ECOS3012
S2 2022
Final Exam
Time Limit: 120 Minutes
This exam contains 5 pages and 5 questions. Total of points is 100.
1. Two players interact with each other. They may be in situation A or situation B. The
payoff tables for the two situations are shown below. Player 2 knows which situation
they are in, but player 1 does not. Player 1 holds the belief that situation A happens
with probability 2/3.
Player 2
C NC
Player 1
C 1, 1 5, 0
NC 0, 5 4, 4
Table 1: Situation A
Player 2
C NC
Player 1
C 1, -1 5, 0
NC 0, 3 2, 2
Table 2: Situation B
(a) (5 points) Model the strategic situation with a game tree.
(b) (10 points) Transform the game tree into a payoff table. Solve for all pure-strategy
Bayesian Nash equilibrium of this game.
ECOS3012 Final Exam - Page 2 of 5
2. Consider a jury made up of five jurors. Each juror casts a sealed vote for either acquitting
(A) or convicting (C) a defendant. The prior probability that the defendant is guilty (G)
is Pr(G) = 0.6, and the prior probability that the defendant is innocent (I) is Pr(I) = 0.4.
Each juror has an i.i.d. private signal si ∈ {sG, sI} which is accurate 80% of the time,
i.e., Pr (sG | G) = Pr (sI | I) = 0.8.
Suppose that a juror’s payoff function is as follows:
u(convict the innocent) = −3, u(aquit the guilty) = −1,
u(convict the guilty) = u(acquit the innocent) = 1.
The defendant is convicted if and only if all five jurors vote for conviction.
(a) (5 points) How would a juror vote if she were the only juror?
(b) (2 points) Suppose all jurors vote truthfully. What is the probability of an innocent
defendant being convicted? What is the probability of a guilty defendant being
acquitted?
(c) (5 points) Is truthful voting a Bayesian Nash equilibrium? Show with calculation.
(d) (8 points) Suppose a symmetric Bayesian Nash equilibrium takes the following
form: vote “C” when the private signal is sG; when the private signal is sI vote
“C” with probability q. What is the condition q must satisfy? (An equation in q
suffices.)
(e) (5 points) In the above BNE, what is the probability of an innocent defendant
being convicted? What is the probability of a guilty defendant being acquitted?
(You do not need work out the value of q. An expression in q suffices.)
ECOS3012 Final Exam - Page 3 of 5
3. A swimming club wants to hand out badges to those who are good at swimming. A
swimmer is equally likely to be of high ability (H) or low ability (L). The swimming
club is only willing to hand out the badge to a swimmer if the probability that she is of
high ability is higher than 0.9. Therefore, the club asks the swimmers to swim certain
distances to demonstrate their ability. Below is a table of costs for both types.
0 km 10 km 20 km 30 km
H 0 3 7 12
L 0 5 12 20
Table 3: Cost of swimming
Wearing the badge generates a utility of 8 for the high type and 10 for the low type.
(a) (10 points) Find a PBE where the swimming club distinguishes the two types.
State the club’s beliefs to support this PBE.
(b) (10 points) Find a PBE where the swimming club does not distinguish the two
types. State the club’s beliefs to support this PBE.
ECOS3012 Final Exam - Page 4 of 5
4. Chris wants to recommend a friend to her company for a job. The company only benefits
from a skilled candidate: If a skilled candidate is hired, the company gets a payoff of 5.
If an unskilled candidate is hired, the company gets a payoff of −20. If no one is hired,
the supervisor’s payoff is 0.
To test the candidate’s skill level, the company offers online tests at three different
difficulty levels. Chris can choose which one to send to her friend.
For the easy test,
x1 = Pr(pass ”Easy”|skilled) = 1, and y1 = Pr(pass ”Easy”|unskilled) = 0.8.
For the medium test,
x2 = Pr(pass ”Medium”|skilled) = 0.9, and y2 = Pr(pass ”Medium”|unskilled) = 0.5.
For the hard test,
x3 = Pr(pass ”Hard”|skilled) = 0.5, and y3 = Pr(pass ”Hard”|unskilled) = 0.1.
Without the test, Chris’ friend is believed (by both Chris and her company) to be skilled
with probability 0.3.
Chris wants to maximize the probability of her friend being hired.
(a) (10 points) Which test will Chris send to her friend? What is the probability of
her friend being hired?
(b) (10 points) If Chris gets to design the test herself, how would she design it? What
is the probability of her friend being hired?
ECOS3012 Final Exam - Page 5 of 5
5. Consider a standard model of information cascades with two true states {H,L} and two
actions {aH , aL}. Ex ante, Pr(H) = 0.5 and Pr(L) = 0.5.
Players payoffs are summarized in the following table.
aH aL
H 1 -1
L 0 1
Each player has an i.i.d. private signal s ∈ {sH , sL}. Pr(sH |H) = Pr(sL|L) = 0.7.
Players choose their action sequentially. Later players can observe the choices, but not
the private signals, of all previous players.
Recall that an information cascade starts when a player k mimics the previous player’s
choice regardless of player k’s own private signal.
Assume that all players adopt a “conformist’s tie-breaking rule”: when indifferent, always
choose what the previous player chose.
(a) (5 points) Does the first player find it strictly more preferable to follow his private
signal?
(b) (5 points) What is the second player’s choice rule (depending on his own private
signal and what the first player choose)?
(c) (5 points) Suppose that the true state is H. What is the probability that a cascade
of the correct action aL eventually occurs?
(d) (5 points) Suppose player 1 is an expert. Her private signal is more accurate than
others: se ∈ {sH , sL}, Pr(se = sH |H) = Pr(se = sL|L) = 0.9. Can an information
cascade of action aL start from player 2? Show your work.