2
Solutions Manual for
AUCTION THEORY*
Alexey Kushnir and Jun Xiao
August 009
Contents
2 Private Value Auctions: A First Look ..................................................................... 2
3 The Revenue Equivalence Principle ......................................................................... 8
4 Qualifications and Extensions ................................................................................. 11
5 Mechanism Design................................................................................................... 17
6 Auctions with Interdependent Values .................................................................... 25
8 Asymmetries and Other Complications................................................................. 34
9 Efficiency and the English Auction ........................................................................ 40
10 Mechanism Design with Interdependent Values .................................................... 43
11 Bidding Rings .......................................................................................................... 48
13 Equilibrium and Efficiency with Private Values ................................................... 52
15 Sequential Sales........................................................................................................ 55
16 Nonidential Objects ................................................................................................. 60
17 Packages and Positions ........................................................................................... 62
1
,V. Krishna, Auction fheory (2nd. Ed.), Elsevier, 2009.
2
,2 Private Value Auctions: A First Look
Problem 2.1 (Pomer distribution) Suppose there are tmo bidders mith private values that are
distributed independently according to the distribution F (x) = xa over [0, 1] mhere a > 0.
Find symmetric equilibrium bidding strategies in a first−price auction.
Solution. Since N = 2, G(x) = F (x) = xa. Thus, using the formula on page 16 of
the text,
∫ x ∫ x a
β (x)
I =x— G (y)dy = x — y dy = a x
O G (x)
a
O x 1+a
Problem 2.2 (Pareto distribution) Suppose there are tmo bidders mith private values that are
distributed independently according to a Pareto distribution F (x) = 1 — (x + 1)—2 over [0,
∞). Find symmetric equilibrium bidding strategies in a first−price auction. Shom by direct
computation that the expected revenues in a first− and second− price auction are the same.
Solution. Again, since N = 2, G (x) = F (x) = 1 — (x + 1)—2. Thus,
∫ x G (y)
β I (x) = x — dy
O G (x)
∫ x
= x— 1 — (y + 1)—2
dy
x O 1 — (x + 1)—2
=
x+2
In the first-price auction, the expected revenue of the seller is
E RI = 2E mI (x)
= 2E∫ G (x) × βI (x)
∞
x
= 2 1 — (x + 1)—2 2 (x + 1)—3 dx
O x+2
= 1/3
Let Y2 be the second highest value, and its density is ƒ2 (y) = 2 (1 — F (y)) g (y)
(see Appendix C).
In a second-price auction, the expected revenue of the seller is
E RII = E [Y2]
∫ ∞
= y2 (y + 1)—2 2 (y + 1)—3 dy
O
= 1/3
Therefore, the expected revenues in the two auctions are the same.
3
,Problem 2.3 (Stochastic dominance) Gonsider an N −bidder first−price auction mith
independent private values. Get β be the symmetric equilibrium bidding strategy mhen mhich
each bidder’s value is distributed according to F on [0, c] . Similarly, let β∗ be the equilibrium
strategy mhen each bidder’s value distribution is F ∗ on [0, c∗] .
a· Shom that if F ∗ dominates F in termsof the reverse hazard rate (see Appendix
B for a definition) then for all x ∈ [0, c] , β∗ (x) ≥ β (x ).
b· By considering F (x) = 3x — x2 on [0, 1 (3 √— 5)] and F ∗ (x) = 3x — 2x2 on
∗ 2
0, 12 , shom that the condition that F first−order stochastically dominates F is not
sufficient to guarantee that β∗ (x) ≥ β (x) .
Solution. Part a. Because G (x) = F (x)N—1 and g (x) = (N — 1) F (x)N—2 ƒ (x) , the
symmetric equilibrium in Proposition 2.2 could be rewritten as follows
∫ x
1
β (x) = yg (y) dy
G (x) O
1 ∫ x
= y (N — 1) F (x)N—2 ƒ (x) dy
[F (x)]N—1 O
∫ x ƒ (y)
= (N — 1) y dy
O F (y)
∫ x
= (N — 1) yσ (y) dy
O
where σ (x) is the reverse hazard rate. Similarly, we have
∫ x
β∗ (x) = (N — 1) yσ∗ (y) dy
O
So it is easy to see that if F ∗ dominates F in terms of reverse hazard rate, then
σ∗ (y) ≥ σ (y) for all y ∈ [0, c] . Therefore β∗ (x) ≥ β (x) for all x ∈ [0, c].
