ECOS3012 Practice Final Exam 1
November 13, 2020
The total mark of this practice exam is 65. How much can you get?
Q1 (20 points) Battle of sexes with uncertainty
After their first date, Ann finds Bob so attractive that she cannot wait to go out with
him again next weekend. However, Ann is not sure if Bob feels the same towards her.
She thinks Bob wants to meet again with probability p and avoid her with probability
1− p. If Bob wants to meet again, they play the game on the left. Otherwise, they play
the game on the right.
Bob
Opera Fight
Ann Opera 2, 1 0, 0Fight 0, 0 1, 2
Bob wants to meet
Bob
Opera Fight
Ann Opera 2, 0 0, 2Fight 0, 1 1, 0
Bob wants to avoid
(a) List all possible pure strategies for Ann and Bob. (4 points)
Let p be the probability that Bob wants to meet.
(b) Find all values of p such that there exists some pure-strategy Bayesian Nash equilib-
rium in which Ann chooses Opera. (8 points)
(c) Find all values of p such that there exists some pure-strategy Bayesian Nash equilib-
rium in which Ann chooses Fight. (8 points)
Q2 (25 points) Consider a jury made up of three 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) = 0.3, and the prior probability that the defendant is innocent (I) is
Pr(I) = 0.7. Each juror has an i.i.d. private signal si ∈ {sG, sI} which is accurate 60%
of the time, i.e., Pr (sG | G) = Pr (sI | I) = 0.6. Each juror prefers conviction if and only
if the probability that the defendant is guilty is at least 0.5.
If each juror votes for acquittal when his private signal is sI and conviction when his
private signal is sG, we say that the jurors are voting truthfully.
Recall: the Bayes’ theorem states that
Pr(A|B) = Pr(B|A) Pr(A)
Pr(B)
.
There are three possible voting rules:
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1. The defendant is convicted only when all three jurors vote for conviction.
2. The defendant is convicted when two or more jurors vote for conviction.
3. The defendant is convicted when one or more jurors vote for conviction.
(a) Under each voting rule, when is a voter pivotal? (2 points)
(b) Under which voting rule do the voters vote truthfully in a Bayesian Nash equilibrium?
Prove your answer. (9 points)
(c) In the case studied in (b), calculate the probability that an innocent defendant is
convicted (Pr (conviction | I)) and the probability that a guilty defendant is acquitted
(Pr (aquittal | G)) in equilibrium. (5 points)
(d) Is there a voting rule under which the voters always vote for acquittal regardless of
their signals in a Bayesian Nash equilibrium? Prove your answer. (9 points)
Q3 (20 points) Jiemai has developed a beta version of an AI software that allegedly can solve
all game-theory questions and she intends to sell it to an ECOS3012 student. However,
lacking performance data, no one knows whether this AI is indeed as good as it claims
to be. Say that the AI is “perfect” if it can correctly solve all questions and “imperfect”
otherwise. The common prior belief is that Pr(perfect) = 0.4 and Pr (imperfect) = 0.6.
The student is willing to purchase this AI software if and only if Pr(perfect) ≥ 0.8.
Jiemai invites the student to test the AI on one sample question free of charge, so that he
can make his purchase decision after observing the AI’s performance. Say that the free
trial is “successful” if the AI gives a correct answer to the sample question and “unsuccess-
ful” otherwise. To increase her chance of selling the AI, Jiemai carefully restricts the type
and difficulty of the sample question so that she controls the conditional probabilities
Pr (successful trial | perfect) and Pr (unsuccessful trial | perfect). The student observes
both the conditional probabilities and the trial outcome.
(a) What values of Pr (successful trial | perfect) and Pr (successful trial | imperfect) should
Jiemai choose? (8 points)
(b) Prior to the start of the trial, what is the unconditional probability that the student
purchases the AI software? (4 points)
Now, suppose that there is an infinite sequence of ECOS3012 students who all want to
purchase the software if and only if Pr(perfect) ≥ 0.8.
The students are ordered in a queue, and each gets an independent free trial with the same
conditional probabilities Pr (successful trial | perfect) and Pr (successful trial | imperfect)
as specified in (a).
The kth student observes whether his own trial is successful and whether each of the
previous k − 1 students has made a purchase or not; he does not observe whether the
previous students’ trials are successful.
(c) Can there be an informational cascade for purchasing the AI software? If so, from
which student can this cascade start at the earliest? If not, please explain. (8 points)
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