171 Econ Fall 2008
Final Exam - Solutions
December 10
You have three hours to take this exam. Please answer all questions. Each question is worth
15 points, with the exception of number 3, which is worth 10 points. Point subtotals are in
brackets. To obtain credit, you must provide arguments or work to support your answer.
1. [15] Find all Nash equilibria of the following game.
L
3, 2
2, 0
1, 1
T
M
B
M
4, 0
3, 3
0, 2
R
1, 1
0, 0
2, 3
Solution : For Player 1, T strictly dominates M , hence we can eliminate M of Player
1. Next, R strictly dominates M for Player 2, hence we eliminate M of Player 2. This
leaves us with
L
3, 2
1, 1
T
B
R
1, 1
2, 3
Of this game, there are two pure-strategy Nash equilibria, (T, L) and (B, R, and one
mixed-strategy Nash equilibrium, 1 = (2/3, 1/3) and 2 = (1/3, 2/3).
2. [15] Consider the extensive-form game represented below, and notice that two of the
values for Player 3s payos are left unspecied, a and b. For all of the questions below,
you may restrict your attention to pure strategies.
1
O
I
2
2, 2, 2 A
C
B
3
4, 3, 1
X
1, 2, 6
Y
7, 5, a
X
3, 2, 1
Y
5, 0, b
(a) [3] One possibility for the missing payos is (5, 0) (that is, a = 5 and b = 0).
Another possibility is (0, 5). Which of these possibilities would make it so that
we can use backwards induction to nd SPNE? (For your answer, write either
(5, 0) or (0, 5).) Very briey state why we can use backwards induction with
these payos, even though this game form has imperfect information.
Solution : (5, 0). With these payos, X dominates Y for Player 3, making it the
only rational choice.
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(b) [6] Find all SPNE of the game, using (a, b) = (5, 0). If, in your answer above, you
stated that these payos allow us to use backwards induction, then use backwards
induction to nd the answer.
Solution : Backwards inducting, we have Player 3 choosing X at her information
set, Player 2 choosing A, and Player 1 choosing I . This yields (I, A, X ) as the
unique SPNE.
(c) [6] Find all SPNE of the game, using (a, b) = (0, 5). If, in your answer to part
(a), you stated that these payos allow us to use backwards induction, then use
backwards induction to nd the answer.
Solution : For these payos, we cannot use backwards induction, so we identify
the Nash equilibria in each subgame. In this case, it is easier to nd the NE of
the sole proper subgame, which begins with Player 2s decision node, than to nd
the NE of the entire game. The subgame can be represented with the following
matrix, which includes the payos only of Players 2 and 3, and has Player 2s
strategies in the rows and 3s strategies in the columns.
A
B
C
X
3, 1
2, 6
2, 1
Y
3, 1
5, 0
0, 5
The unique PSNE of this game is (A, X ), which means that in any SPNE of the
overall game, Players 2 and 3 must play these strategies. Player 1 can backwards
induct from this point, realizing that Player 2 will choose A, so Player 1s rational
choice is I . Thus, the SPNE strategy prole, as in the above question, is (I, A, X ).
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 1/2 and Game (b) with probability 1/2. Find a pure-strategy Bayesian
Nash equilibrium of this Bayesian game.
U
D
L
2, 2
0, 0
R
0, 0
4, 4
U
D
Game (a)
L
0, 2
4, 0
R
2, 0
0, 4
Game (b)
Figure 1: Player 1 knows which game is being played, but Player 2 does not.
Solution : First we convert this game into Bayesian normal form (see the gure on the
next page), with separate payos for each type of Player 1, one that knows that the
game is Game (a), and one that knows that it is Game (b). This allows us to quickly
verify that the strategy prole (DU, R) is the only pure-strategy BNE.
2
UU
UD
DU
DD
L
(2, 0), 2
(2, 4), 1
(0, 0), 1
(0, 4), 0
R
(0, 2), 0
(0, 0), 2
(4, 2), 2
(4, 0), 4
4. [15] Consider the two player normal-form game G shown below.