Part b. Obviously, F ∗ (x) ≤F (x), so F ∗ stochastically dominates F . The
distributions F and F ∗ are illustrated in Figure S2.1, where the solid line represents
F and the dashed line represents F ∗.
4
, 1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0.0 0.1 0.2 0.3 0.4 0.5
x
Figure S2.1
Suppose there are two bidders, then
∫ x G (y)
dy
β (x) = x — G (x)
O
∫ x
3y — y2
= x— dy
O 3x — x2
1 2x — 9
= x
6 x—3
√
for x ∈ 0, 1 3 — 5 . Similarly,
2
∫ x
3y — 2y2
β∗ (x) = x — dy
O 3x — 2x2
1 x
= (8x — 9)
6 2x — 3
√
for x ∈ 0, 1 . It is easy to see that β∗ (x) < β (x) for x ∈ (0, 1 3 — 5 ]. The bidding
2 2
strategies β and β∗ are plotted in Figure S2.2, where β is the solid line and β∗ is the
dashed line.
5
, 0.20
0.15
0.10
0.05
0.00
0.0 0.1 0.2 0.3 0.4 0.5
x
Figure S2.2
Problem 2.4 (Mixed auction) Gonsider an N −bidder auction mhich is a “mixture of a
first− and second−price auction in the sense that the highest bidder mins and pays a convex
combination of his omn bid and the second−highest bid. Precisely, there is a fixed ∈ (0, 1) such
that upon minning, bidder i pays bi + (1 — ) (max j /= i bj) . Find a symmetric equilibrium
bidding strategy in such an auction mhen all bidders’ values are distributed according to F.
Solution. Suppose all bidders other than 1 follow the strategy β. The expected
payoff of bidder i from bidding b when his value is x is
П (b, x)) = G β—1 (b) [x — b — (1 — )E [β (Y1) | β (Y1) ≤ b1]]
" , β—1(b) #
β(y)g(y)dy
= G β—1 (b) x — b — (1 — ) O
G(β—1 (b))
∫ β—1(b)
= G β—1 (b) (x — b) — (1 — ) β(y)g(y)dy
O
Maximizing this with respect to b yields the first-order condition:
g β—1 (b)
0 = (x — b) — G β—1 (b)
βJ β—1 (b)
1
— (1 — ) bg(β—1 (b))
βJ β—1 (b)
At a symmetric equilibrium, b = β (x), so the first-order condition becomes
1 1
G (x) βJ (x) + g (x) β (x) = xg (x)
6
, Using G (x)(1/ )—1 as the integrating factor, the solution to the above differential
equation is easily seen to be
∫x
1
β (x) = yg (y) dy
G (x) O
where G Ξ G1/ and g = GJ .
Problem 2.5 (Resale) Gonsider a tmo−bidder first−price auction in mhich bidders’ values
are distributed according to F. Get β be the symmetric equilibrium (as derived in Proposition
2.2). Nom suppose that after the auction is over, both the losing and minning bids are
publicly announced. In addition, there is the possibility of post− auction resale: fhe minner
of the auction may, if he so mishes, offer the object to the other bidder at a fixed
“take−it−or−leave−it price of p. If the other bidder agrees, then the object changes hands and the
losing bidder pays the minning bidder p. Othermise, the object stays mith the minning bidder
and no money changes hands. fhe possibility of post−auction resale in this manner is
commonly knomn to both bidders prior to participating in the auction. Shom that β
remains an equilibrium even if resale is allomed. In particular, shom that a bidder mith
value x cannot gain by bidding an amount b > β (x) even mhen he has the option of
reselling the object to the other bidder.
