ECON 383
Practice Problems from Chapter 9
8, 9, 10(a), 10(b)
H. K. Chen (SFU)
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Chapter 9 — Ex.8(a)
8. Three bidders with independent private values vi ∼ U [0, 1], i = 1, 2, 3,
participate in a second price auction.
(a) Suppose all bidders behave rationally. Which bidder (in terms of
values) wins the auction and how much does this bidder pay (again in
terms of the bidder’s values)?
Without loss of generality, assume v1 < v2 < v3
H. K. Chen (SFU)
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Chapter 9 — Ex.8(a)
8. Three bidders with independent private values vi ∼ U [0, 1], i = 1, 2, 3,
participate in a second price auction.
(a) Suppose all bidders behave rationally. Which bidder (in terms of
values) wins the auction and how much does this bidder pay (again in
terms of the bidder’s values)?
Without loss of generality, assume v1 < v2 < v3
Since a rational bidder would bid truthfully, the winner is
and he pays
.
H. K. Chen (SFU)
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Chapter 9 — Ex.8(a)
8. Three bidders with independent private values vi ∼ U [0, 1], i = 1, 2, 3,
participate in a second price auction.
(a) Suppose all bidders behave rationally. Which bidder (in terms of
values) wins the auction and how much does this bidder pay (again in
terms of the bidder’s values)?
Without loss of generality, assume v1 < v2 < v3
Since a rational bidder would bid truthfully, the winner is
bidder 3 and he pays v2 .
H. K. Chen (SFU)
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Chapter 9 — Ex.8(b)
8(b) Suppose bidder 3 irrationally bids more than his true value for the
object; in particular, bidder 3’s bid is (v3 + 1)/2. All other bidders know
that bidder 3 is irrational in this way, although they don’t know bidder 3’s
actual value for the object. How does this affect the behavior of the other
bidders?
Focus on bidder 1’s behavior without loss of generality.
Let b3 = (v3 + 1)/2 and b̄ = max{b3 , b2 } = highest competing bid
H. K. Chen (SFU)
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Chapter 9 — Ex.8(b)
8(b) Suppose bidder 3 irrationally bids more than his true value for the
object; in particular, bidder 3’s bid is (v3 + 1)/2. All other bidders know
that bidder 3 is irrational in this way, although they don’t know bidder 3’s
actual value for the object. How does this affect the behavior of the other
bidders?
Focus on bidder 1’s behavior without loss of generality.
Let b3 = (v3 + 1)/2 and b̄ = max{b3 , b2 } = highest competing bid
Truthful bidding: b1 = v1
H. K. Chen (SFU)
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Chapter 9 — Ex.8(b)
8(b) Suppose bidder 3 irrationally bids more than his true value for the
object; in particular, bidder 3’s bid is (v3 + 1)/2. All other bidders know
that bidder 3 is irrational in this way, although they don’t know bidder 3’s
actual value for the object. How does this affect the behavior of the other
bidders?
Focus on bidder 1’s behavior without loss of generality.
Let b3 = (v3 + 1)/2 and b̄ = max{b3 , b2 } = highest competing bid
Truthful bidding: b1 = v1
positive payoff if v1 > b̄, and zero payoff if v1 ≤ b̄
H. K. Chen (SFU)
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Chapter 9 — Ex.8(b)
8(b) Suppose bidder 3 irrationally bids more than his true value for the
object; in particular, bidder 3’s bid is (v3 + 1)/2. All other bidders know
that bidder 3 is irrational in this way, although they don’t know bidder 3’s
actual value for the object. How does this affect the behavior of the other
bidders?
Focus on bidder 1’s behavior without loss of generality.
Let b3 = (v3 + 1)/2 and b̄ = max{b3 , b2 } = highest competing bid
Truthful bidding: b1 = v1
positive payoff if v1 > b̄, and zero payoff if v1 ≤ b̄
Over-bidding: b1 > v1
H. K. Chen (SFU)
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Chapter 9 — Ex.8(b)
