ECON 383
Practice Problems from Chapter 9
1, 3, 5, 6
H. K. Chen (SFU)
ECON 383
1 / 10
Chapter 9 — Ex.1
1. Consider a second price sealed-bid auction with an unknown number of
bidders. You know there will be either two or three other bidders (besides
your firm) participating in the auction. All bidders have independent,
private values for the good. Your value is c. What bid should you submit,
and how does it depend on the number of other bidders in the auction?
H. K. Chen (SFU)
ECON 383
2 / 10
Chapter 9 — Ex.1
1. Consider a second price sealed-bid auction with an unknown number of
bidders. You know there will be either two or three other bidders (besides
your firm) participating in the auction. All bidders have independent,
private values for the good. Your value is c. What bid should you submit,
and how does it depend on the number of other bidders in the auction?
You should bid c, your true value, regardless of the number of other
bidders
H. K. Chen (SFU)
ECON 383
2 / 10
Chapter 9 — Ex.1
1. Consider a second price sealed-bid auction with an unknown number of
bidders. You know there will be either two or three other bidders (besides
your firm) participating in the auction. All bidders have independent,
private values for the good. Your value is c. What bid should you submit,
and how does it depend on the number of other bidders in the auction?
You should bid c, your true value, regardless of the number of other
bidders
Two things are important in a second price auction:
H. K. Chen (SFU)
ECON 383
2 / 10
Chapter 9 — Ex.1
1. Consider a second price sealed-bid auction with an unknown number of
bidders. You know there will be either two or three other bidders (besides
your firm) participating in the auction. All bidders have independent,
private values for the good. Your value is c. What bid should you submit,
and how does it depend on the number of other bidders in the auction?
You should bid c, your true value, regardless of the number of other
bidders
Two things are important in a second price auction:
Your bid, which influences your chance of winning
H. K. Chen (SFU)
ECON 383
2 / 10
Chapter 9 — Ex.1
1. Consider a second price sealed-bid auction with an unknown number of
bidders. You know there will be either two or three other bidders (besides
your firm) participating in the auction. All bidders have independent,
private values for the good. Your value is c. What bid should you submit,
and how does it depend on the number of other bidders in the auction?
You should bid c, your true value, regardless of the number of other
bidders
Two things are important in a second price auction:
Your bid, which influences your chance of winning
The second highest bid, which determines how much you pay if you win
H. K. Chen (SFU)
ECON 383
2 / 10
Chapter 9 — Ex.1
1. Consider a second price sealed-bid auction with an unknown number of
bidders. You know there will be either two or three other bidders (besides
your firm) participating in the auction. All bidders have independent,
private values for the good. Your value is c. What bid should you submit,
and how does it depend on the number of other bidders in the auction?
You should bid c, your true value, regardless of the number of other
bidders
Two things are important in a second price auction:
Your bid, which influences your chance of winning
The second highest bid, which determines how much you pay if you win
There’s only one second highest bid, regardless the number of bidders
H. K. Chen (SFU)
ECON 383
2 / 10
Chapter 9 — Ex.1
1. Consider a second price sealed-bid auction with an unknown number of
bidders. You know there will be either two or three other bidders (besides
your firm) participating in the auction. All bidders have independent,
private values for the good. Your value is c. What bid should you submit,
and how does it depend on the number of other bidders in the auction?
You should bid c, your true value, regardless of the number of other
bidders
Two things are important in a second price auction:
Your bid, which influences your chance of winning
The second highest bid, which determines how much you pay if you win
There’s only one second highest bid, regardless the number of bidders
Therefore, in a second price auction, bidding truthfully is a weakly
dominant strategy regardless of the number of bidders in the auction
H. K. Chen (SFU)
ECON 383
2 / 10
Chapter 9 — Ex.3(a)
3. All bidders in a second price auction have private independent values
vi ∈ {0, 1}, each value realizes with equal probability.
(a) Suppose i ∈ {1, 2}. Show that the seller’s expected revenue is 1/4.
