacchio b.
Suppose you have the opportunity to acquire a more fuel efficient fleet. After getting the new fleet your cost per seat offered will be only
$200, while Epsilon’s cost per seat will be unchanged. Suppose you acquire the new fleet, and Epsilon does not. If both firms simultaneously choose the quantity of seats to place on the New
Amsterdam to Los Diablos route, what will be the Nash equilibrium number of seats each firm offers? What is each firm’s daily operating profit in equilibrium? [7 points]
Solution:
Now Untidy’s best response function is q * u
=
1200
q
2 e .
Epsilon’s is:
These intersect at q u q
* e
=
900
q u
2
= 500 and q
.
e
= 200. The average fare is $700. This gives a profit of (700 – 200)(500) =
$250,000 for Untidy and (700– 500)(200) = $40,000 for
Epsilon. c.
Now we explore what would happen if both firms have a new fleet a cost per seat of $200. Again both firms choose their quantities of seats per day simultaneously. What will be the Nash equilibrium number of seats each firm offers? What is each firm’s daily operating profit in equilibrium? [5 points]
Solution:
The best response functions are q
* u
=
1200
q
These intersect at q
2 u e
, q
= 400
* e
=
1200 and q
q u .
e
2
= 400. The average fare is $600. This gives a profit of (600 – 200)(400) =
$160,000 for each airline.
4

ECON/TIM/CMPS 166a
Final Exam Fall 2014
Friedman & Musacchio d.
Now we will explore the decision of both airlines to invest in a new fleet. For both airlines, a new fleet costs $100 million (for the entire fleet, not per plane). Suppose conditions (interest rates, etc.) are such that to convert daily profits into a net present value one must multiply by 1000. (e.g. a $1,000 a day has a Net Present Value of $1 million.)
Thus
 If both firms buy a new fleet, their long-term payoff is 1000 times their daily profit calculated in part D minus $100 million.
 If neither firm buys a new fleet, their long-term payoff is just
1,000 times their daily profit in part A.
 If one firm buys a new fleet and the other doesn’t, the firm that buys gets 1,000 times the daily profit of the firm with the cost advantage in part B minus $100 million. The other firm just gets 1,000 times the daily profit of the firm with the cost disadvantage in part B.
If both firms make the Buy vs. Not Buy decision simultaneously, write a 2x2 game matrix to describe this game. Keep your numbers in millions for this part. [5 points]
Solution:
NB B
NB
B
90, 90
150, 40
40, 150
60, 60 e.
Is there a Pure Nash equilibrium to the game you constructed in D? If so, what is it? [3 points]
Yes. It is (B,B).
5

ECON/TIM/CMPS 166a
Final Exam Fall 2014
Friedman & Musacchio
3.
Consider a variation of the traditional Hawk-Dove evolutionary game. In this variation, if two Doves meet, they cooperate to increase the value of the resources they share by an amount a. In particular, the game matrix has the form
Hawk Dove
Hawk
Dove
V
-
2 c
,
V
0, V
-
2 c
V
+
V, 0 a
,
V
+ a
2 2
Here, V is the value of the shared resource, and c is the cost of Hawks fighting each other. [17 points] a.
Under what condition is it an ESS for everyone to play Hawk? Express your answer as a condition on V, c, and a. [5 points]
(Your answer will look something like this: “(H,H) is an ESS if V> c/a or a< c 2 .” Of course your conditions will be different!)
If V>c, then (H,H) is a strict NE and thus a ESS.
If V=c, then a 2 nd condition must be met:
P(H,D) > P(D,D)
For this to be true, we need a<V.
In summary, (H,H) is an ESS if
V>c, or
V=c and a<V. b.
Under what condition is it an ESS for everyone to play Dove? [5
points]
If a>V, then (D,D) is a strict NE.
If a=V then the 2 nd condition must be met,
P(D,H) > P(H,H)
For this to be true, we need c>V.
In summary, (D,D) is an ESS if a>V, or a=V and c>V.
6

ECON/TIM/CMPS 166a
Final Exam Fall 2014
Friedman & Musacchio c.
Suppose V=10, c=20, and a=5. Are there any pure ESS? Is there a mixed ESS? If so, find it and demonstrate that it meets the necessary conditions to be an ESS. [7 points]
There are no pure ESS since the conditions from part a or b are not met. The game matrix with the numbers substituted is
Hawk
Dove
Hawk
-5,-5
0, 10
Dove
10, 0
7.5, 7.5
To have a mixed ESS, we need the population to be split in proportions such that Hawk and dove get the same payoffs. Let p be the fraction of Hawks.
P(H,p) = P(D,p)
-5p + 10 (1-p) = 0p + 7.5(1-p)
-7.5 p = -2.5
p = 2.5/7.5 = 1/3
To demonstrate that it is an ESS, we need to verify stability. When there are mostly Hawks (above 1/3), it is more attractive to be a Dove. Thus the system will be drawn toward our candidate ESS. Similarly, when there are mostly Dove (above 2/3) it is more attractive to be a Hawk. This will push the system toward the candidate ESS again.
In summary, the mixed ESS is for 1/3 of the population to be Hawk, and 2/3 Dove.
7

