171 Econ Spring 2010
Final Exam
June 8
You have three hours to take this exam. Please answer all 5 questions, for a maximum of
100 points. Point totals and subtotals are indicated in brackets. To obtain credit, you must
provide arguments or work to support your answer.
1. [10] Find all Nash equilibria of the following game.
A
B
C
X
2, 1
3, 3
1, 2
Y
4, 2
0, 0
2, 8
Z
2, 0
1, 1
5, 1
1
1
Solution : There are three equilibria. (B, X ) and (A, Y ) are PSNE and (( 4 , 5 , 0), ( 1 , 2 , 0))
5
2
is a mixed-strategy Nash equilibrium.
1
2. [20] Consider the extensive-form game represented below.
1
A
C
B
2
X
5, 2
Y
2, 6
X
6, 2
Y
6, 2
2, 6
(a) [5] Which solution concept is the appropriate one to apply to this game? Find
the set of equilibria using that concept.
Solution : Perfect-Bayesian Equilibrium is the appropriate solution concept (because it discards equilibria that are not sequentially rational). The only kind of
3
PBE is of the following form: (A, (p, 1 p)), where p = Pr(X ) [ 1 , 4 ], and player
4
1
2s beliefs are given by 2 (B ) = 2 (C ) = 2 .
(b) [5] Suppose that we delete the information set between player 2s two decision
nodes. (Plugging the values (a, b, c) = (5, 6, 2) into the gure below shows the
extensive form of this game.) In other words, suppose that 2 can actually observe
whether 1 chose B or C . What solution concept should we apply now? Find the
unique equilibrium under that concept.
Solution : Now we should apply SPNE, which yields (A, XY ).
(c) [5] Now let (a, b, c) = (3, 1, 3). Find all PSNE of this game.
Solution : The PSNE are (A, XX ), (A, XY ), and (C, XY ).
(d) [5] Which of these are subgame-perfect?
Solution : (A, XY ) and (C, XY ) are subgame-perfect.
1
A
a, 2
C
B
2
X
2
Y
2, 6
6, 2
2
X
b, 2
Y
c, 6
3. [10] Suppose two players play one of the two normal-form games shown below. Player
1 knows which game is being played, but player 2 thinks that it is Game (a) with
probability 2/3 and Game (b) with probability 1/3. Find a pure-strategy Bayesian
Nash equilibrium of this Bayesian game.
U
D
L
0, 1
1, 0
R
1, 0
2, 1
U
D
Game (a)
L
0, 1
1, 0
R
1, 0
0, 1
Game (b)
Figure 1: Player 1 knows which game is being played, but Player 2 does not.
Solution : (DU, R) is the unique, pure-strategy BNE. You can verify this by looking at
the Bayesian normal form below.
UU
UD
DU
DD
L
0, 1
12
,
33
21
,
33
1, 0
R
1, 0
21
,
33
52
,
33
4
,1
3
Figure 2: The Bayesian normal form.
3
4. [30] In the signaling game represented below, there are two types of Player 1, smart
and dumb, the probabilities of which are 0.4 and 0.6, respectively. Player 1 is in college
and can either ((Q)uit or (G)raduate. Player 2 is a prospective employer and can either
(N)ot hire or (H)ire Player 1. Player 2s payo does not depend upon 1s education,
only her intelligence. Player 1s payo depends partly on her education: both types
benet from completing their education, but the smart type gets more out of it. Player
1s payo also depends on 2s hiring decision: the smart type wants a job but the weak
type does not.
0, 0
1, 1
N
H
.4
Q
1s
G
2, 1
0, 0
2
N
H
c
.6
2
Q
1d
G
N
H
N
H
2, 0
3, 1
3, 1
1, 0
(a) [10] Find a separating PBE.
Solution : The strategy prole is (GQ, NH ), where player 1s strategy rst states
the action of the smart type, then of the dumb type, and player 2s strategy rst
states the response to Q and then the response to G. The accompanying beliefs
are 2 (s|Q) = 0 and 2 (s|G) = 1.
(b) [10] Find a pooling PBE.
Solution : The strategy prole is (GG, NN ) and the accompanying beliefs are
1
2 (s|Q) = p 2 and 2 (s|G) = .4.