H. K. Chen (SFU)
ECON 383
3 / 10
Chapter 9 — Ex.3(a)
3. All bidders in a second price auction have private independent values
vi ∈ {0, 1}, each value realizes with equal probability.
(a) Suppose i ∈ {1, 2}. Show that the seller’s expected revenue is 1/4.
Consider all four possible pairs of bidder values:
( v1 , v2 )
(0, 0)
(0, 1)
(1, 0)
(1, 1)
H. K. Chen (SFU)
Prob
1/4
1/4
1/4
1/4
selling price
ECON 383
expected revenue
3 / 10
Chapter 9 — Ex.3(a)
3. All bidders in a second price auction have private independent values
vi ∈ {0, 1}, each value realizes with equal probability.
(a) Suppose i ∈ {1, 2}. Show that the seller’s expected revenue is 1/4.
Consider all four possible pairs of bidder values:
( v1 , v2 )
(0, 0)
(0, 1)
(1, 0)
(1, 1)
H. K. Chen (SFU)
Prob
1/4
1/4
1/4
1/4
selling price
0
ECON 383
expected revenue
3 / 10
Chapter 9 — Ex.3(a)
3. All bidders in a second price auction have private independent values
vi ∈ {0, 1}, each value realizes with equal probability.
(a) Suppose i ∈ {1, 2}. Show that the seller’s expected revenue is 1/4.
Consider all four possible pairs of bidder values:
( v1 , v2 )
(0, 0)
(0, 1)
(1, 0)
(1, 1)
H. K. Chen (SFU)
Prob
1/4
1/4
1/4
1/4
selling price
0
ECON 383
expected revenue
0
3 / 10
Chapter 9 — Ex.3(a)
3. All bidders in a second price auction have private independent values
vi ∈ {0, 1}, each value realizes with equal probability.
(a) Suppose i ∈ {1, 2}. Show that the seller’s expected revenue is 1/4.
Consider all four possible pairs of bidder values:
( v1 , v2 )
(0, 0)
(0, 1)
(1, 0)
(1, 1)
H. K. Chen (SFU)
Prob
1/4
1/4
1/4
1/4
selling price
0
0
ECON 383
expected revenue
0
0
3 / 10
Chapter 9 — Ex.3(a)
3. All bidders in a second price auction have private independent values
vi ∈ {0, 1}, each value realizes with equal probability.
(a) Suppose i ∈ {1, 2}. Show that the seller’s expected revenue is 1/4.
Consider all four possible pairs of bidder values:
( v1 , v2 )
(0, 0)
(0, 1)
(1, 0)
(1, 1)
H. K. Chen (SFU)
Prob
1/4
1/4
1/4
1/4
selling price
0
0
0
ECON 383
expected revenue
0
0
0
3 / 10
Chapter 9 — Ex.3(a)
3. All bidders in a second price auction have private independent values
vi ∈ {0, 1}, each value realizes with equal probability.
(a) Suppose i ∈ {1, 2}. Show that the seller’s expected revenue is 1/4.
Consider all four possible pairs of bidder values:
( v1 , v2 )
(0, 0)
(0, 1)
(1, 0)
(1, 1)
H. K. Chen (SFU)
Prob
1/4
1/4
1/4
1/4
selling price
0
0
0
1
ECON 383
expected revenue
0
0
0
3 / 10
Chapter 9 — Ex.3(a)
3. All bidders in a second price auction have private independent values
vi ∈ {0, 1}, each value realizes with equal probability.
(a) Suppose i ∈ {1, 2}. Show that the seller’s expected revenue is 1/4.
Consider all four possible pairs of bidder values:
( v1 , v2 )
(0, 0)
(0, 1)
(1, 0)
(1, 1)
H. K. Chen (SFU)
Prob
1/4
1/4
1/4
1/4
selling price
0
0
0
1
ECON 383
expected revenue
0
0
0
1/4
3 / 10
Chapter 9 — Ex.3(a)
3. All bidders in a second price auction have private independent values
vi ∈ {0, 1}, each value realizes with equal probability.
