Homework 3
David Easley and Jon Kleinberg
Due in Class September 29, 2010
As noted on the course home page, homework solutions must be submitted by upload to the
CMS site, at
https://cms.csuglab.cornell.edu/
. This means that you should write these
up in an electronic format (Word ﬁles, PDF ﬁles, and most other formats can be uploaded to
CMS).
Homework will be due at the start of class on the due date, and the CMS site will stop accepting
homework uploads after this point. We cannot accept late homework except for University-
approved excuses (which include illness, a family emergency, or travel as part of a University
sports team or other University activity).
Reading:
The questions below are primarily based on the material in Chapters 9, 10 and
11 of the book.
(1)
In this problem we will examine a second-price, sealed-bid auction for a single item.
Assume that there are three bidders who have independent, private values
v
i
; each is a random
number independently and uniformly distributed on the interval [0
,
1], and each bidder knows
his or her own value, but not the other values.
You are bidder 1 and your value for the item is
v
1
= 1
/
2. You know that bidder 2 bids
optimally given her value for the item. You know that bidder 3 seems to like to win the auction
even if winning requires him to pay more than his value for the item. In particular, bidder
3 always bids 1
/
4 more than his value for the item. How much should you bid? Provide an
explanation for your answer; a formal proof is not necessary.
(2)
(This is Exercise 10 from Chapter 9.)
In this problem we will examine a second-price,
sealed-bid auction for a single item. We’ll consider a case in which true values for the item may
diﬀer across bidders, and it requires extensive research by a bidder to determine her own true
value for an item — maybe this is because the bidder needs to determine her ability to extract
value from the item after purchasing it (and this ability may diﬀer from bidder to bidder).
There are three bidders. Bidders 1 and 2 have values
v
1
and
v
2
, each of which is a random
number independently and uniformly distributed on the interval [0
,
1]. Through having per-
formed the requisite level of research, bidders 1 and 2 know their own values for the item,
v
1
and
v
2
, respectively, but they do not know each other’s value for the item.
Bidder 3 has not performed enough research to know his own true value for the item. He
does know that he and bidder 2 are extremely similar, and therefore that his true value
v
3
is
exactly equal to the true value
v
2
of bidder 2. The problem is that bidder 3 does not know this
value
v
2
(nor does he know
v
1
).
1