Problem Set III
Game Theory and Asymmetric Information (B)
1. Consider a bank B and an entrepreneur E, who applies for a loan to finance an
investment project. The entrepreneur has private information about her type: she can
be productive (P) or not-productive (N). While the entrepreneur knows her own type,
the bank has only a prior probability distribution over entrepreneur types: the bank
assesses a probability ¼ that the entrepreneur is of the productive type and of the
unproductive type with probability ¾. The application can be for a short-term or a
long-term loan. The bank can either approve a loan application or reject it. If the bank
approves a short-term loan application, both the bank and the entrepreneur (of either
type) receive a payoff of $1 (i.e., the payoffs are $1, $1). If the bank approves a longterm loan application (long-term loan carries a higher interest rate), both the
entrepreneur and bank receive a payoff of $2 if the entrepreneur is of the productive
type (i.e., the payoffs are $2, $2); both the bank and the entrepreneur receive a
negative payoff of one dollar (i.e., the payoffs $-1, $-1) if the entrepreneur is
unproductive, since the unproductive entrepreneur is likely to fail in implementing the
long-term project, thus wasting his own effort and the bank’s money. If any loan
application is rejected, both the entrepreneur and bank receive a payoff of zero dollars
each (payoffs: 0, 0).
(a) Draw the extensive form of the above game.
(b) Show that there are two perfect Bayesian equilibria of this game.
(i)
A separating PBE where the productive entrepreneur applies for a long-term
loan and the unproductive entrepreneur applies for a short-term loan, and the
bank accepts all loan applications, inferring the entrepreneur’s true type with
probability 1 from the type of loan she requests.
(ii)
A pooling PBE where both types of entrepreneurs apply only for a short-term
loan, with the bank accepting all short-term loan applications and rejecting all
long-term loan applications (and inferring that any loan applicant (short-term
or long-term) is of the productive type with probability ¼ and unproductive
with probability ¾).
(c) Apply the Cho-Kreps intuitive criterion to the above equilibria. Which of the
above equilibria survives the Cho-Kreps intuitive criterion? Provide detailed
computations to justify your answer.
Assume for the following three questions that both buyers and sellers are risk-neutral, so
that they want to maximize their expected payoff.
2. Consider an example in the spirit of Akerlof. A buyer and a seller can potentially
trade a good (car) of uncertain quality; the good is equally likely to be a lemon (=L),
or a peach (=H). The buyer has reservation values b=14 for a lemon and b=42 for a
peach. The seller has reservation values s=0 for a lemon and s=40 for a peach.
(a) Suppose neither the buyer nor the seller can tell the true quality of the car. What
quality levels of cars will be traded, and at what range of prices (if at all)?
(b) Now assume that the seller knows the true quality of the car while the buyer does
not. What quality levels of cars will be traded at what prices (if at all)?
3. Now consider a modified version of Question 2. The good (car) may have three
different quality levels: it is equally likely to be a lemon (=L), a melon (=M), or a
peach (=H). The buyer has values b=14 for a lemon, b=28 for a melon, and b=42 for
a peach. The seller has values s=0 for a lemon, s=20 for a lemon, and s=40 for a
peach.
(a) As a starting point, suppose neither the seller nor the buyer has any information
on the quality of the car. At what price or range of prices can full trade take place,
if at all? Here full trade means that all three qualities of cars will be traded.
(b) Now suppose the seller becomes partially informed: she now knows a lemon
when she sees one, but can’t distinguish a melon from a peach. The buyers’
information level remains the same as before. Now, is it possible to find a price at
which lemons, melons, and peaches can all be traded? Is it possible to find a price
at which trade can take place partially? Here partial trade means some qualities of
cars are traded while other qualities are not: e.g., only lemons are traded or only
melons are traded.
(c) Now suppose the seller is perfectly informed. The buyers’ information level
remains the same as before. Now, is it possible to find a price at which full trade
will take place? Is it possible to find a price at which trade can take place
partially?
(d) In Akerlof, we find that as the extent of asymmetric information (i.e., the
difference in information levels between buyers and sellers) increases, the extent
of trade decreases. Is this still true if we compare parts (b) and (c)? Could you
give a reason for why this (or is not) the case?
4. You find a very old piece of furniture (antique) in your attic. You don’t know the true
value of this antique, but only that this value is $100,000 with probability , and
$50,000 with probability 1-. You want to sell this to an antique dealer, who, you
know, will be able to assess the true value of the object with probability 1. Assume
that you plan to make a take-it-or-leave-it offer at price P to the dealer. If the dealer
accepts the offer, he has to pay the price P and buy the object. If the offer is rejected,
you have to keep the antique furniture in your living room as a decorative piece,
which will give you a (personal) reservation value of $30,000.
(a) What is the offer price you should set if  = 0.8? Focus only on the pure strategy
equilibrium.
(b) What is the optimal offer price you should set if =0.2? Focus only on the pure
strategy equilibrium.
(c) Suppose there are two dealers in the market and both know the true value of the
object, and there is no collusion between the two. If you auction the furniture
using a second-price sealed-bid auction (and both antique dealers participate),
what will be the selling price? What is your expected payoff? Justify your answer
with an (informal) proof.