(a) Suppose i ∈ {1, 2}. Show that the seller’s expected revenue is 1/4.
Consider all four possible pairs of bidder values:
( v1 , v2 )
(0, 0)
(0, 1)
(1, 0)
(1, 1)
H. K. Chen (SFU)
Prob selling price
1/4
0
0
1/4
1/4
0
1/4
1
Total
ECON 383
expected revenue
0
0
0
1/4
1/4
3 / 10
Chapter 9 — Ex.3(b)
3(b) What is the seller’s expected revenue if there are three bidders?
H. K. Chen (SFU)
ECON 383
4 / 10
Chapter 9 — Ex.3(b)
3(b) What is the seller’s expected revenue if there are three bidders?
There are eight possible combinations of bidder values:
(v1 , v2 , v3 )
(0, 0, 0)
(0, 0, 1)
(0, 1, 0)
(0, 1, 1)
(1, 0, 0)
(1, 0, 1)
(1, 1, 0)
(1, 1, 1)
H. K. Chen (SFU)
Prob
1/8
1/8
1/8
1/8
1/8
1/8
1/8
1/8
selling price
ECON 383
expected revenue
4 / 10
Chapter 9 — Ex.3(b)
3(b) What is the seller’s expected revenue if there are three bidders?
There are eight possible combinations of bidder values:
(v1 , v2 , v3 )
(0, 0, 0)
(0, 0, 1)
(0, 1, 0)
(0, 1, 1)
(1, 0, 0)
(1, 0, 1)
(1, 1, 0)
(1, 1, 1)
H. K. Chen (SFU)
Prob
1/8
1/8
1/8
1/8
1/8
1/8
1/8
1/8
selling price
0
ECON 383
expected revenue
0
4 / 10
Chapter 9 — Ex.3(b)
3(b) What is the seller’s expected revenue if there are three bidders?
There are eight possible combinations of bidder values:
(v1 , v2 , v3 )
(0, 0, 0)
(0, 0, 1)
(0, 1, 0)
(0, 1, 1)
(1, 0, 0)
(1, 0, 1)
(1, 1, 0)
(1, 1, 1)
H. K. Chen (SFU)
Prob
1/8
1/8
1/8
1/8
1/8
1/8
1/8
1/8
selling price
0
0
ECON 383
expected revenue
0
0
4 / 10
Chapter 9 — Ex.3(b)
3(b) What is the seller’s expected revenue if there are three bidders?
There are eight possible combinations of bidder values:
(v1 , v2 , v3 )
(0, 0, 0)
(0, 0, 1)
(0, 1, 0)
(0, 1, 1)
(1, 0, 0)
(1, 0, 1)
(1, 1, 0)
(1, 1, 1)
H. K. Chen (SFU)
Prob
1/8
1/8
1/8
1/8
1/8
1/8
1/8
1/8
selling price
0
0
0
ECON 383
expected revenue
0
0
0
4 / 10
Chapter 9 — Ex.3(b)
3(b) What is the seller’s expected revenue if there are three bidders?
There are eight possible combinations of bidder values:
(v1 , v2 , v3 )
(0, 0, 0)
(0, 0, 1)
(0, 1, 0)
(0, 1, 1)
(1, 0, 0)
(1, 0, 1)
(1, 1, 0)
(1, 1, 1)
H. K. Chen (SFU)
Prob
1/8
1/8
1/8
1/8
1/8
1/8
1/8
1/8
selling price
0
0
0
1
ECON 383
expected revenue
0
0
0
1/8
4 / 10
Chapter 9 — Ex.3(b)
3(b) What is the seller’s expected revenue if there are three bidders?
There are eight possible combinations of bidder values:
(v1 , v2 , v3 )
(0, 0, 0)
(0, 0, 1)
(0, 1, 0)
(0, 1, 1)
(1, 0, 0)
(1, 0, 1)
(1, 1, 0)
(1, 1, 1)
H. K. Chen (SFU)
Prob
1/8
1/8
1/8
1/8
1/8
1/8
1/8
1/8
selling price
0
0
0
1
0
ECON 383
expected revenue
0
0
0
1/8
0
4 / 10
Chapter 9 — Ex.3(b)
3(b) What is the seller’s expected revenue if there are three bidders?
