Simon Fraser University
Fall 2015
Econ 383 D100 Auction Theory Midterm Exam
Instructor: Songzi Du
Thursday October 29, 2015, 12:30 – 2:20 PM
Write your name, SFU ID number, and tutorial section number both on the exam
booklets and on the questionnaire. Hand in both the exam booklets and this
questionnaire. But note that only the exam booklets are graded.
• Name:
• SFU ID number:
• Tutorial section number:
General instructions
1. This is a closed-book exam: no books, notes, computer, cellphone, internet, or other
aids. A scientific, non-graphing calculator is allowed.
2. If you use decimals in calculation, keep two decimal places.
3. Write clearly. Illegible answers will receive no credit.
4. Show your work! Partial credits are given. Answers without proper explanation/calculation
will be penalized.
5. A request for regrade can be accepted only if the exam is written with a pen.
1
1. (15 points) There are three buyers {x, y, z} and three houses {a, b, c}. The buyers’
valuations are given below (assume each buyer wants only one house).
Buyer
x
y
z
value for house a value for house b value for house c
3
1
2
3
2
1
4
3
1
For this problem, assume integer prices.
a. Do prices pa = 2, pb = 0, pc = 0 clear the market? Explain.
b. There are six possible matchings of buyers to houses (x − a means buyer x is matched
to house a). Calculate the total value (i.e., social welfare) for each matching. Which one
maximizes the total value?
Matching
Total value
x − a, y − b, z − c
x − a, y − c, z − b
x − b, y − a, z − c
x − b, y − c, z − a
x − c, y − a, z − b
x − c, y − b, z − a
c. Run the ascending price auction on these values. What are the (market-clearing) prices
and matching given by the auction?
2. (10 points) There is one object for sale, and three bidders, whose values for the object
are independently and uniformly distributed over [0, 1]. Consider a second price auction with
a reserve price r.
(a) Calculate the seller’s expected revenue as a function of r, Rev(r), when bidders play
their dominant strategy. Hint: consider the following cases: (i) all three values are below r;
(ii) all three values are above r; (iii) one value is above r, and the other two values are below
r (there are three ways for this to happen); (iv) two values are above r, and the other value
is below r (there are three ways for this to happen).
2
(b) Calculate the derivative of the expected revenue with respect to r, Rev0 (r), and show
that Rev0 (r) = 3(1 − 2r)r2 . Show ALL steps in your derivation.
(c) What is the r that maximizes the expected revenue?
3. (10 points) There is one object for sale, and four bidders, whose values for the object
are independently and uniformly distributed over [0, 1]. Consider an average price auction
where the highest bidder wins the object and pays the average of all bids (all other bidders
do no pay). The equilibrium bidding strategy is si (vi ) = Avi for a constant A. Derive this
constant A.
4. (5 points) Consider an average price auction of a single object, with three bidders.
Is truthful bidding a dominant strategy? Explain.
3