C
D
E
C
8, 8
11, 2
2, 3
D
2, 11
5, 5
2, 3
E
3, 2
3, 2
0, 0
G
(a) [7] Suppose this game is repeated T times, and each players payo is the undiscounted sum of the her payo in each period. Find all SPNE of this nitely
repeated game.
Solution : We begin by analyzing the stage game. First we can eliminate E for
both players, because is it is dominated by D . The remaining game is Prisoners
Dilemma, with the unique equilibrium, (D, D ), being Pareto inferior to (C, C ).
Because the game is nite, regardless of the history of the game, only (D, D )
can be played in the nal stage. Backwards inducting, this is true in any earlier
stage as well. Thus, in the unique SPNE, each player chooses D in every stage,
regardless of the history of the game.
(b) [8] Now suppose that G is repeated innitely, with discount factor < 1. Graphically show what average discounted payos are possible in this game in the limit
as approaches 1.
Solution : We begin by graphing the set of feasible average discounted payos,
which is the convex hull of the set of possible payo proles. Then we calculate
the minmax payo, which is 3 for each player. The shaded area in the gure (at
the end of this document) shows the set of average discounted payos that are
possible in equilibrium, which is simply the set of feasible payos for which the
payo of each player exceeds her minmax payo of 3. The vertical and horizontal
boundaries of the shaded area are not included.
5. [15] A worker interacts with a rm as follows. First the worker decides how much
to invest in developing her skills. Let x 0 denote the workers investment choice,
and suppose that the investment entails a personal cost to the worker of x2 . If the
worker is employed by the rm, her investment generates return ax for the rm. If
she decides to work for herself, her investment generates a return of bx that she keeps.
The numbers a and b are positive constants with a > b, which means that investment
is more productive in the rm.
The rm observes the workers investment, then oers the worker a wage w . The
worker then accepts or rejects the wage. If the worker accepts the wage, she works at
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the rm and payos for the worker and the rm are w x2 and ax w , respectively.
If the worker rejects the oer, she works on her own and her payo is bx x2 , while
the rms payo is zero.
(a) [5] Find and report the subgame perfect equilibrium for this game. What level of
investment does the worker choose?
Solution : Backwards inducting, we begin with the workers decision.
Given accept/reject the investment level x that she already chose, the workers payo after
rejecting the rms oer is bx x2 , and her payo after accepting the rms oer
is w x2 . Thus, it is rational for her to accept whenever x bx. In the previous
stage, the rm makes the wage oer. Any wage below bx is rejected, yielding
a payo of 0 for the rm, and any higher wage is accepted, yielding a payo of
ax w for the rm. The rm chooses the minimum acceptable wage, w = bx,
which is accepted and brings prots of (a b)x > 0. In the initial stage, the
worker chooses her investment knowing that she will end up employed by the rm
at wage bx, with payo bx x2 . Maximizing this payo, she chooses x = b/2.
In summary, the SPNE is as follows: the worker chooses x = b/2, the rm oers
w = bx and the worker accepts any oer w bx, rejecting otherwise.
(b) [5] Now suppose that the worker gets to propose the wage (after making her
investment) and that the rm chooses to accept or reject. What level of investment
prevails in equilibrium?
Solution : Using backwards induction, the rm will accept any w ax. In the
previous stage, the worker will oer the highest acceptable wage, w = ax, because
the payo for working under this wage, ax x2 , is higher than the payo for
working for herself, bx x2 . Given that she will be employed and earn payo
ax x2 , in the rst stage the worker maximizes her utility by choosing x = a/2.
So the SPNE is as follows: the worker chooses x = a/2, proposes q = ax, and the
rm accepts any w ax, rejecting otherwise.
(c) [5] State which of these two arrangements leads to a higher total payo (the sum
of the payos of the worker and the rm) and provide a brief, intuitive explanation
for why this makes sense.
Solution : When the rm oers the wage, the total payo is (ax w ) + (w x2 ) =
2
2
2
ax x2 = ab b4 . When the worker makes the oer, the total payo is a2 a4 .