8(b) Suppose bidder 3 irrationally bids more than his true value for the
object; in particular, bidder 3’s bid is (v3 + 1)/2. All other bidders know
that bidder 3 is irrational in this way, although they don’t know bidder 3’s
actual value for the object. How does this affect the behavior of the other
bidders?
Focus on bidder 1’s behavior without loss of generality.
Let b3 = (v3 + 1)/2 and b̄ = max{b3 , b2 } = highest competing bid
Truthful bidding: b1 = v1
positive payoff if v1 > b̄, and zero payoff if v1 ≤ b̄
Over-bidding: b1 > v1
payoff is negative if b̄ ∈ (v1 , b1 ], otherwise same as truthful bidding
H. K. Chen (SFU)
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Chapter 9 — Ex.8(b)
8(b) Suppose bidder 3 irrationally bids more than his true value for the
object; in particular, bidder 3’s bid is (v3 + 1)/2. All other bidders know
that bidder 3 is irrational in this way, although they don’t know bidder 3’s
actual value for the object. How does this affect the behavior of the other
bidders?
Focus on bidder 1’s behavior without loss of generality.
Let b3 = (v3 + 1)/2 and b̄ = max{b3 , b2 } = highest competing bid
Truthful bidding: b1 = v1
positive payoff if v1 > b̄, and zero payoff if v1 ≤ b̄
Over-bidding: b1 > v1
payoff is negative if b̄ ∈ (v1 , b1 ], otherwise same as truthful bidding
Under-bidding: b1 < v1
H. K. Chen (SFU)
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Chapter 9 — Ex.8(b)
8(b) Suppose bidder 3 irrationally bids more than his true value for the
object; in particular, bidder 3’s bid is (v3 + 1)/2. All other bidders know
that bidder 3 is irrational in this way, although they don’t know bidder 3’s
actual value for the object. How does this affect the behavior of the other
bidders?
Focus on bidder 1’s behavior without loss of generality.
Let b3 = (v3 + 1)/2 and b̄ = max{b3 , b2 } = highest competing bid
Truthful bidding: b1 = v1
positive payoff if v1 > b̄, and zero payoff if v1 ≤ b̄
Over-bidding: b1 > v1
payoff is negative if b̄ ∈ (v1 , b1 ], otherwise same as truthful bidding
Under-bidding: b1 < v1
payoff is zero if b̄ ∈ (b1 , v1 ], otherwise same as truthful bidding
H. K. Chen (SFU)
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Chapter 9 — Ex.8(b)
8(b) Suppose bidder 3 irrationally bids more than his true value for the
object; in particular, bidder 3’s bid is (v3 + 1)/2. All other bidders know
that bidder 3 is irrational in this way, although they don’t know bidder 3’s
actual value for the object. How does this affect the behavior of the other
bidders?
Focus on bidder 1’s behavior without loss of generality.
Let b3 = (v3 + 1)/2 and b̄ = max{b3 , b2 } = highest competing bid
Truthful bidding: b1 = v1
positive payoff if v1 > b̄, and zero payoff if v1 ≤ b̄
Over-bidding: b1 > v1
payoff is negative if b̄ ∈ (v1 , b1 ], otherwise same as truthful bidding
Under-bidding: b1 < v1
payoff is zero if b̄ ∈ (b1 , v1 ], otherwise same as truthful bidding
Thus truthful bidding is still a weakly dominant strategy for both
bidders 1 and 2
H. K. Chen (SFU)
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Chapter 9 — Ex.8(c)
8(c) What effect does bidder 3’s irrational behavior have on the expected
payoffs of bidder 1?
H. K. Chen (SFU)
ECON 383
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Chapter 9 — Ex.8(c)
8(c) What effect does bidder 3’s irrational behavior have on the expected
payoffs of bidder 1?
Bidder 1’s expected payoff is
Pr(v1 ≥ max{v2 , b3 }) v1 − E [max{v2 , b3 }|v1 ≥ max{v2 , b3 }]
|
{z
}
{z
}
|
probability of winning
H. K. Chen (SFU)
expected 2nd highest bid
conditional on winning
ECON 383
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Chapter 9 — Ex.8(c)
8(c) What effect does bidder 3’s irrational behavior have on the expected
payoffs of bidder 1?
Bidder 1’s expected payoff is
Pr(v1 ≥ max{v2 , b3 }) v1 − E [max{v2 , b3 }|v1 ≥ max{v2 , b3 }]