Solution. First, let us consider the resale stage. Suppose bidder 1 wins the auction
and the announced bids are b1 and b2. Hence bidder 1 can recover bidder 2's private
value by x2 = β2—1(b2).Therefore bidder 1 suggests the price x2 which extracts all the
surplus from bidder 2 if x2 ≥ x1, and does not offer otherwise. Then bidder 1's payoff
is max(x1 — b1, x2 — b1). If bidder 1 loses the auction, he gets zero payoff because
bidder 2 offers price x1 to him and extracts all the surplus.
Second, now we move to the auction stage. Let β (x) = F (x) 1 , x yƒ (y)dy be the
O
symmetric equilibrium without resale. We are going to show that any deviation of
bidder 1 from β (x) is not profitable. Suppose bidder 1 deviates by bidding β (z)
when his private value is x, while bidder 2 still plays β (x2). Bidder 1's ex ante
expected payoff is
(x — β(z)) F (z) if x, ≥ z
П1 (z, x) = x
(x — β(z)) F (x) + x (y — β(z)) ƒ (y)dy if x < z
If x2 < z ≤ x there is no resale. If z > x ≥ x2, bidder 1 does not offer to bidder 2
and his payoff remains the same. If z ≥ x2 ≥ x, bidder 1 sells to bidder 2 and the
7
,payoff after resale is x2 — β(z). Note that
∫ x
(x — β(z)) F (x) + (y — β(z)) ƒ (y)dy
∫x x
= xF (x) + yƒ (y)dy — β(z)F (z)
x
∫x ∫x
= xF (x) + yƒ (y)dy — yƒ (y)dy
x ∫ x O
1
= xF (x) — F (x) yƒ (y)dy
F (x) O
= (x — β(x)) F (x)
so we have
(x — β(z)) F (z) if x ≥ z
П
1 (z, x) =
(x — β(x)) F (x) if x < z
which is not more than П1(x, x). So no deviation strictly increases a bidder's payoff
and β (x) is still an equilibrium in the presence of resale.
8
,3 The Revenue Equivalence Principle
Problem 3.1 (War of attrition) Gonsider a tmo−bidder war of attrition in mhich the
bidder mith the highest bid mins the object but both bidders pay the losing bid. Bidders’
values independently and identically distributed according to F.
a· Use the revenue equivalence principle to derive a symmetric equilibrium bidding strategy
in the mar of attrition.
b· Directly compute the symmetric equilibrium bidding strategy and the sellers’ revenue
mhen the bidders’ values are uniformly distributed on [0, 1].
Solution. Part a. Suppose that there is a symmetric, increasing equilibrium of the
mar of attrition, β, such that the expected payment of a bidder with value 0 is 0.
Since the assumptions of Proposition 3.1 are satisfied, we must have that for all x,
the expected payment is ∫
x
m (x) = yƒ (y) dy
O
On the other hand, we also have
m (x) = E [β (Y1) | Y1 < x]
1 ∫x
= β (y) ƒ (y) dy
F (x) O
where Y1 is the bid from the other bidder. Combining the two equations, we have
∫ x ∫ x
1
yƒ (y) dy = β (y) ƒ (y) dy
O F (x) O
Differentiating both sides with respect to x and rearranging this, we get
∫x
β (x) = F (x) x + yƒ (y) dy
O
Part b. Suppose that bidder 1 has valuation x. He chooses b to maximize his expected
payoff
1 ∫ β—1(b)
—1
F β (b) x — β (y) ƒ (y) dy
F β—1 (b) O
where the first term is the product of his probability of winning and his valuation,
and the second term is his expected payment.
