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Practice Problems from Chapter 15
4, 5, 6
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Chapter 15 — Ex.4(a)
4. Suppose ad slots a, b have clickthrough rates of 4 and 3, respectively.
Advertisers x, y, z have values per click at 4, 3, 1, respectively.
(a) If the slots are allocated using a VCG procedure, what assignment of
slots will occur and what prices will the advertisers pay?
H. K. Chen (SFU)
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Chapter 15 — Ex.4(a)
4. Suppose ad slots a, b have clickthrough rates of 4 and 3, respectively.
Advertisers x, y, z have values per click at 4, 3, 1, respectively.
(a) If the slots are allocated using a VCG procedure, what assignment of
slots will occur and what prices will the advertisers pay?
Since both slots and advertisers can be ranked, the VCG procedure
results in an assortative matching: {a-x, b-y}.
H. K. Chen (SFU)
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Chapter 15 — Ex.4(a)
4. Suppose ad slots a, b have clickthrough rates of 4 and 3, respectively.
Advertisers x, y, z have values per click at 4, 3, 1, respectively.
(a) If the slots are allocated using a VCG procedure, what assignment of
slots will occur and what prices will the advertisers pay?
Since both slots and advertisers can be ranked, the VCG procedure
results in an assortative matching: {a-x, b-y}.
Price paid by x for a is
social welfare without
others’ welfare with
pa,x =
−
x in the mechanism
x in the mechanism
a
= VBS−x − VBS−
−x
= (ra vy + rb vz ) − (rb vy )
= (12 + 3) − (9)
=6
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Chapter 15 — Ex.4(a)
4. Suppose ad slots a, b have clickthrough rates of 4 and 3, respectively.
Advertisers x, y, z have values per click at 4, 3, 1, respectively.
(a) If the slots are allocated using a VCG procedure, what assignment of
slots will occur and what prices will the advertisers pay?
Price paid by y for b is
b
pb,y = VBS−y − VBS−
−y
= (ra vx + rb vz ) − (ra vx )
= (16 + 3) − (16)
=3
H. K. Chen (SFU)
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Chapter 15 — Ex.4(a)
4. Suppose ad slots a, b have clickthrough rates of 4 and 3, respectively.
Advertisers x, y, z have values per click at 4, 3, 1, respectively.
(a) If the slots are allocated using a VCG procedure, what assignment of
slots will occur and what prices will the advertisers pay?
Price paid by y for b is
b
pb,y = VBS−y − VBS−
−y
= (ra vx + rb vz ) − (ra vx )
= (16 + 3) − (16)
=3
The prices paid reflect the externality an advertiser imposes on the
other advertisers.
H. K. Chen (SFU)
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Chapter 15 — Ex.4(b)
4. Suppose ad slots a, b have clickthrough rates of 4 and 3, respectively.
Advertisers x, y, z have values per click at 4, 3, 1, respectively.
(b) Suppose now there is a third ad slot, c, with rc = 2. Using the VCG
procedure, find the resulting assignment and prices paid by each advertiser.
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Chapter 15 — Ex.4(b)
4. Suppose ad slots a, b have clickthrough rates of 4 and 3, respectively.
Advertisers x, y, z have values per click at 4, 3, 1, respectively.
(b) Suppose now there is a third ad slot, c, with rc = 2. Using the VCG
procedure, find the resulting assignment and prices paid by each advertiser.
Obviously, optimal assignment is {a-x, b-y, c-z}
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Chapter 15 — Ex.4(b)
4. Suppose ad slots a, b have clickthrough rates of 4 and 3, respectively.
Advertisers x, y, z have values per click at 4, 3, 1, respectively.
(b) Suppose now there is a third ad slot, c, with rc = 2. Using the VCG
procedure, find the resulting assignment and prices paid by each advertiser.
Obviously, optimal assignment is {a-x, b-y, c-z}
Price paid by x for a:
a
pa,x = VBS−x − VBS−
−x
= (ra vy + rb vz ) − (rb vy + rc vz )
= (12 + 3) − (9 + 2)
=4
H. K. Chen (SFU)
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Chapter 15 — Ex.4(b)
4. Suppose ad slots a, b have clickthrough rates of 4 and 3, respectively.
Advertisers x, y, z have values per click at 4, 3, 1, respectively.
(b) Suppose now there is a third ad slot, c, with rc = 2. Using the VCG
procedure, find the resulting assignment and prices paid by each advertiser.
Price paid by y for b:
b
pb,y = VBS−y − VBS−
−y
= 3−2
=1
H. K. Chen (SFU)
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Chapter 15 — Ex.4(b)
4. Suppose ad slots a, b have clickthrough rates of 4 and 3, respectively.
Advertisers x, y, z have values per click at 4, 3, 1, respectively.