There are eight possible combinations of bidder values:
(v1 , v2 , v3 )
(0, 0, 0)
(0, 0, 1)
(0, 1, 0)
(0, 1, 1)
(1, 0, 0)
(1, 0, 1)
(1, 1, 0)
(1, 1, 1)
H. K. Chen (SFU)
Prob
1/8
1/8
1/8
1/8
1/8
1/8
1/8
1/8
selling price
0
0
0
1
0
1
ECON 383
expected revenue
0
0
0
1/8
0
1/8
4 / 10
Chapter 9 — Ex.3(b)
3(b) What is the seller’s expected revenue if there are three bidders?
There are eight possible combinations of bidder values:
(v1 , v2 , v3 )
(0, 0, 0)
(0, 0, 1)
(0, 1, 0)
(0, 1, 1)
(1, 0, 0)
(1, 0, 1)
(1, 1, 0)
(1, 1, 1)
H. K. Chen (SFU)
Prob
1/8
1/8
1/8
1/8
1/8
1/8
1/8
1/8
selling price
0
0
0
1
0
1
1
ECON 383
expected revenue
0
0
0
1/8
0
1/8
1/8
4 / 10
Chapter 9 — Ex.3(b)
3(b) What is the seller’s expected revenue if there are three bidders?
There are eight possible combinations of bidder values:
(v1 , v2 , v3 )
(0, 0, 0)
(0, 0, 1)
(0, 1, 0)
(0, 1, 1)
(1, 0, 0)
(1, 0, 1)
(1, 1, 0)
(1, 1, 1)
H. K. Chen (SFU)
Prob
1/8
1/8
1/8
1/8
1/8
1/8
1/8
1/8
selling price
0
0
0
1
0
1
1
1
ECON 383
expected revenue
0
0
0
1/8
0
1/8
1/8
1/8
4 / 10
Chapter 9 — Ex.3(b)
3(b) What is the seller’s expected revenue if there are three bidders?
There are eight possible combinations of bidder values:
(v1 , v2 , v3 )
(0, 0, 0)
(0, 0, 1)
(0, 1, 0)
(0, 1, 1)
(1, 0, 0)
(1, 0, 1)
(1, 1, 0)
(1, 1, 1)
H. K. Chen (SFU)
Prob selling price
1/8
0
0
1/8
1/8
0
1/8
1
0
1/8
1
1/8
1/8
1
1/8
1
Total
ECON 383
expected revenue
0
0
0
1/8
0
1/8
1/8
1/8
1/2
4 / 10
Chapter 9 — Ex.3(c)
3(c) Explain why the seller’s expected revenue increases as the number of
bidders increases.
H. K. Chen (SFU)
ECON 383
5 / 10
Chapter 9 — Ex.3(c)
3(c) Explain why the seller’s expected revenue increases as the number of
bidders increases.
As the number of bidders increases, it becomes more likely that the
second highest value among them is 1 instead of 0
H. K. Chen (SFU)
ECON 383
5 / 10
Chapter 9 — Ex.3(c)
3(c) Explain why the seller’s expected revenue increases as the number of
bidders increases.
As the number of bidders increases, it becomes more likely that the
second highest value among them is 1 instead of 0
As a general observation, when the number of bidders increases, it is
less likely for all bidders except one to have a value of 0
H. K. Chen (SFU)
ECON 383
5 / 10
Chapter 9 — Ex.3(c)
3(c) Explain why the seller’s expected revenue increases as the number of
bidders increases.
