Simon Fraser University
Spring 2014
Econ 383 D300 Auction Theory Midterm Exam
Instructor: Songzi Du
Monday March 3, 2014, 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.
Do not turn over this page until instruction to start is given.
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1.
(5 points) There are four buyers {x, y, z, w} and four houses {a, b, c, d}. The
valuations are given in Table 1. Suppose that pa = 1, pb = 1, pc = 2, and pd = 2. Are
these prices market-clearing? Explain.
Buyer
x
y
z
w
value for house a value for house b value for house c value for house d
1
1
4
4
0
1
7
0
3
3
5
6
0
1
5
3
Table 1:
2.
(10 points) There are five students {x, y, z, w, u} and five available dormitories
{a, b, c, d, e}. The valuations are given in Table 2. Find the optimal matching, i.e., one that
maximizes social welfare.
Student
x
y
z
w
u
value for dorm a value for dorm b value for dorm c value for dorm d value for dorm e
3
2
2
1
0
4
1
2
1
0
3
2
0
0
0
1
2
2
0
2
0
3
2
1
2
Table 2:
3.
(10 points) There are four buyers {x, y, z, w} and four houses {a, b, c, d}. The
valuations are given in Table 3. Find a set of market-clearing prices and a perfect matching
given by the prices.
Buyer
x
y
z
w
value for house a value for house b value for house c value for house d
3
2
3
3
1
3
2
1
3
1
1
1
3
0
2
2
Table 3:
4. (10 points) Consider the following simultaneous-move game with two players. Each
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player simultaneously chooses a bid from the set {1 dollar , 50 dollars, 100 dollars }. The
player with the higher bid wins a prize of 100 dollars; the other player (the loser) does not
receive any prize. If the two bids are the same, neither player receives the prize. Each player
must pay his own bid, whether or not he wins the prize. (The loser pays too.) The payoff of
each player is simply his net winnings. For example, if player 1 bids 100 and player 2 bids
50, then player 1 gets 100 − 100 = 0 and player 2 gets 0 − 50 = −50; if player 1 bids 50 and
player 2 bids 50, then each gets 0 − 50 = −50.
(i) Write the payoff matrix of this game.
(ii) Find the pure-strategy Nash equilibria of this game.
5. (15 points) There are 10 people in a small town. The valuation of person i (1 ≤ i ≤
10) for a new park, vi , is either $0 or $100 and is his private information. The town decides
whether or not to build the park with the following voting game. Each person votes “yes”
or “no” (simultaneously) on the park, and the park will be built if and only if at least five
people vote “yes.” If the park will not be built, nobody pays anything, so everyone’s payoff
is zero. If the park will be built, then person i’s payoff is vi − Pi , where Pi is the amount
that he pays for the project.
(i) Consider first the following “naive” payment scheme if the park will be built: those
who have voted “yes” each pays $90, and those who have voted “no” pay nothing. The
truthful strategy (for any person) is to vote “yes” if one’s value is $100 and to vote “no”
if one’s value is $0. Argue that the truthful strategy is not a dominant strategy in the
voting game given the naive payment scheme.
(ii) Now consider a more “sophisticated” payment scheme if the park will be built: those
who have voted “no” still pay nothing; if person i has voted “yes,” then he pays nothing
if there are at least five other people who have voted “yes” (so the park will be built
even without i’s vote), and he pays $90 if there are only four other people who have
voted “yes” (so the part will not be built without i’s vote). Argue that the truthful
strategy is a dominant strategy in the voting game given this sophisticated payment
scheme.
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