ECON/TIM/CMPS 166a
Final Exam Fall 2014
Friedman & Musacchio
4.
On the planet Materialistic-land, there exists a species of human-like creatures that are very materialistic. In particular sex Y strongly prefers to partner with sex X creatures that are wealthy. Nature makes only 1/3 of sex
X creatures wealthy and the rest are poor. Everyone on the planet knows these proportions.
A particular sex X creature (player X) wants to win the affections of a particular sex Y creature (player Y). Player X knows whether it has been made wealthy or poor, and it must decide whether to buy a luxury space ship or not. Player Y will not see directly whether X is wealthy or poor, but Y will see whether X chooses to buy a luxury space ship. After seeing this, Y must decide whether to accept or reject X.
Player X, whether X is rich or poor, gets 10 units of utility if accepted by Y, and 0 units if rejected. It costs 5 units of utility for a wealthy X to buy a luxury space ship, while it costs C>5 units of utility for a poor X to buy a luxury space ship since it would take a much greater fraction of a poor X’s wealth to buy one. In summary:
Buy-Accept
Wealthy X 10 – 5
Poor X 10 – C
OUTCOMES:
Not Buy Accept
10
10
Buy Reject
-5
-C
Not Buy Reject
0
0
Player Y gets a utility of 10 if it accepts a wealthy player X, and a utility of -10 if it accepts a poor player X.
Write the game tree for this game. Take care to mark information sets correctly! [15 points] a.
Under what condition on C is it a Perfect Bayesian Equilibrium (PBE) for player X to have a separating strategy of buying only if wealthy and for Y to accept only those who buy a luxury space ship? [7 points]
If X is pursuing a separating strategy, Y should believe that X is wealthy if X buys, and believe X is poor otherwise. With these beliefs, Y should accept X’s that buy and reject X’s that don’t. A wealthy X thus gets accepted after buying, giving a net payoff of 5, vs. 0 for not buying. Thus wealthy X’s should buy. A poor X gets a payoff of 10 - C for buying and 0 for not. For the poor X to find it best not to buy, it should be that
C ≥ 10.
8

ECON/TIM/CMPS 166a
Final Exam Fall 2014
Friedman & Musacchio b.
Suppose C =7. For this C, is there a PBE? If so, find it. [8 points]
From a, we see that for this value of C, it is not possible to have the separating strategy be part of a
PBE. We could consider the reverse separating strategy – that poor buys and wealthy doesn’t, but that seems unlikely to work. Indeed a quick analysis would verify that. If the X’s pool, then Y can’t learn anything, and needs to decide based on the prior probabilities.
The expected payoff from accepting a “random” X is
(2/3)(-10) + (1/3)(10) = -10/3, vs. 0 for rejecting. Thus Y should reject. Given that, it’s not worthwhile for X of either type to buy the spaceship.
In summary, the PBE is: wealthy X: Not Buy, Poor X: Not Buy
Y: Believe X is wealthy with chance 1/3 no matter if they buy or not. Reject X whether they buy or not.
9

ECON/TIM/CMPS 166a
Final Exam Fall 2014
Friedman & Musacchio
5.
Two weeks ago the business press reported major declines in telecom stocks triggered by Sprint’s large price cut for wireless data plan. The other two large firms in this industry are ATT and Verizon. [20 points] a.
Write out a static 3-player game where the players simultaneously decide whether to charge a high (H) or low (L) price. Payoffs for ATT and Verizon are R(H)=2-NL when they choose H, where NL=number of competitors choosing L; and are R(L)=1+NH when that player chooses L, where NH=number of its competitors choosing H. Sprint's payoffs are R(H)-1 or R(L)-1 when it plays H or L. [5 points]
Answer:
S:H V:H V:L
A:H
A:L
2,2,1
3 ,1,0
1, 3 ,0
2 , 2 ,-1
S:L
A:H
V:H
1,1, 2
V:L
0, 2 , 1
A:L 2 ,0, 1 1 , 1 , 0 b.
Find all NE for the static game. Which of them (if any) is in strictly dominant strategies? Which of them (if any) is efficient? [5 points]
Answer: Only one NE: (L,L,L), as seen from BR in bold above. It is in strictly dominant strategies since for each player the payoff difference is
R(H) - R(L) = 2-NL - (1+ NH) = 1-(NL+ NH)
= 1-2 = -1<0, i.e., L is strictly dominant for each player.
This NE is not efficient: the payoff sum is 2, while for
(H,H,H) the payoff sum is 5>2.
10

ECON/TIM/CMPS 166a
Final Exam Fall 2014
Friedman & Musacchio c.
Now consider the game you just wrote out as the stage game of a repeated game with no definite final period. Explain briefly how each player’s discount factor affects its payoff in the repeated game, and what determines its discount factor. [5 points]
Answer: In the repeated game the payoff is
$\sum_t^{\infty} d^t \pi_t$, where $\pi_t$ is the stage game payoff in part a. Hence a higher discount factor
$d \in [0,1]$ implies that a rational player will give greater weight to future payoffs relative to the present period payoff.
In turn, $d=\fraq{q}{1+r}$, where r is the interest rate and q is the continuation probability. That is, a player's d is higher with a higher survival probability to the next period, and a lower interest rate r.
When d is sufficiently large for all players, then cooperation on H is sustainable as a NE of the repeated game via Grim Trigger (or TfT) strategies.
{The actual calculation of the threshold value of d is not requested, but students missing other parts of the argument can get partial credit for making the calculation.} d.
Some analysts said that Sprint’s probability of surviving declined substantially earlier this year. Could that have any connection to
Sprint’s decision to play L two weeks ago? [5 points]
Answer: Yes. If q declined enough to drive d beneath the threshold value for Sprint, then they would rationally play their stage-game dominant strategy, triggering a price war. This would lower efficiency (i.e., industry profit) and reduce stock prices.
11