|
{z
}
{z
}
|
probability of winning
expected 2nd highest bid
conditional on winning
Bidder 3’s over bidding affects bidder 1’s expected payoff in two ways
(compared to the case when bidder 3 bids truthfully):
H. K. Chen (SFU)
ECON 383
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Chapter 9 — Ex.8(c)
8(c) What effect does bidder 3’s irrational behavior have on the expected
payoffs of bidder 1?
Bidder 1’s expected payoff is
Pr(v1 ≥ max{v2 , b3 }) v1 − E [max{v2 , b3 }|v1 ≥ max{v2 , b3 }]
|
{z
}
{z
}
|
probability of winning
expected 2nd highest bid
conditional on winning
Bidder 3’s over bidding affects bidder 1’s expected payoff in two ways
(compared to the case when bidder 3 bids truthfully):
reduces bidder 1’s probability of winning
H. K. Chen (SFU)
ECON 383
4/9
Chapter 9 — Ex.8(c)
8(c) What effect does bidder 3’s irrational behavior have on the expected
payoffs of bidder 1?
Bidder 1’s expected payoff is
Pr(v1 ≥ max{v2 , b3 }) v1 − E [max{v2 , b3 }|v1 ≥ max{v2 , b3 }]
|
{z
}
{z
}
|
probability of winning
expected 2nd highest bid
conditional on winning
Bidder 3’s over bidding affects bidder 1’s expected payoff in two ways
(compared to the case when bidder 3 bids truthfully):
reduces bidder 1’s probability of winning
increases bidder 1’s expected payment when he wins
H. K. Chen (SFU)
ECON 383
4/9
Chapter 9 — Ex.8(c)
8(c) What effect does bidder 3’s irrational behavior have on the expected
payoffs of bidder 1?
Bidder 1’s expected payoff is
Pr(v1 ≥ max{v2 , b3 }) v1 − E [max{v2 , b3 }|v1 ≥ max{v2 , b3 }]
|
{z
}
{z
}
|
probability of winning
expected 2nd highest bid
conditional on winning
Bidder 3’s over bidding affects bidder 1’s expected payoff in two ways
(compared to the case when bidder 3 bids truthfully):
reduces bidder 1’s probability of winning
increases bidder 1’s expected payment when he wins
Overall, bidder 1’s expected payoff decreases
H. K. Chen (SFU)
ECON 383
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Chapter 9 — Ex.9(a)
9. Two bidders in a second price auction have independent private values
vi ∈ {1, 2}, i = 1, 2. Each value realizes with equal probability. Let x be
the price.
(a) Show that the seller’s expected revenue is 5/4.
H. K. Chen (SFU)
ECON 383
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Chapter 9 — Ex.9(a)
9. Two bidders in a second price auction have independent private values
vi ∈ {1, 2}, i = 1, 2. Each value realizes with equal probability. Let x be
the price.
(a) Show that the seller’s expected revenue is 5/4.
Pr(x = 1) =
H. K. Chen (SFU)
, and Pr(x = 2) =
ECON 383
5/9
Chapter 9 — Ex.9(a)
9. Two bidders in a second price auction have independent private values
vi ∈ {1, 2}, i = 1, 2. Each value realizes with equal probability. Let x be
the price.
(a) Show that the seller’s expected revenue is 5/4.
Pr(x = 1) = 3/4, and Pr(x = 2) = 1/4
H. K. Chen (SFU)
ECON 383
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Chapter 9 — Ex.9(a)
9. Two bidders in a second price auction have independent private values
vi ∈ {1, 2}, i = 1, 2. Each value realizes with equal probability. Let x be
the price.
(a) Show that the seller’s expected revenue is 5/4.
Pr(x = 1) = 3/4, and Pr(x = 2) = 1/4
Therefore, expected revenue is
1
5
3
(1) + (2) =
4
4
4
H. K. Chen (SFU)
ECON 383
5/9
Chapter 9 — Ex.9(b)
9(b) Suppose seller sets a publicly known reserve price R with 1 < R < 2.
What is the seller’s expected revenue as a function of R?
Truthful bidding is still optimal, despite that R is publicly announced
H. K. Chen (SFU)
ECON 383
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Chapter 9 — Ex.9(b)
9(b) Suppose seller sets a publicly known reserve price R with 1 < R < 2.
What is the seller’s expected revenue as a function of R?
Truthful bidding is still optimal, despite that R is publicly announced
There are three possible values of the selling price x:


0 if
x = 2 if


R
H. K. Chen (SFU)
ECON 383
6/9
Chapter 9 — Ex.9(b)
9(b) Suppose seller sets a publicly known reserve price R with 1 < R < 2.
What is the seller’s expected revenue as a function of R?
Truthful bidding is still optimal, despite that R is publicly announced
There are three possible values of the selling price x:


0 if v1 , v2 < R
x = 2 if


R
H. K. Chen (SFU)
ECON 383
6/9
Chapter 9 — Ex.9(b)
9(b) Suppose seller sets a publicly known reserve price R with 1 < R < 2.
What is the seller’s expected revenue as a function of R?
Truthful bidding is still optimal, despite that R is publicly announced
There are three possible values of the selling price x:


0 if v1 , v2 < R
x = 2 if v1 , v2 > R


R
H. K. Chen (SFU)
ECON 383
6/9
Chapter 9 — Ex.9(b)
9(b) Suppose seller sets a publicly known reserve price R with 1 < R < 2.
What is the seller’s expected revenue as a function of R?
Truthful bidding is still optimal, despite that R is publicly announced
There are three possible values of the selling price x:


0 if v1 , v2 < R
x = 2 if v1 , v2 > R


R
otherwise
H. K. Chen (SFU)
ECON 383
6/9
Chapter 9 — Ex.9(b)
9(b) Suppose seller sets a publicly known reserve price R with 1 < R < 2.
What is the seller’s expected revenue as a function of R?
Truthful bidding is still optimal, despite that R is publicly announced
There are three possible values of the selling price x:


0 if v1 , v2 < R
x = 2 if v1 , v2 > R


R
otherwise
So expected revenue is
Pr(v1 , v2 > R)(2) + (1 − Pr(v1 , v2 < R) − Pr(v1 , v2 > R))(R)
H. K. Chen (SFU)
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Chapter 9 — Ex.9(b)
9(b) Suppose seller sets a publicly known reserve price R with 1 < R < 2.
What is the seller’s expected revenue as a function of R?
Truthful bidding is still optimal, despite that R is publicly announced
There are three possible values of the selling price x:


0 if v1 , v2 < R
x = 2 if v1 , v2 > R


R
otherwise
So expected revenue is
Pr(v1 , v2 > R)(2) + (1 − Pr(v1 , v2 < R) − Pr(v1 , v2 > R))(R)
1
1 1
= (2) + 1 − −
(R)
4
4 4
H. K. Chen (SFU)
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Chapter 9 — Ex.9(b)
9(b) Suppose seller sets a publicly known reserve price R with 1 < R < 2.
What is the seller’s expected revenue as a function of R?
Truthful bidding is still optimal, despite that R is publicly announced
There are three possible values of the selling price x:


0 if v1 , v2 < R
x = 2 if v1 , v2 > R


R
otherwise
So expected revenue is
Pr(v1 , v2 > R)(2) + (1 − Pr(v1 , v2 < R) − Pr(v1 , v2 > R))(R)
1
1 1
= (2) + 1 − −
(R)
4
4 4
1+R
=
2
H. K. Chen (SFU)
ECON 383
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Chapter 9 — Ex.9(c)
9(c) Show that a seller who wants to maximize expected revenue would
never set a reserve price R that is more than 1 and less than 1.5.
Recall that expected payoff is
1+R
,
2
H. K. Chen (SFU)
ECON 383
R ∈ (1, 2)
7/9
Chapter 9 — Ex.9(c)
9(c) Show that a seller who wants to maximize expected revenue would
never set a reserve price R that is more than 1 and less than 1.5.
Recall that expected payoff is
1+R
,
2
R ∈ (1, 2)
To maximize expected payoff, seller would choose R
H. K. Chen (SFU)
ECON 383
7/9
Chapter 9 — Ex.9(c)
9(c) Show that a seller who wants to maximize expected revenue would
never set a reserve price R that is more than 1 and less than 1.5.
Recall that expected payoff is
1+R
,
2
R ∈ (1, 2)
To maximize expected payoff, seller would choose R
as close to 2 as possible
H. K. Chen (SFU)
ECON 383
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Chapter 9 — Ex.9(c)
9(c) Show that a seller who wants to maximize expected revenue would
never set a reserve price R that is more than 1 and less than 1.5.
Recall that expected payoff is
1+R
,
2
R ∈ (1, 2)
To maximize expected payoff, seller would choose R
as close to 2 as possible
Obviously then, R ∈ (1, 1.5) is never optimal.
H. K. Chen (SFU)
ECON 383
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Chapter 9 — Ex.10(a)
10. Two bidders in a second price auction, with vi ∈ {1, 7} where
i ∈ {1, 2}, each value realizes with equal probability.
(a) For each pair of values (v1 , v2 ), what bid will each bidder submit, what
price will the winning bidder pay, and how much profit will the winning
bidder earn?
( v1 , v2 )
(1, 1)
(1, 7)
(7, 1)
(7, 7)
H. K. Chen (SFU)
(b1 , b2 )
winner
ECON 383
price
profit
8/9
Chapter 9 — Ex.10(a)
10. Two bidders in a second price auction, with vi ∈ {1, 7} where
i ∈ {1, 2}, each value realizes with equal probability.
(a) For each pair of values (v1 , v2 ), what bid will each bidder submit, what
price will the winning bidder pay, and how much profit will the winning
bidder earn?
( v1 , v2 )
(1, 1)
(1, 7)
(7, 1)
(7, 7)
H. K. Chen (SFU)
(b1 , b2 )
(1, 1)
winner
ECON 383
price
profit
8/9
Chapter 9 — Ex.10(a)
10. Two bidders in a second price auction, with vi ∈ {1, 7} where
i ∈ {1, 2}, each value realizes with equal probability.
(a) For each pair of values (v1 , v2 ), what bid will each bidder submit, what
price will the winning bidder pay, and how much profit will the winning
bidder earn?
( v1 , v2 )
(1, 1)
(1, 7)
(7, 1)
(7, 7)
H. K. Chen (SFU)
(b1 , b2 )
(1, 1)
winner
1 or 2
ECON 383
price
profit
8/9
Chapter 9 — Ex.10(a)
10. Two bidders in a second price auction, with vi ∈ {1, 7} where
i ∈ {1, 2}, each value realizes with equal probability.
(a) For each pair of values (v1 , v2 ), what bid will each bidder submit, what
price will the winning bidder pay, and how much profit will the winning
bidder earn?
( v1 , v2 )
(1, 1)
(1, 7)
(7, 1)
(7, 7)
H. K. Chen (SFU)
(b1 , b2 )
(1, 1)
winner
1 or 2
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price
1
profit
8/9
Chapter 9 — Ex.10(a)
10. Two bidders in a second price auction, with vi ∈ {1, 7} where
i ∈ {1, 2}, each value realizes with equal probability.
(a) For each pair of values (v1 , v2 ), what bid will each bidder submit, what
price will the winning bidder pay, and how much profit will the winning
bidder earn?
( v1 , v2 )
(1, 1)
(1, 7)
(7, 1)
(7, 7)
H. K. Chen (SFU)
(b1 , b2 )
(1, 1)
winner
1 or 2
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price
1
profit
0
8/9
Chapter 9 — Ex.10(a)
10. Two bidders in a second price auction, with vi ∈ {1, 7} where
i ∈ {1, 2}, each value realizes with equal probability.
(a) For each pair of values (v1 , v2 ), what bid will each bidder submit, what
price will the winning bidder pay, and how much profit will the winning
bidder earn?
( v1 , v2 )
(1, 1)
(1, 7)
(7, 1)
(7, 7)
H. K. Chen (SFU)
(b1 , b2 )
(1, 1)
(1, 7)
winner
1 or 2
ECON 383
price
1
profit
0
8/9
Chapter 9 — Ex.10(a)
10. Two bidders in a second price auction, with vi ∈ {1, 7} where