As the number of bidders increases, it becomes more likely that the
second highest value among them is 1 instead of 0
As a general observation, when the number of bidders increases, it is
less likely for all bidders except one to have a value of 0
In other words, it becomes very unlikely for the second highest bid to
be 0 as there are more and more bidders
H. K. Chen (SFU)
ECON 383
5 / 10
Chapter 9 — Ex.3(c)
3(c) Explain why the seller’s expected revenue increases as the number of
bidders increases.
As the number of bidders increases, it becomes more likely that the
second highest value among them is 1 instead of 0
As a general observation, when the number of bidders increases, it is
less likely for all bidders except one to have a value of 0
In other words, it becomes very unlikely for the second highest bid to
be 0 as there are more and more bidders
Consequently, the selling price of 1 occurs more frequently, which
results in a higher expected revenue for the seller
H. K. Chen (SFU)
ECON 383
5 / 10
Chapter 9 — Ex.3(c)
3(c) Explain why the seller’s expected revenue increases as the number of
bidders increases.
Formally, for any n bidders, selling price is 0 only if the profile of
values (v1 , . . . , vn ) has no more than one 1.
H. K. Chen (SFU)
ECON 383
6 / 10
Chapter 9 — Ex.3(c)
3(c) Explain why the seller’s expected revenue increases as the number of
bidders increases.
Formally, for any n bidders, selling price is 0 only if the profile of
values (v1 , . . . , vn ) has no more than one 1.
There are 2n possible profiles of bidder values, among them, n + 1
profiles have no more than one 1
H. K. Chen (SFU)
ECON 383
6 / 10
Chapter 9 — Ex.3(c)
3(c) Explain why the seller’s expected revenue increases as the number of
bidders increases.
Formally, for any n bidders, selling price is 0 only if the profile of
values (v1 , . . . , vn ) has no more than one 1.
There are 2n possible profiles of bidder values, among them, n + 1
profiles have no more than one 1
Thus, the expected revenue (as a function of number of bidders) is
n+1
R(n) = 1 − n
×1
2
|
{z
}
Pr(selling price6=0)
H. K. Chen (SFU)
ECON 383
6 / 10
Chapter 9 — Ex.3(c)
3(c) Explain why the seller’s expected revenue increases as the number of
bidders increases.
Formally, for any n bidders, selling price is 0 only if the profile of
values (v1 , . . . , vn ) has no more than one 1.
There are 2n possible profiles of bidder values, among them, n + 1
profiles have no more than one 1
Thus, the expected revenue (as a function of number of bidders) is
n+1
R(n) = 1 − n
×1
2
|
{z
}
Pr(selling price6=0)
It is obvious to see that limn→∞ R(n) = 1
H. K. Chen (SFU)
ECON 383
6 / 10
Chapter 9 — Ex.5
5. One seller and two bidders interact in a second price auction. Seller
values the object at s while buyers 1 and 2 value it at v1 , v2 , respectively.
All three values are independent and private. Suppose both buyers know
that the seller will submit his own sealed bid of s, but they don’t know the
value of s. Is it optimal for the buyers to bid truthfully?
H. K. Chen (SFU)
ECON 383
7 / 10
Chapter 9 — Ex.5
5. One seller and two bidders interact in a second price auction. Seller
values the object at s while buyers 1 and 2 value it at v1 , v2 , respectively.
All three values are independent and private. Suppose both buyers know
that the seller will submit his own sealed bid of s, but they don’t know the
value of s. Is it optimal for the buyers to bid truthfully?
This is the same as having a third buyer entering the auction.
H. K. Chen (SFU)
ECON 383
7 / 10
Chapter 9 — Ex.5
5. One seller and two bidders interact in a second price auction. Seller
values the object at s while buyers 1 and 2 value it at v1 , v2 , respectively.
All three values are independent and private. Suppose both buyers know
that the seller will submit his own sealed bid of s, but they don’t know the
value of s. Is it optimal for the buyers to bid truthfully?
This is the same as having a third buyer entering the auction.
Based on the reasoning in Ex.1, both buyers 1 and 2 should still bid
truthfully.