2
Subtracting the former from the latter and simplifying yields ( ab )2 > 0, so
2
the total payo is greater when the worker proposes the wage. Intuitively, the
player that proposes the wage holds all the bargaining power, eectively playing
an ultimatum game with the other player and keeping the entire surplus value
created by the relationship. Knowing this, the worker has more incentive to invest
when she is the proposer and can reap the surplus created by her investment. On
the other hand, when the rm proposes the wage, it can exploit that fact that
the workers investment is already sunk and oer the minimum acceptable wage.
This lowers the workers incentive to invest in the rst place.
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1
L
R
M
2
X
3, 0
Y
X
Y
0, 1 0, 1
2, 2
3, 0
6. [15] Consider the following extensive form game:
(a) [5] This game has no pure-strategy perfect Bayesian equilibrium. Explain how we
know this.
Solution : The game has no pure-strategy Nash equilibria, so it cannot have a
pure-strategy PBE.
(b) [10] Find the mixed-strategy perfect Bayesian equilibria.
Solution : The unique form of mixed-strategy PBE is as follows: 1 = R, 2 =
(q, 1 q ), with q [1/3, 2/3], and 2 (L) = 2 (M ) = 1/2.
7. [15] In the signaling game represented below, there are two types of Player 1, strong
and weak, the probabilities of which are 0.9 and 0.1, respectively. Player 1 can consume
(B)eer or (Q)uiche for breakfast, with the strong type preferring B and the weak type
preferring Q. Player 2 can either (F)ight or (N)ot. Player 1s payo is the sum of two
elements: she obtains two units if Player 2 does not ght, and one unit if she consumes
her preferred breakfast. Player 2s payo does not depend upon Player 1s breakfast,
it is 1 if he ghts the weak type, or if he does not ght the strong type. Otherwise, it
is zero.
2, 1
0, 0
3, 0
1, 1
N
F
Q
1s
B
.9
2
N
c
.1
2
F
Q
1w
B
N
F
N
F
3, 1
1, 0
2, 0
0, 1
(a) [5] This game has no separating perfect Bayesian equilibria. Write down the
two separating strategies for Player 1, and for each one, explain why it can not
be a part of an equilibrium. Then give an intuitive explanation, based on the
above verbal description of the game, of why we should not expect any separating
equilibria.
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Solution : The two separating strategies are BQ and QB , where the rst letter
indicates the choice of the strong type and the second letter indicates the choice
of the weak type. Player 2s best response to BQ is F N , where the rst letter
indicates the response to Q and the second letter indicates the response to B .
However, given these responses, the weak type has an incentive to deviate by
imitating the strong type and choosing B . Player 2s best response to QB is NF .
Again, the weak type has an incentive to imitate the high type. If the two types
of Player 1 separate, Player 2 will be able to identify each type by her action
and will always ght the weak type while not ghting the strong type. Because
missing her preferred breakfast costs each type 1, while avoiding a ght earns 2,
it is always worth it for the weak type to switch breakfasts in order to imitate the
strong type and avoid the ght.
(b) [10] This game has two types of pooling perfect Bayesian equilibria: those that
involve both types of Player 1 pooling on B and those that involve pooling on Q.
Give one example of the second type of equilibrium.
Solution : If Player 1s chooses QQ, Player 2s best response following Q is N
because with probability .9 it was chosen by the strong type. If Player 2 observes
B , she assigns probability at least 0.5 that Player 1 is weak, and chooses F in
response. In summary, (QQ, NF ), 2 (w |Q) = .1, 2 (w |B ) 1/2.
Bonus +5 points In the equilibrium from part (b), given the weak types equilibrium payo, she
would never want to deviate and choose B , no matter what she thought Player
2s response would be. Using this fact, make an argument as to why the strong
type of Player 1 might want to deviate, making this equilibrium unreasonable.
Solution : The weak types payo in this equilibrium is 3. By deviating and choosing B , the weak type cannot attain a payo higher than 2. Thus, is is unreasonable
to conclude that a deviation came from a the weak type. Knowing this, the strong
type might want to deviate, saying to Player 2, You should understand that I am
the strong type, because the weak type could never prot by deviating. Therefore
you should not ght me. This speech would be credible and so the strong type
could protably deviate, rendering this equilibrium unreasonable.
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