Simon Fraser University
Fall 2015
Econ 302 D200 Midterm Exam Solution
Instructor: Songzi Du
Tuesday November 3, 2015, 8:30 – 10:20 AM
1. (10 points) You and your n − 1 roommates each have five hours of free time you
could spend cleaning your apartment. You all dislike cleaning, but you all like having a
clean apartment: each person’s payoff is the total hours spent (by everyone) cleaning, minus
a number c times the hours spent (individually) cleaning. That is, each roommate i gets:
ui (s1 , s2 , . . . , sn ) = −csi +
n
X
sj ,
j=1
where 0 ≤ sj ≤ 5 is the hours spent by roommate j cleaning. Assume everyone chooses
simultaneously how much time to spend cleaning.
(a) Find the Nash equilibrium if c < 1.
(b) Find the Nash equilibrium if c > 1.
(c) Set n = 5 and c = 2. Is the Nash equilibrium outcome Pareto efficient? If not, can you
find an outcome in which everyone is better off than in the Nash equilibrium outcome?
Solution:
Part (a): If c < 1, then for each player i, si = 5 is the strictly dominant strategy. So the
Nash equilibrium is every player i chooses si = 5.
Part (b): If c > 1, then for each player i, si = 0 is the strictly dominant strategy. So the
Nash equilibrium is every player i chooses si = 0.
Part (c): The Nash equilibrium outcome here of every player i choosing si = 0 is not
Pareto efficient. If everyone chooses si = 5, then everyone would be better off than in the
Nash equilibrium outcome, because −5 × 2 + 5 × 5 > −0 × 2 + 0 × 5.
2. (10 points) Player 1 is a police officer who must decide whether to patrol the streets
or to hang out at the coffee shop. His payoff from hanging out at the coffee shop is 10, while
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his payoff from patrolling the streets depends on whether he catches a robber, who is player
2. If the robber prowls the streets then the police officer will catch him and obtain a payoff
of 20. If the robber stays in his hideaway then the officer’s payoff is 0. The robber must
choose between staying hidden or prowling the streets. If he stays hidden then his payoff
is 0, while if he prowls the streets his payoff is −10 if the officer is patrolling the streets
and 10 if the officer is at the coffee shop. Suppose the officer and the robber make decision
simultaneously.
(a) Write the payoff matrix for this game.
(b) Find the Nash equilibrium (pure and mixed) of this game. What are the players’ expected
payoffs in the Nash equilibria?
Solution:
Part (a):
Prowl
Hide
Patrol
20, -10
0, 0
Coffee
10, 10
10, 0
Part (b): Clearly, there is no pure-strategy Nash equilibrium.
Suppose the officer plays p Patrol + (1 − p) Coffee and the robber plays q Prowl + (1 −
q) Hide. Then for the officer to be indifferent between Patrol and Coffee, we must have:
20q = 10, i.e., q = 1/2; for the robber to be indifferent between Prowl and Hide, we must
have: −10p + 10(1 − p) = 0, i.e., p = 1/2. This gives the mixed-strategy Nash equilibrium.
3. (15 points) Two staff managers in a student dormitory, the house manager (player
1) and kitchen manager (player 2), must select a resident assistant from a pool of three
candidates: {a, b, c}. Player 1 prefers a to b, and b to c. Player 2 prefers b to a, and a
to c. The process that is imposed on them is as follows: First, the house manager vetoes
one of the candidates and announces the veto to the central office for staff selection and to
the kitchen manager. Next the kitchen manager vetoes one of the remaining two candidates
and announces it to the central office. Finally the director of the central office assigns the
remaining candidate to be the resident assistant.
(Veto means to reject.)
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(a) Model this as an extensive-form game (using a game tree) in which a player’s mostpreferred candidate gives a payoff of 2, the second most-preferred candidate gives a
payoff of 1, and the last candidate gives 0. Draw the game tree.
(b) Find and describe the subgame-perfect equilibrium for this game.
(c) Now assume that before the two players play the game, player 2 can send an alienating
e-mail to one of the candidates, which would result in that candidate withdrawing her
application. Would player 2 choose to do this, and if so, with which candidate? Explain
by considering the subgame perfect equilibrium outcome following the withdraw of each
candidate.
Solution:
Part (a):
Part (b): The subgame perfect equilibrium is: Player 2 vetoes c if Player 1 has vetoed a,
vetoes c if Player 1 has vetoed b, vetoes a if Player 1 has vetoed c; and Player 1 vetoes b in
the first stage.
Part (c): If a withdraws, then in the SPE of the game player 1 vetoes c, so b is selected in
the end. If b withdraws, then in the SPE player 1 vetoes c, so a is selected. If c withdraws,
then in the SPE player 1 vetoes b, so a is selected. Clearly, player 2 wants a to withdraw,
so he should send the alienating email to a.
4. (10 points) Suppose that there are 200 symmetric firms each with a constant marginal
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cost of 2 and a fixed cost of 5. They first decide (simultaneously) whether to enter or not
enter a market. A firm that does not enter gets payoff 0. A firm i that enters the market
sees the other firms that have entered, plays a Cournot (quantity competition) game with
the others given a demand Q = 60 − P , and earns a payoff of P · qi − 2qi − 5. Find the
subgame perfect equilibria in pure strategy. What are the firms’ profits in these equilibria?