H. K. Chen (SFU)
ECON 383
7 / 10
Chapter 9 — Ex.6(a)
6. One object is being sold using a second price auction. There are several
bidders with independent private values drawn from a distribution on
[0, 1]. Consider the possibility of collusion between two bidders who know
each others’ value for the object. The objective of the colluding bidders is
to maximize their joint payoffs. All bids should be within the [0, 1] range.
(a) Suppose there are only two bidders. What bids should they submit?
H. K. Chen (SFU)
ECON 383
8 / 10
Chapter 9 — Ex.6(a)
6. One object is being sold using a second price auction. There are several
bidders with independent private values drawn from a distribution on
[0, 1]. Consider the possibility of collusion between two bidders who know
each others’ value for the object. The objective of the colluding bidders is
to maximize their joint payoffs. All bids should be within the [0, 1] range.
(a) Suppose there are only two bidders. What bids should they submit?
Since loser’s payoff is 0, maximizing joint payoffs is the same as
maximizing the winner’s payoff
H. K. Chen (SFU)
ECON 383
8 / 10
Chapter 9 — Ex.6(a)
6. One object is being sold using a second price auction. There are several
bidders with independent private values drawn from a distribution on
[0, 1]. Consider the possibility of collusion between two bidders who know
each others’ value for the object. The objective of the colluding bidders is
to maximize their joint payoffs. All bids should be within the [0, 1] range.
(a) Suppose there are only two bidders. What bids should they submit?
Since loser’s payoff is 0, maximizing joint payoffs is the same as
maximizing the winner’s payoff
.
Since vi is given, winner’s payoff vi − p is maximized when
H. K. Chen (SFU)
ECON 383
8 / 10
Chapter 9 — Ex.6(a)
6. One object is being sold using a second price auction. There are several
bidders with independent private values drawn from a distribution on
[0, 1]. Consider the possibility of collusion between two bidders who know
each others’ value for the object. The objective of the colluding bidders is
to maximize their joint payoffs. All bids should be within the [0, 1] range.
(a) Suppose there are only two bidders. What bids should they submit?
Since loser’s payoff is 0, maximizing joint payoffs is the same as
maximizing the winner’s payoff
Since vi is given, winner’s payoff vi − p is maximized when p = 0 .
H. K. Chen (SFU)
ECON 383
8 / 10
Chapter 9 — Ex.6(a)
6. One object is being sold using a second price auction. There are several
bidders with independent private values drawn from a distribution on
[0, 1]. Consider the possibility of collusion between two bidders who know
each others’ value for the object. The objective of the colluding bidders is
to maximize their joint payoffs. All bids should be within the [0, 1] range.
(a) Suppose there are only two bidders. What bids should they submit?
Since loser’s payoff is 0, maximizing joint payoffs is the same as
maximizing the winner’s payoff
Since vi is given, winner’s payoff vi − p is maximized when p = 0 .
Given the objective, the winner should be
H. K. Chen (SFU)
ECON 383
8 / 10
Chapter 9 — Ex.6(a)
6. One object is being sold using a second price auction. There are several
bidders with independent private values drawn from a distribution on
[0, 1]. Consider the possibility of collusion between two bidders who know
each others’ value for the object. The objective of the colluding bidders is
to maximize their joint payoffs. All bids should be within the [0, 1] range.
(a) Suppose there are only two bidders. What bids should they submit?
Since loser’s payoff is 0, maximizing joint payoffs is the same as
maximizing the winner’s payoff
Since vi is given, winner’s payoff vi − p is maximized when p = 0 .
Given the objective, the winner should be bidder with higher vi
H. K. Chen (SFU)
ECON 383
8 / 10
Chapter 9 — Ex.6(a)
6. One object is being sold using a second price auction. There are several
bidders with independent private values drawn from a distribution on
[0, 1]. Consider the possibility of collusion between two bidders who know
each others’ value for the object. The objective of the colluding bidders is
to maximize their joint payoffs. All bids should be within the [0, 1] range.
(a) Suppose there are only two bidders. What bids should they submit?