Because F (x) = x and ƒ (x) = 1, the expected payoff becomes
∫ β—1 (b)
1
β—1 (b) x — —1 β (y) dy
β (b) O
Maximizing with respect to b yields the first-order condition:
1
, β—1(b)
—β β—1 (b) r β—1 (b) + r
1 β (y) dy
1 x+ β (β —1 (b)) β (β —1 (b)) O
r
0 = β β—1 (b)
9
, β—1
2
(b)
1
0
Solutions Manual for
AUCTION THEORY*
Alexey Kushnir and Jun Xiao
August 009
Contents
2 Private Value Auctions: A First Look ..................................................................... 2
3 The Revenue Equivalence Principle ......................................................................... 8
4 Qualifications and Extensions ................................................................................. 11
5 Mechanism Design................................................................................................... 17
6 Auctions with Interdependent Values .................................................................... 25
8 Asymmetries and Other Complications................................................................. 34
9 Efficiency and the English Auction ........................................................................ 40
10 Mechanism Design with Interdependent Values .................................................... 43
11 Bidding Rings .......................................................................................................... 48
13 Equilibrium and Efficiency with Private Values ................................................... 52
15 Sequential Sales........................................................................................................ 55
16 Nonidential Objects ................................................................................................. 60
17 Packages and Positions ........................................................................................... 62
1
,V. Krishna, Auction fheory (2nd. Ed.), Elsevier, 2009.
2
,2 Private Value Auctions: A First Look
Problem 2.1 (Pomer distribution) Suppose there are tmo bidders mith private values that are
distributed independently according to the distribution F (x) = xa over [0, 1] mhere a > 0.
Find symmetric equilibrium bidding strategies in a first−price auction.
Solution. Since N = 2, G(x) = F (x) = xa. Thus, using the formula on page 16 of
the text,
∫ x ∫ x a
β (x)
I =x— G (y)dy = x — y dy = a x
O G (x)
a
O x 1+a
Problem 2.2 (Pareto distribution) Suppose there are tmo bidders mith private values that are
distributed independently according to a Pareto distribution F (x) = 1 — (x + 1)—2 over [0,
∞). Find symmetric equilibrium bidding strategies in a first−price auction. Shom by direct
computation that the expected revenues in a first− and second− price auction are the same.
Solution. Again, since N = 2, G (x) = F (x) = 1 — (x + 1)—2. Thus,
∫ x G (y)
β I (x) = x — dy
O G (x)
∫ x
= x— 1 — (y + 1)—2
dy
x O 1 — (x + 1)—2
=
x+2
In the first-price auction, the expected revenue of the seller is
E RI = 2E mI (x)
= 2E∫ G (x) × βI (x)
∞
x
= 2 1 — (x + 1)—2 2 (x + 1)—3 dx
O x+2
= 1/3
Let Y2 be the second highest value, and its density is ƒ2 (y) = 2 (1 — F (y)) g (y)
(see Appendix C).
In a second-price auction, the expected revenue of the seller is
E RII = E [Y2]
∫ ∞
= y2 (y + 1)—2 2 (y + 1)—3 dy
O
= 1/3
Therefore, the expected revenues in the two auctions are the same.
3
,Problem 2.3 (Stochastic dominance) Gonsider an N −bidder first−price auction mith
independent private values. Get β be the symmetric equilibrium bidding strategy mhen mhich
each bidder’s value is distributed according to F on [0, c] . Similarly, let β∗ be the equilibrium
strategy mhen each bidder’s value distribution is F ∗ on [0, c∗] .
a· Shom that if F ∗ dominates F in termsof the reverse hazard rate (see Appendix
B for a definition) then for all x ∈ [0, c] , β∗ (x) ≥ β (x ).
b· By considering F (x) = 3x — x2 on [0, 1 (3 √— 5)] and F ∗ (x) = 3x — 2x2 on
∗ 2
0, 12 , shom that the condition that F first−order stochastically dominates F is not
sufficient to guarantee that β∗ (x) ≥ β (x) .
Solution. Part a. Because G (x) = F (x)N—1 and g (x) = (N — 1) F (x)N—2 ƒ (x) , the
symmetric equilibrium in Proposition 2.2 could be rewritten as follows
∫ x
1
β (x) = yg (y) dy
G (x) O
1 ∫ x
= y (N — 1) F (x)N—2 ƒ (x) dy
[F (x)]N—1 O
∫ x ƒ (y)
= (N — 1) y dy
O F (y)
∫ x
= (N — 1) yσ (y) dy
O
where σ (x) is the reverse hazard rate. Similarly, we have
∫ x
β∗ (x) = (N — 1) yσ∗ (y) dy
O
So it is easy to see that if F ∗ dominates F in terms of reverse hazard rate, then
σ∗ (y) ≥ σ (y) for all y ∈ [0, c] . Therefore β∗ (x) ≥ β (x) for all x ∈ [0, c].