Solution:
In the subgame where n firms have entered the market, we first solve for the (symmetric)
Nash equilibrium in which each of the firms produces quantity q. Thus, q solves
max(60 − qi − (n − 1)q)qi − 2qi ,
qi
i.e., the first order condition:
60 − qi − (n − 1)q − qi − 2
= 0,
qi =q
⇐⇒
q=
58
.
n+1
n
In this equilibrium, the market price is P = 60 − n+1
58, and the payoff of each firm is
√
2
58
− 5. Notice that if n = 58/ 5 − 1 ≈ 24.94, each firm gets 0 payoff. If there are
(n+1)2
n > 24.94 firms in the market, then each gets a negative payoff, so they would want to leave
the market; thus n > 24.94 firms in the market is not an equilibrium. If there are n ≤ 23
firms in the market, then each gets a positive payoff; moreover, even if one additional firm
enters, we still have n + 1 ≤ 24 firms, so the payoff is still positive; thus n ≤ 23 firms in the
market is also not an equilibrium.
Therefore, the subgame perfect equilibrium is:
1. Only 24 firms enter the market in the first stage; in any subgame with n firms in the
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market, each of the n firms produces quantity q = n+1
58. In this SPE, each of the 24
582
firms gets a payoff of 252 − 5 ≈ 0.3824.
5. (10 points) Consider the following dynamic game: Player 1 can choose to play it safe
(denote this choice by S), in which case both he and player 2 get a payoff of 3 each, or he
can risk playing a game with player 2 (denote this choice by R). If he chooses R then they
play the following simultaneous-move game:
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A
B
C
8, 0
0, 2
D
6, 6
2, 2
(a) Draw a game tree that represents this game.
(b) Find and describe all subgame perfect equilibria (pure and mixed) of this game.
Solution:
Part (a):
Part (b): Clearly there is no pure-strategy Nash equilibrium in the subgame. Consider
the mixed-strategy profile (pC + (1 − p)D, qA + (1 − q)B). For player 1 to be indifferent
between C and D, we must have 8q = 6q + 2(1 − q), i.e., q = 1/2. For player 2 to be
indifferent between A and B, we must have 6(1 − p) = 2, i.e., p = 2/3. In this equilibrium
of the subgame, player 1 gets an expected payoff of 4 and player 2 gets an expected payoff
of 2. Thus, player 1 prefers the risky action R in the beginning.
In summary, the subgame perfect equilibrium is player 1 choosing (R, 2/3C + 1/3D) and
player 2 choosing 1/2A + 1/2B.
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6. (10 points) Find and describe all pure-strategy subgame perfect equilibria (if any)
in the following games. You can restrict attention to equilibrium in which player 2 chooses
B-No. Show the steps that you use to find the SPE.
In this game player 1 is either a good type or a bad type, each is realized with probability
1/2.
Solution:
We go through the four possibilities for the strategy of player 1.
1. Suppose player 1 plays (Good-A, Bad-A).
i. (A-Yes, B-No) gives player 2 expected payoff −1/4.
ii. (A-No, B-No) gives player 2 expected payoff 0.
Thus, the best response of player 2 is (A-No, B-No).
Now suppose player 2 plays (A-No, B-No). Then Good-A is a best response for the
good type of player 1; and Bad-A is a best response for the bad type of player 1.
Therefore, player 1 playing (Good-A, Bad-A) and player 2 playing (A-No, B-No) is a
SPE.
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2. Suppose player 1 plays (Good-A, Bad-B).
i. (A-Yes, B-No) gives player 2 expected payoff 3/4.
ii. (A-No, B-No) gives player 2 expected payoff 1/2.
Thus, the best response of player 2 is (A-Yes, B-No).
Now suppose player 2 plays (A-Yes, B-No). Then Bad-A is better than Bad-B for the
bad type of player 1. This is inconsistent with the initial guess.
3. Suppose player 1 plays (Good-B, Bad-A).
i. (A-Yes, B-No) gives player 2 expected payoff 0.
ii. (A-No, B-No) gives player 2 expected payoff 1/2.
Thus, the best response of player 2 is (A-No, B-No).
Now suppose player 2 plays (A-No, B-No). Then Good-B is a best response for the
good type of player 1; and Bad-A is a best response for the bad type of player 1.
Therefore, player 1 playing (Good-B, Bad-A) and player 2 playing (A-No, B-No) is a
SPE.
4. Suppose player 1 plays (Good-B, Bad-B).
i. (A-Yes, B-No) gives player 2 expected payoff 1.
ii. (A-No, B-No) gives player 2 expected payoff 1.
Thus, both (A-Yes, B-No) and (A-No, B-No) are best responses for player 2.
Suppose player 2 plays (A-Yes, B-No). Then Good-A is better than Good-B for the
good type of player 1. This is inconsistent with the initial guess.
Suppose player 2 plays (A-No, B-No). Then Good-B is a best response for the good
type of player 1; and Bad-B is a best response for the bad type of player 1.
Therefore, player 1 playing (Good-B, Bad-B) and player 2 playing (A-No, B-No) is a
SPE.
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