Since loser’s payoff is 0, maximizing joint payoffs is the same as
maximizing the winner’s payoff
Since vi is given, winner’s payoff vi − p is maximized when p = 0 .
Given the objective, the winner should be bidder with higher vi
Therefore, the two bids should be
bL
and bH
where bL is the bid submitted by the bidder with lower value, and bH
the bid submitted by the bidder with higher value
H. K. Chen (SFU)
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Chapter 9 — Ex.6(a)
6. One object is being sold using a second price auction. There are several
bidders with independent private values drawn from a distribution on
[0, 1]. Consider the possibility of collusion between two bidders who know
each others’ value for the object. The objective of the colluding bidders is
to maximize their joint payoffs. All bids should be within the [0, 1] range.
(a) Suppose there are only two bidders. What bids should they submit?
Since loser’s payoff is 0, maximizing joint payoffs is the same as
maximizing the winner’s payoff
Since vi is given, winner’s payoff vi − p is maximized when p = 0 .
Given the objective, the winner should be bidder with higher vi
Therefore, the two bids should be
bL = 0
and bH > 0
where bL is the bid submitted by the bidder with lower value, and bH
the bid submitted by the bidder with higher value
H. K. Chen (SFU)
ECON 383
8 / 10
Chapter 9 — Ex.6(b)
6(b) Suppose there is a third bidder who is not part of the collusion. Does
the existence of this bidder change the optimal bids for the two colluding
bidders?
H. K. Chen (SFU)
ECON 383
9 / 10
Chapter 9 — Ex.6(b)
6(b) Suppose there is a third bidder who is not part of the collusion. Does
the existence of this bidder change the optimal bids for the two colluding
bidders?
We can treat the two colluding bidders (L and H) as a single player
with two choice variables:
H. K. Chen (SFU)
ECON 383
9 / 10
Chapter 9 — Ex.6(b)
6(b) Suppose there is a third bidder who is not part of the collusion. Does
the existence of this bidder change the optimal bids for the two colluding
bidders?
We can treat the two colluding bidders (L and H) as a single player
with two choice variables:
bH to influence the probability of winning; and
bL to minimize the payment, conditional on winning
H. K. Chen (SFU)
ECON 383
9 / 10
Chapter 9 — Ex.6(b)
6(b) Suppose there is a third bidder who is not part of the collusion. Does
the existence of this bidder change the optimal bids for the two colluding
bidders?
We can treat the two colluding bidders (L and H) as a single player
with two choice variables:
bH to influence the probability of winning; and
bL to minimize the payment, conditional on winning
What should be the third bidder’s strategy?
H. K. Chen (SFU)
ECON 383
.
9 / 10
Chapter 9 — Ex.6(b)
6(b) Suppose there is a third bidder who is not part of the collusion. Does
the existence of this bidder change the optimal bids for the two colluding
bidders?
We can treat the two colluding bidders (L and H) as a single player
with two choice variables:
bH to influence the probability of winning; and
bL to minimize the payment, conditional on winning
What should be the third bidder’s strategy?
Bid truthfully
.
Bidding truthfully is still a dominant strategy for the third bidder.
H. K. Chen (SFU)
ECON 383
9 / 10
Chapter 9 — Ex.6(b)
6(b) Suppose there is a third bidder who is not part of the collusion. Does
the existence of this bidder change the optimal bids for the two colluding
bidders?
We can treat the two colluding bidders (L and H) as a single player
with two choice variables:
bH to influence the probability of winning; and
bL to minimize the payment, conditional on winning
What should be the third bidder’s strategy?
Bid truthfully
.
Bidding truthfully is still a dominant strategy for the third bidder.
What should be the optimal bL ?
H. K. Chen (SFU)
ECON 383
9 / 10
Chapter 9 — Ex.6(b)
6(b) Suppose there is a third bidder who is not part of the collusion. Does
the existence of this bidder change the optimal bids for the two colluding
bidders?
We can treat the two colluding bidders (L and H) as a single player
with two choice variables:
bH to influence the probability of winning; and
bL to minimize the payment, conditional on winning
What should be the third bidder’s strategy?
