Econ 4400
Major Homework
This is to be done individually (without help from anyone else), but you
may consult notes, books, videos, etc.
Consider the following simplified view of a jury weighing evidence (treat
the jury as a single decision maker). A defendant in a court case is
accused of committing a crime. Independent of guilt (G) or innocence (I),
with probability 3/4, the defendant possesses evidence. This evidence is
in the set {d1, d2, d3}. When he possesses evidence, the probability of
which document it is depends on whether he is guilty or innocent. We
denote the probabilities with which di is realized, conditional on I or G,
respectively, (and conditional on evidence being realized) by pIi and pGi.
Suppose these are given by pI3 = 3/8, pI2 = 3/8, pI1 = 1/4, pG3 = 1/8, pG2 =
1/8, and pG1 = 3/4.
Assume the prior belief that the defendant is innocent is 3/8. Suppose
that the jury believes that both guilty and innocent types of the
defendant do not disclose d1, but both do disclose d2 and d3. What is the
jury’s posterior belief that the defendant is innocent when it observes
no document disclosed?
A particular model of used car comes in four categories: perfect, good,
bad, and hopeless. There are equal numbers of cars in each category.
The market consists of at least two potential buyers and sellers of the
car. Assume that there are equal numbers of buyers and sellers. All
agents are interested in maximizing the expected surplus they obtain
from having a car net of any payments they give or receive. They obtain
surplus of zero if they do not own a car. The table below gives both the
buyer and the seller’s valuation of each type of car.
Category Buyer Seller
Perfect 48 46
Good 44 42
Bad 40 38
Hopeless 36 34
So, for example, if a buyer pays the price 44 for a perfect car, he obtains
net surplus of 48 − 44 = 4. If a seller receives the price 48 for her
hopeless car, then she obtains surplus of 48 − 34 = 14.
The game works as follows: Simultaneously each buyer announces a
price that he is willing to pay for a car. Each seller decides whether or
not to sell her car (and to whom to sell it). If more than one seller
accepts the price of the same buyer, then one of the sellers is randomly
selected to sell the car.
Find the equilibrium price (or prices) assuming that the sellers know the
value of the car that they own, and buyers do not. Describe which cars
are sold.
Consider the following game with nature:
Does this game have any separating perfect Bayesian equilibrium? Show
your analysis and, if there is such an equilibrium, report it (only one is
required).
Consider the following simplified view of a jury weighing evidence (treat
the jury as a single decision maker). A defendant in a court case is
accused of committing a crime. Assume there is potentially one piece of
relevant evidence, which is denoted by d. If the defendant is guilty, d is
realized with probability pG = 1/8. If the defendant is innocent, d is
realized with probability pI = 1/2.
Assume that the jury’s prior probability that the defendant is innocent is
5/8. That is, before observing anything pertaining to the evidence
realization, the jury assigns
a prior probability of 5/8 to the defendant being innocent. Assume that
the jury forms an updated belief about the defendant’s innocence/guilt
using Bayes’ rule.
Assume that if d is realized, the defendant chooses whether to disclose
it. Suppose that the jury chooses an action a ∈ [0, 1] and has a payoff
given by uJ = −[a − θ]2
, where θ = 1 if the defendant is actually innocent
and θ = 0 if the defendant is actually guilty. Note that this payoff
function implies that the jury wishes to match its action a to its posterior
(or updated) belief that the defendant is innocent. Further, assume that
the defendant’s payoff is increasing in a so the defendant prefers that
the jury have a higher posterior to a lower one.
Assume that the defendant knows his own type (innocent or guilty) and
observes whether d exists (is realized). If d exists, the defendant chooses
whether to disclose it. The jury only observes whether d is disclosed –
not whether the defendant is guilty or innocent. When the defendant
does not disclose d, we can say that the defendant disclosed ∅. Let b(d)
denote the jury’s posterior belief that the defendant is innocent when
the defendant discloses d. Analogously, let b(∅) denote the jury’s
posterior belief that the defendant is innocent when the defendant
discloses ∅.
Finally, suppose that, when he possesses, each type of defendant
randomizes as to whether to disclose d with probability 1/2. That is,
when either type of defendant possesses d, he discloses it with
probability 1/2.
(a) Find b(d).
(b) Can the described randomization by each type of defendant be part
of a PBE? Explain. Simple intuition is fine.
Consider the following game of incomplete information played by a
worker (W) and a firm (F).
The worker has private information about her level of ability. With
probability p she is a high-ability type (H) and with probability 1 − p she
is a low-ability type (L). After observing her own type, the worker
decides whether to obtain a costly education (E) or not (N); think of E as
getting a degree. The firm observes the worker’s education but the firm
does not observe the worker’s quality type. The firm then decides
whether to employ the worker in an important managerial job (M) or in
a much less important job (C). The payoffs are represented in the above
extensive form.
Is there a separating perfect Bayesian equilibrium in which the highability type worker obtains an education and the low-ability type does
not? If yes, fully describe such an equilibrium. If no, prove there is not
such an equilibrium.
Are there any pooling perfect Bayesian equilibria in which both types of
workers take the same actions (pure strategies) regarding education? If
yes, fully describe all such equilibria. If these depend on p, describe how.
If no, prove there is not such an equilibrium.
If there are equilibria in both (a) and (b), which does the firm prefer?
Explain.
If there are equilibria in both (a) and (b), which does the low-ability
worker prefer? Explain.
If there are equilibria in both (a) and (b), which does the high-ability
worker prefer? Explain.
(f) In light of your answers to (c), (d), and (e), is it possible to rank the
pooling and signaling/separating equilibria from a social welfare
perspective? Explain.