(c) [10 ] Find an equilibrium in which one type of player 1 mixes, playing both Q
and G with positive probability.
Solution : The strategy prole is as follows. The smart type plays G and the dumb
2
type mixes, playing G with probability 3 and Q with probability 1 . Following Q,
3
player 2 chooses N and following player G, 2 mixes, playing N and H with equal
probability. The accompanying beliefs are 2 (s|Q) = 0 and 2 (s|G) = .5.
4
5. [30] The government decides to auction o the rights to drill all of the oil under Sulphur
Mountain. Ilse and Junjun decide to participate in the auction and, not knowing
exactly how much oil is underground, each hires a consultant to estimate the size of
the oil reserves. Consultants are expensive, though, and Ilse and Junjun can each only
aord to pay the consultant to estimate the oil under one side of the mountain.
Ilses consultant estimates that there are ei dollars worth of oil under the north side
of the mountain and Junjuns consultant estimates that there are ej dollars worth of
oil under the south side of the mountain, where ei and ej are both drawn uniformly
from the interval [0, 1]. Thus, the total value of the oil is v = ei + ej , but because each
person keeps her own estimate secret from the other, they each only know their own
estimate. The only thing that each person knows about the other persons estimate is
that it is drawn uniformly from the unit interval.
(a) [20] Suppose that the government holds a second-price auction for the oil rights. I
claim that there is a (Bayesian) Nash Equilibrium in which each player bids twice
her estimate. Verify this claim, by showing that if Junjuns strategy is bj = 2ej ,
then Ilses best response is bi = 2ei . To help you, Ive broken down the process
into the following steps:
1. [4] Write down the probability that Ilse wins as a function of her bid, bi (given
Junjuns strategy).
Solution : Pr(win) = Pr(bj < bi ) = Pr(2ej < bi ) = Pr(ej <
bi
)
2
=
bi
2
2. [4] Write down the expected price that Ilse pays if she wins.
Solution : Winning means that bj < bi , so the price would be bj . The expectation of bj given that bj < bi is b2i .
3. [4] Write down the expected value of Junjuns estimate, ej if Ilse wins.
b
Solution : Because bj = 2ej , the expected value of ej is E [ 2j |bj < bi ], which
can be rewritten as 1 E [bj |bj < bi ]. This is equal to 1 times your answer to
2
2
part 2, so the answer is b4i .
4. [4] Using these three calculations, write down Ilses expected payo for bidding bi , given ei .
Solution :
EUi (bi |ei ) = Pr(win)E [payo|win] + Pr(lose)E [payo|lose].
Because the payo from losing is zero, this simplies to EUi (bi |ei ) = Pr(win)E [payo|win].
The payo from winning is the value v = ei + ej minus the price. Plugging
in the answers from above we get
EUi (bi |ei ) =
bi bi
1
b2
bi
[ei + ] = [ei bi i ]
2
4
2
2
4
5
5. [4] Show that the bi that maximizes this expected payo is bi = 2ei .
Solution :
EUi
ei bi
= = 0.
bi
2
4
The solution to this rst-order condition is b = 2ei , so the strategy is a besti
response as claimed.
(b) [10 ] Now suppose that the auction is a rst-price auction. Is the Nash equilibrium bidding strategy going to result in higher or lower bids? Find the NE
bidding strategy and show why it works.
Solution : In a rst-price auction, the optimal bidding strategy typically yields
lower bids than in a second-price auction because the bidders trade o the probability of winning with the magnitude of the price they will have to pay if they
win. The optimal strategy in this case is for each person to bid her own estimate.
(So bi = ei and bj = ej .) We can verify this with the same ve steps:
The probability of winning is Pr(bj < bi ) = Pr(ej < bi ) = bi .
Winning implies that bj < bi . Because its a rst-price auction, the price is bi .
The expected value of ej given that Ilse wins is the same as the expected value
of bj given that bj < bi , because its taken as given that Junjuns strategy is
to bid her estimate. Given that all values are equally likely, this come out to
be bi .
2
bi
bi
EUi (bi |ei ) = Pr(win)E [payo|win] = bi E [ei +ej bi ] = bi [ei + bi ] = bi [ei ].
2
2
EUi
= ei bi = 0,
bi
so b = ei .
i
6