Simon Fraser University
Spring 2015
Econ 383 D100 Auction Theory Final Exam
Instructor: Songzi Du
Thursday April 16, 2015, 8:30 – 11:30 AM
Write your name, SFU ID number, and tutorial section number on both the exam
booklets and the questionnaire. Hand in both the exam booklets and the
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.
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1. (20 points) There are three buyers {1, 2, 3} and three houses {a, b, c}. The buyers’
values are tabulated below:
Buyer value for house a value for house b value for house c
1
4
3
1
2
3
2
2
3
2
1
0
Table 1
In parts (i)–(iv) you will calculate the allocation and payments in the VCG mechanism
given truthful reports of value.
(i) There are six matchings that assign buyers to distinct houses. Calculate the total value
(i.e., social welfare) for each of them. What is the optimal (social-welfare maximizing)
matching? This is the matching implemented by the VCG mechanism.
1
1
1
1
1
1
–
–
–
–
–
–
Matching
a, 2 – b, 3 – c
a, 2 – c, 3 – b
b, 2 – a, 3 – c
b, 2 – c, 3 – a
c, 2 – a, 3 – b
c, 2 – b, 3 – a
Total value
(ii) Consider buyer 1. What is the total value of others when buyer 1 is present in the
VCG mechanism? What is the total value of others when buyer 1 is not present in the
VCG mechanism? (To answer the second question calculate others’ total value in all
matchings without buyer 1, and choose the optimal one.) Finally, what is the payment
that buyer 1 makes in the VCG mechanism?
Matching
2 – b, 3 – c
2 – c, 3 – b
2 – a, 3 – c
2 – c, 3 – a
2 – a, 3 – b
2 – b, 3 – a
Total value without buyer 1
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(iii) Repeat (ii) for buyer 2.
Matching
1 – a, 3 – c
1 – a, 3 – b
1 – b, 3 – c
1 – b, 3 – a
1 – c, 3 – b
1 – c, 3 – a
Total value without buyer 2
(iv) Repeat (ii) for buyer 3.
Matching
1 – a, 2 – b
1 – a, 2 – c
1 – b, 2 – a
1 – b, 2 – c
1 – c, 2 – a
1 – c, 2 – b
Total value without buyer 3
(v) Run the ascending-price auction on the values in Table 1. (Consider only integer prices.)
For each step of the auction, write down the current prices and the preferred-object
graph, and identify the constricted set of buyers and the over-demanded set of houses.
What are the market-clearing prices and the matching from this auction?
2. (10 points) Suppose that there are three advertisers {1, 2, 3} and three advertisement
slots {1, 2, 3}, and each advertiser wants only one slot. Assume multiplicative valuation
vi,j = ri vj , where ri is slot i’s clickthrough rate and vj is buyer j’s revenue per click. And
assume it is common knowledge that r1 = 8, r2 = 7, r3 = 1, v1 = 10, v2 = 6 and v3 = 1.
Consider the generalized second price (GSP) auction; let bj be buyer j’s bid on a per-click
basis.
(i) Construct a Nash equilibrium of the GSP auction that is revenue-equivalent to the
VCG mechanism, and verify that no bidder has an incentive to deviate (i.e., it is indeed a
Nash equilibrium).
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(ii) Is b1 = 10, b2 = 4 and b3 = 1 a Nash equilibrium of the GSP auction? Why or why
not?
3. (5 points) Consider a third-price auction of a single object (the highest bidder gets
the object and pays the third highest bid; the others do not pay). Give an example (values
and bids) that shows that truthful bidding is not a dominant strategy in this auction, and
explain your example.
4. (10 points) Consider a second-price auction of a single object. Suppose that bidders
bid truthfully in the auction.
(i) There is a reserve price r between 0 and 1, and a single bidder whose value is randomly
drawn from the uniform distribution on [0, 1]. Calculate the expected revenue from the
auction as a function of r. What is the reserve price r that maximizes the revenue, and
what is this maximum revenue?
(ii) Now suppose there is no reserve price (r = 0). There are two bidders whose values are
randomly and independently drawn from the uniform distribution on [0, 1]. Calculate
the expected revenue from the auction. Is it higher than or lower than the maximum
revenue from part (i)?
5. (10 points) There are 20 people in a small town. The valuation of person i (where
1 ≤ i ≤ 20) 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 is built if and only if at least 10
people vote “yes.” If the park is not built, nobody pays, so everyone’s payoff is zero. If the
park is 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 is 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.
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(ii) Now consider a more “sophisticated” payment scheme if the park is built: those who
have voted “no” still pay nothing; if person i has voted “yes,” then he pays nothing if
there are at least 10 other people who have also voted “yes” (so the park is built even
without i’s vote), and he pays $90 if there are exactly 9 other people who have also
voted “yes” (so the park is not built without i’s vote). Argue that the truthful strategy
is a dominant strategy in the voting game given this sophisticated payment scheme.
(iii) Can you think of any problem with the sophisticated payment scheme?
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