Simon Fraser University
Spring 2015
Econ 383 D100 Auction Theory Midterm Exam
Instructor: Songzi Du
Thursday March 5, 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.
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1. 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
2
2
3
0
2
7
1
1
4
a. (5 points) Do prices pa = 2, pb = 1, pc = 1 clear the market? Explain.
b. (10 points) 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 of them.
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. (10 points) Run the ascending price auction on the valuations. What are the (marketclearing) prices and matching given by the auction?
2.
(10 points) There are one object and two bidders, whose values for the object
are independently and uniformly distributed over [0, 1]. Consider a second price auction
with reserve price r = 0.4. Calculate the seller’s expected revenue when bidders play their
dominant strategy. What is the probability that the object is not sold in the auction?
3.
(10 points) There are one object and three 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. The equilibrium
bidding strategy is si (vi ) = Avi for a constant A. Derive the constant A.
4. (10 points) There are two players who simultaneously select numbers between zero
and one. Suppose player 1 chooses s1 and player 2 chooses s2 . If si < sj , then player i gets a
2
payoff of (si + sj )/2 and player j gets 1 − (si + sj )/2. (For example, if s1 = 0.4 and s2 = 0.5,
then player 1 gets 0.45 and player 2 gets 0.55.) If s1 = s2 , then both players get a payoff of
1/2.
a. If player 2 chooses s2 < 1/2, what s1 should player 1 choose? Explain.
b. If player 2 chooses s2 > 1/2, what s1 should player 1 choose? Explain.
c. What is the pure-strategy Nash equilibrium? Explain.
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