Part b. Obviously, F ∗ (x) ≤F (x), so F ∗ stochastically dominates F . The
distributions F and F ∗ are illustrated in Figure S2.1, where the solid line represents
F and the dashed line represents F ∗.
4
, 1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0.0 0.1 0.2 0.3 0.4 0.5
x
Figure S2.1
Suppose there are two bidders, then
∫ x G (y)
dy
β (x) = x — G (x)
O
∫ x
3y — y2
= x— dy
O 3x — x2
1 2x — 9
= x
6 x—3
√
for x ∈ 0, 1 3 — 5 . Similarly,
2
∫ x
3y — 2y2
β∗ (x) = x — dy
O 3x — 2x2
1 x
= (8x — 9)
6 2x — 3
√
for x ∈ 0, 1 . It is easy to see that β∗ (x) < β (x) for x ∈ (0, 1 3 — 5 ]. The bidding
2 2
strategies β and β∗ are plotted in Figure S2.2, where β is the solid line and β∗ is the
dashed line.
5
, 0.20
0.15
0.10
0.05
0.00
0.0 0.1 0.2 0.3 0.4 0.5
x
Figure S2.2
Problem 2.4 (Mixed auction) Gonsider an N −bidder auction mhich is a “mixture of a
first− and second−price auction in the sense that the highest bidder mins and pays a convex
combination of his omn bid and the second−highest bid. Precisely, there is a fixed ∈ (0, 1) such
that upon minning, bidder i pays bi + (1 — ) (max j /= i bj) . Find a symmetric equilibrium
bidding strategy in such an auction mhen all bidders’ values are distributed according to F.
Solution. Suppose all bidders other than 1 follow the strategy β. The expected
payoff of bidder i from bidding b when his value is x is
П (b, x)) = G β—1 (b) [x — b — (1 — )E [β (Y1) | β (Y1) ≤ b1]]
" , β—1(b) #
β(y)g(y)dy
= G β—1 (b) x — b — (1 — ) O
G(β—1 (b))
∫ β—1(b)
= G β—1 (b) (x — b) — (1 — ) β(y)g(y)dy
O
Maximizing this with respect to b yields the first-order condition:
g β—1 (b)
0 = (x — b) — G β—1 (b)
βJ β—1 (b)
1
— (1 — ) bg(β—1 (b))
βJ β—1 (b)
At a symmetric equilibrium, b = β (x), so the first-order condition becomes
1 1
G (x) βJ (x) + g (x) β (x) = xg (x)
6
, Using G (x)(1/ )—1 as the integrating factor, the solution to the above differential
equation is easily seen to be
∫x
1
β (x) = yg (y) dy
G (x) O
where G Ξ G1/ and g = GJ .
Problem 2.5 (Resale) Gonsider a tmo−bidder first−price auction in mhich bidders’ values
are distributed according to F. Get β be the symmetric equilibrium (as derived in Proposition
2.2). Nom suppose that after the auction is over, both the losing and minning bids are
publicly announced. In addition, there is the possibility of post− auction resale: fhe minner
of the auction may, if he so mishes, offer the object to the other bidder at a fixed
“take−it−or−leave−it price of p. If the other bidder agrees, then the object changes hands and the
losing bidder pays the minning bidder p. Othermise, the object stays mith the minning bidder
and no money changes hands. fhe possibility of post−auction resale in this manner is
commonly knomn to both bidders prior to participating in the auction. Shom that β
remains an equilibrium even if resale is allomed. In particular, shom that a bidder mith
value x cannot gain by bidding an amount b > β (x) even mhen he has the option of
reselling the object to the other bidder.