i ∈ {1, 2}, each value realizes with equal probability.
(a) For each pair of values (v1 , v2 ), what bid will each bidder submit, what
price will the winning bidder pay, and how much profit will the winning
bidder earn?
( v1 , v2 )
(1, 1)
(1, 7)
(7, 1)
(7, 7)
H. K. Chen (SFU)
(b1 , b2 )
(1, 1)
(1, 7)
winner
1 or 2
2
ECON 383
price
1
profit
0
8/9
Chapter 9 — Ex.10(a)
10. Two bidders in a second price auction, with vi ∈ {1, 7} where
i ∈ {1, 2}, each value realizes with equal probability.
(a) For each pair of values (v1 , v2 ), what bid will each bidder submit, what
price will the winning bidder pay, and how much profit will the winning
bidder earn?
( v1 , v2 )
(1, 1)
(1, 7)
(7, 1)
(7, 7)
H. K. Chen (SFU)
(b1 , b2 )
(1, 1)
(1, 7)
winner
1 or 2
2
ECON 383
price
1
1
profit
0
6
8/9
Chapter 9 — Ex.10(a)
10. Two bidders in a second price auction, with vi ∈ {1, 7} where
i ∈ {1, 2}, each value realizes with equal probability.
(a) For each pair of values (v1 , v2 ), what bid will each bidder submit, what
price will the winning bidder pay, and how much profit will the winning
bidder earn?
( v1 , v2 )
(1, 1)
(1, 7)
(7, 1)
(7, 7)
H. K. Chen (SFU)
(b1 , b2 )
(1, 1)
(1, 7)
(7, 1)
winner
1 or 2
2
ECON 383
price
1
1
profit
0
6
8/9
Chapter 9 — Ex.10(a)
10. Two bidders in a second price auction, with vi ∈ {1, 7} where
i ∈ {1, 2}, each value realizes with equal probability.
(a) For each pair of values (v1 , v2 ), what bid will each bidder submit, what
price will the winning bidder pay, and how much profit will the winning
bidder earn?
( v1 , v2 )
(1, 1)
(1, 7)
(7, 1)
(7, 7)
H. K. Chen (SFU)
(b1 , b2 )
(1, 1)
(1, 7)
(7, 1)
winner
1 or 2
2
1
ECON 383
price
1
1
1
profit
0
6
6
8/9
Chapter 9 — Ex.10(a)
10. Two bidders in a second price auction, with vi ∈ {1, 7} where
i ∈ {1, 2}, each value realizes with equal probability.
(a) For each pair of values (v1 , v2 ), what bid will each bidder submit, what
price will the winning bidder pay, and how much profit will the winning
bidder earn?
( v1 , v2 )
(1, 1)
(1, 7)
(7, 1)
(7, 7)
H. K. Chen (SFU)
(b1 , b2 )
(1, 1)
(1, 7)
(7, 1)
(7, 7)
winner
1 or 2
2
1
ECON 383
price
1
1
1
profit
0
6
6
8/9
Chapter 9 — Ex.10(a)
10. Two bidders in a second price auction, with vi ∈ {1, 7} where
i ∈ {1, 2}, each value realizes with equal probability.
(a) For each pair of values (v1 , v2 ), what bid will each bidder submit, what
price will the winning bidder pay, and how much profit will the winning
bidder earn?
( v1 , v2 )
(1, 1)
(1, 7)
(7, 1)
(7, 7)
H. K. Chen (SFU)
(b1 , b2 )
(1, 1)
(1, 7)
(7, 1)
(7, 7)
winner
1 or 2
2
1
1 or 2
ECON 383
price
1
1
1
profit
0
6
6
8/9
Chapter 9 — Ex.10(a)
10. Two bidders in a second price auction, with vi ∈ {1, 7} where
i ∈ {1, 2}, each value realizes with equal probability.
(a) For each pair of values (v1 , v2 ), what bid will each bidder submit, what
price will the winning bidder pay, and how much profit will the winning
bidder earn?
( v1 , v2 )
(1, 1)
(1, 7)
(7, 1)
(7, 7)
H. K. Chen (SFU)
(b1 , b2 )
(1, 1)
(1, 7)
(7, 1)
(7, 7)
winner
1 or 2
2
1
1 or 2
ECON 383
price
1
1
1
7
profit
0
6
6
8/9
Chapter 9 — Ex.10(a)
10. Two bidders in a second price auction, with vi ∈ {1, 7} where
i ∈ {1, 2}, each value realizes with equal probability.
(a) For each pair of values (v1 , v2 ), what bid will each bidder submit, what
price will the winning bidder pay, and how much profit will the winning
bidder earn?
( v1 , v2 )
(1, 1)
(1, 7)
(7, 1)
(7, 7)
H. K. Chen (SFU)
(b1 , b2 )
(1, 1)
(1, 7)
(7, 1)
(7, 7)
winner
1 or 2
2
1
1 or 2
ECON 383
price
1
1
1
7
profit
0
6
6
0
8/9
Chapter 9 — Ex.10(b)
10(b) Calculate the expected profit for each bidder and the expected
revenue for the seller.
(v1 , v2 )
(1, 1)
(1, 7)
(7, 1)
(7, 7)
H. K. Chen (SFU)
( b1 , b2 )
(1, 1)
(1, 7)
(7, 1)
(7, 7)
Average
winner
1 or 2
2
1
1 or 2
price
1
1
1
7
ECON 383
profit of Bidder 1
0
0
6
0
9/9
Chapter 9 — Ex.10(b)
10(b) Calculate the expected profit for each bidder and the expected
revenue for the seller.
(v1 , v2 )
(1, 1)
(1, 7)
(7, 1)
(7, 7)
( b1 , b2 )
(1, 1)
(1, 7)
(7, 1)
(7, 7)
Average
winner
1 or 2
2
1
1 or 2
price
1
1
1
7
2.5
profit of Bidder 1
0
0
6
0
1.5
A bidder’s ex ante expected profit is 1.5
Seller’s expected revenue is 2.5
H. K. Chen (SFU)
ECON 383
9/9