Bid truthfully
.
Bidding truthfully is still a dominant strategy for the third bidder.
What should be the optimal bL ?
If won (bH > v3 ), then any bL ∈ [0, v3 ] is optimal: colluders have to
pay more if bL > v3
H. K. Chen (SFU)
ECON 383
9 / 10
Chapter 9 — Ex.6(b)
6(b) Suppose there is a third bidder who is not part of the collusion. Does
the existence of this bidder change the optimal bids for the two colluding
bidders?
We can treat the two colluding bidders (L and H) as a single player
with two choice variables:
bH to influence the probability of winning; and
bL to minimize the payment, conditional on winning
What should be the third bidder’s strategy?
Bid truthfully
.
Bidding truthfully is still a dominant strategy for the third bidder.
What should be the optimal bL ?
If won (bH > v3 ), then any bL ∈ [0, v3 ] is optimal: colluders have to
pay more if bL > v3
If lost (bH ≤ v3 ), then any bL ∈ [0, bH ] is optimal: payoff is zero
anyway
H. K. Chen (SFU)
ECON 383
9 / 10
Chapter 9 — Ex.6(b)
6(b) Suppose there is a third bidder who is not part of the collusion. Does
the existence of this bidder change the optimal bids for the two colluding
bidders?
We can treat the two colluding bidders (L and H) as a single player
with two choice variables:
bH to influence the probability of winning; and
bL to minimize the payment, conditional on winning
What should be the third bidder’s strategy?
Bid truthfully
.
Bidding truthfully is still a dominant strategy for the third bidder.
What should be the optimal bL ? bL ∈ [0, min{v3 , bH }]
If won (bH > v3 ), then any bL ∈ [0, v3 ] is optimal: colluders have to
pay more if bL > v3
If lost (bH ≤ v3 ), then any bL ∈ [0, bH ] is optimal: payoff is zero
anyway
H. K. Chen (SFU)
ECON 383
9 / 10
Chapter 9 — Ex.6(b)
6(b) Suppose there is a third bidder who is not part of the collusion. Does
the existence of this bidder change the optimal bids for the two colluding
bidders?
Given truthful bidding of bidder 3, what is the optimal bH ?
H. K. Chen (SFU)
ECON 383
10 / 10
Chapter 9 — Ex.6(b)
6(b) Suppose there is a third bidder who is not part of the collusion. Does
the existence of this bidder change the optimal bids for the two colluding
bidders?
Given truthful bidding of bidder 3, what is the optimal bH ?
bH = vH
The auction is essentially between bidders H and 3. So the usual
argument for truthful bidding being weakly dominant applies
H. K. Chen (SFU)
ECON 383
10 / 10
Chapter 9 — Ex.6(b)
6(b) Suppose there is a third bidder who is not part of the collusion. Does
the existence of this bidder change the optimal bids for the two colluding
bidders?
Given truthful bidding of bidder 3, what is the optimal bH ?
bH = vH
The auction is essentially between bidders H and 3. So the usual
argument for truthful bidding being weakly dominant applies
In summary, with a non-colluding bidder, the two colluders should bid
bL ∈ [0, min{v3 , bH }]
H. K. Chen (SFU)
ECON 383
and bH = vH
10 / 10
Chapter 9 — Ex.6(b)
6(b) Suppose there is a third bidder who is not part of the collusion. Does
the existence of this bidder change the optimal bids for the two colluding
bidders?
Given truthful bidding of bidder 3, what is the optimal bH ?
bH = vH
The auction is essentially between bidders H and 3. So the usual
argument for truthful bidding being weakly dominant applies
In summary, with a non-colluding bidder, the two colluders should bid
bL ∈ [0, min{v3 , bH }]
and bH = vH
Note that bL = 0 and bH = vH are an optimal colluding strategy
regardless of whether there exists a non-colluding bidder.
H. K. Chen (SFU)
ECON 383
10 / 10