STAT 155 Midterm Solutions
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Instructions: This test will be available on Tuesday 07/22/2020 at 4pm on
b-courses/pages/midterm. The solutions must be submitted to Gradescope by
10pm on Wednesday 07/22/2020. Late submissions will not be accepted. All
work must be shown in order to receive full credit. Attempt as much as you
can; partial credit will be awarded for incomplete solutions with intermediate
steps written out or explained clearly. This exam is open book, open notes but
no other resources. Feel free to email us during the midterm if you have any
concerns.
Problem # Points Your score
1 2
2 20
3 15
4 8
5 20
Total 65
c Copyright 2020 by Adam Lucas
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1. Statistics and the entire academic enterprise are based on one quality –
integrity. We are all part of a community that doesn’t fabricate evidence,
doesn’t fudge data, doesn’t present other people’s work as our own, doesn’t
lie and cheat. You trust that we will treat you fairly and with respect.
We trust that you will treat us and your fellow students fairly and with
respect. Please transcribe the UC Berkeley’s Honor Code below
and sign your name next to it:1
“I certify that all solutions will be entirely my own and that I will not
consult or share information with other people during the exam. I promise
I will act with honesty, integrity, and respect for others.”
1If you decide to type your solutions (e.g. using LATEX), you can type the answer to this
question too.
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2. Suppose there are k identical items for sale and n bidders. A bidder i
can only receive items from some publicly known set Si (for example, you
may imagine that initially items are stored in many different storehouses
scattered across the country, and each bidder can only order items from
storehouses that are less than 100 miles away from her location). The
utility of bidder i is vixi pi, where xi is the number of items that he got,
vi is his private valuation, and pi is his payment.
(a) Suppose each bidder can get several items from her set. Find the
welfare-maximizing DSIC auction.
Solution. To maximize welfare each item i should be allocated to the
highest bidder among the set {j : i ∈ Sj} i.e. among the bidders who
are able to order this item. (Another way to describe this allocation
rule is the following: first give the all the items that the highest
bidder can get to the highest bidder, then give all remaining items
that the second highest bidder can get to the second highest bidder
and so on). The corresponding DSIC auction is the following: each
item i is sold to the highest bidder from the set {j : i ∈ Sj} at the
price equal to the second highest bid from that set. In other words,
each item is sold separately in the second-price auction among the
bidders who can get it. It is dominant to bid truthfully in each
of those single-item auctions, so it is dominant to bid truthfully in
our auction. Moreover, each of those auctions yields non-
negative
utility for bidding truthfully, and thus so does our overall auction.
Therefore, our proposed auction is DSIC.
For the following parts suppose that each bidder only can
buy one item, and consider the following auction:
Bidders are served one by one, in the order of their bids
starting from the highest;
At the turn of bidder i he gets the first unsold item from Si.
If there are no such items, bidder i doesn’t get anything,
and all the bidders who got items but haven’t paid yet
pay the bid of bidder i.
For example: Suppose there are four bidders and three items,
S1 = S2 = {1, 2}, S3 = {1}, S4 = {3}, b1 = 4, b2 = 3,
b3 = 2 and b1 = 1. The first bidder will get item 1
since this is the first unsold item in S1, the second will
then get item 2, the third will get nothing, and the last
bidder will get item 3. Bidders 1 and 2 will pay 2, and
bidders 3 and 4 will not pay anything.
(b) Is the allocation rule of this auction monotone
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Solution. Yes, if all items from your set were already sold before
your turn, then decreasing your bid cannot help as it will only post-
pone your turn and all the items will still be sold.
(c) Is this allocation welfare-maximizing
Solution. No it’s not. Suppose there are two bidders and two items,
S1 = {1, 2}, S2 = {1}, b1 = 2 and b2 = 1. In the auction above the
first bidder will get item 1, and the second will get nothing. However,
the welfare-maximizing allocation is ”second bidder gets item 1, first
bidder gets item 2”.
(d) Is this auction DSIC
Solution. No. Consider the case when there are 3 bidders and two
items, S1 = {1}, S2 = S3 = {2}, b1 = 3, b2 = 2 and b3 = 1 , the
highest bidder will get item 1, and will pay the third highest price 1.
But she can avoid paying anything by decreasing her bid to b1 = 0.5
to be the lowest. If she does that, she will still get her item (as no
one else can buy it), but for free.
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3. There is one unit of infinitely divisible good for sale, and there are two
bidders who are willing to buy some shares of it. The utility of bidder i
when he gains yi of the good is vi log(1 + yi) pi where pi is the payment
and vi is the private valuation of bidder i.
(a) Represent this in the form of single-parameter environment. Note
that in a single parameter environment, there is a feasible set X of
vectors (x1, …, xn), each bidder has private valuations vi, and utility
is linear in both xi and vi.
Solution. Let’s do the change of variables: xi = log(1 + yi). Then
the utility of i-th bidder after getting the good is vixi, and the feasible
set is X = {x : ex1 + ex2 ≤ 3, x1 ≥ 0, x2 ≥ 0}.
(b) What is the welfare-maximizing allocation rule Is it monotone
Solution. We need to maximize the following quantity:
b1 log(1 + y1) + b2 log(2 y1)
in y1 ∈ [0, 1]. Take the derivative:
b1
1 + y1
b2
2 y1 .
the derivative is decreasing in y1, and is zero at y1 =
2b1 b2
b1+b2
. Thus,
the allocation that maximizes utility is
y1(b1, b2) =