Solution. First, let us consider the resale stage. Suppose bidder 1 wins the auction
and the announced bids are b1 and b2. Hence bidder 1 can recover bidder 2's private
value by x2 = β2—1(b2).Therefore bidder 1 suggests the price x2 which extracts all the
surplus from bidder 2 if x2 ≥ x1, and does not offer otherwise. Then bidder 1's payoff
is max(x1 — b1, x2 — b1). If bidder 1 loses the auction, he gets zero payoff because
bidder 2 offers price x1 to him and extracts all the surplus.
Second, now we move to the auction stage. Let β (x) = F (x) 1 , x yƒ (y)dy be the
O
symmetric equilibrium without resale. We are going to show that any deviation of
bidder 1 from β (x) is not profitable. Suppose bidder 1 deviates by bidding β (z)
when his private value is x, while bidder 2 still plays β (x2). Bidder 1's ex ante
expected payoff is
(x — β(z)) F (z) if x, ≥ z
П1 (z, x) = x
(x — β(z)) F (x) + x (y — β(z)) ƒ (y)dy if x < z
If x2 < z ≤ x there is no resale. If z > x ≥ x2, bidder 1 does not offer to bidder 2
and his payoff remains the same. If z ≥ x2 ≥ x, bidder 1 sells to bidder 2 and the
7
,payoff after resale is x2 — β(z). Note that
∫ x
(x — β(z)) F (x) + (y — β(z)) ƒ (y)dy
∫x x
= xF (x) + yƒ (y)dy — β(z)F (z)
x
∫x ∫x
= xF (x) + yƒ (y)dy — yƒ (y)dy
x ∫ x O
1
= xF (x) — F (x) yƒ (y)dy
F (x) O
= (x — β(x)) F (x)
so we have
(x — β(z)) F (z) if x ≥ z
П
1 (z, x) =
(x — β(x)) F (x) if x < z
which is not more than П1(x, x). So no deviation strictly increases a bidder's payoff
and β (x) is still an equilibrium in the presence of resale.
8
,3 The Revenue Equivalence Principle
Problem 3.1 (War of attrition) Gonsider a tmo−bidder war of attrition in mhich the
bidder mith the highest bid mins the object but both bidders pay the losing bid. Bidders’
values independently and identically distributed according to F.
a· Use the revenue equivalence principle to derive a symmetric equilibrium bidding strategy
in the mar of attrition.
b· Directly compute the symmetric equilibrium bidding strategy and the sellers’ revenue
mhen the bidders’ values are uniformly distributed on [0, 1].
Solution. Part a. Suppose that there is a symmetric, increasing equilibrium of the
mar of attrition, β, such that the expected payment of a bidder with value 0 is 0.
Since the assumptions of Proposition 3.1 are satisfied, we must have that for all x,
the expected payment is ∫
x
m (x) = yƒ (y) dy
O
On the other hand, we also have
m (x) = E [β (Y1) | Y1 < x]
1 ∫x
= β (y) ƒ (y) dy
F (x) O
where Y1 is the bid from the other bidder. Combining the two equations, we have
∫ x ∫ x
1
yƒ (y) dy = β (y) ƒ (y) dy
O F (x) O
Differentiating both sides with respect to x and rearranging this, we get
∫x
β (x) = F (x) x + yƒ (y) dy
O
Part b. Suppose that bidder 1 has valuation x. He chooses b to maximize his expected
payoff
1 ∫ β—1(b)
—1
F β (b) x — β (y) ƒ (y) dy
F β—1 (b) O
where the first term is the product of his probability of winning and his valuation,
and the second term is his expected payment.
Because F (x) = x and ƒ (x) = 1, the expected payoff becomes
∫ β—1 (b)
1
β—1 (b) x — —1 β (y) dy
β (b) O
Maximizing with respect to b yields the first-order condition:
1
, β—1(b)
—β β—1 (b) r β—1 (b) + r
1 β (y) dy
1 x+ β (β —1 (b)) β (β —1 (b)) O
r
0 = β β—1 (b)
9
, β—1
2
(b)
1
0