(b) Suppose now there is a third ad slot, c, with rc = 2. Using the VCG
procedure, find the resulting assignment and prices paid by each advertiser.
Price paid by y for b:
b
pb,y = VBS−y − VBS−
−y
= 3−2
=1
Price paid by z for c: pc,z = 0
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Chapter 15 — Ex.4(c)
4(c) What revenue will the search engine receive from the VCG Procedure
in parts (a) and (b)? If you were running the search engine, given this set
of advertisers and slots, and could choose whether to create slot c or not,
what would you do? Why?
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Chapter 15 — Ex.4(c)
4(c) What revenue will the search engine receive from the VCG Procedure
in parts (a) and (b)? If you were running the search engine, given this set
of advertisers and slots, and could choose whether to create slot c or not,
what would you do? Why?
Revenue in (a): R(a) = pa,x + pb,y = 6 + 3 = 9
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Chapter 15 — Ex.4(c)
4(c) What revenue will the search engine receive from the VCG Procedure
in parts (a) and (b)? If you were running the search engine, given this set
of advertisers and slots, and could choose whether to create slot c or not,
what would you do? Why?
Revenue in (a): R(a) = pa,x + pb,y = 6 + 3 = 9
Revenue in (b): R(b) = pa,x + pb,y + pc,z = 4 + 1 + 0 = 5
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Chapter 15 — Ex.4(c)
4(c) What revenue will the search engine receive from the VCG Procedure
in parts (a) and (b)? If you were running the search engine, given this set
of advertisers and slots, and could choose whether to create slot c or not,
what would you do? Why?
Revenue in (a): R(a) = pa,x + pb,y = 6 + 3 = 9
Revenue in (b): R(b) = pa,x + pb,y + pc,z = 4 + 1 + 0 = 5
A revenue-maximizing search engine manager would not create slot c
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Chapter 15 — Ex.4(c)
4(c) What revenue will the search engine receive from the VCG Procedure
in parts (a) and (b)? If you were running the search engine, given this set
of advertisers and slots, and could choose whether to create slot c or not,
what would you do? Why?
Revenue in (a): R(a) = pa,x + pb,y = 6 + 3 = 9
Revenue in (b): R(b) = pa,x + pb,y + pc,z = 4 + 1 + 0 = 5
A revenue-maximizing search engine manager would not create slot c
Pricing in the VCG mechanism is based on the externality an agent
causes on the other agents
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Chapter 15 — Ex.4(c)
4(c) What revenue will the search engine receive from the VCG Procedure
in parts (a) and (b)? If you were running the search engine, given this set
of advertisers and slots, and could choose whether to create slot c or not,
what would you do? Why?
Revenue in (a): R(a) = pa,x + pb,y = 6 + 3 = 9
Revenue in (b): R(b) = pa,x + pb,y + pc,z = 4 + 1 + 0 = 5
A revenue-maximizing search engine manager would not create slot c
Pricing in the VCG mechanism is based on the externality an agent
causes on the other agents
Existence of a valuable slot c means that when z doesn’t get slot b
(due to the presence of either x or y), z’s payoff is not zero
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Chapter 15 — Ex.4(c)
4(c) What revenue will the search engine receive from the VCG Procedure
in parts (a) and (b)? If you were running the search engine, given this set
of advertisers and slots, and could choose whether to create slot c or not,
what would you do? Why?
Revenue in (a): R(a) = pa,x + pb,y = 6 + 3 = 9
Revenue in (b): R(b) = pa,x + pb,y + pc,z = 4 + 1 + 0 = 5
A revenue-maximizing search engine manager would not create slot c
Pricing in the VCG mechanism is based on the externality an agent
causes on the other agents
Existence of a valuable slot c means that when z doesn’t get slot b
(due to the presence of either x or y), z’s payoff is not zero
Thus the externality caused by either x or y is dampened by a slot c
that has positive value to z. But this means both x and y are paying
less. Therefore it’s not optimal to create slot c.
H. K. Chen (SFU)
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Chapter 15 — Ex.5(a)
5. Consider two ad slots a, b, with ra = 12, rb = 5; and two advertisers,
x, y, with vx = 5, vy = 4.
(a) Compute the socially optimal allocation and the VCG prices for it.
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Chapter 15 — Ex.5(a)
5. Consider two ad slots a, b, with ra = 12, rb = 5; and two advertisers,
x, y, with vx = 5, vy = 4.
(a) Compute the socially optimal allocation and the VCG prices for it.
Socially optimal allocation is {a-x, b-y}
H. K. Chen (SFU)
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Chapter 15 — Ex.5(a)
5. Consider two ad slots a, b, with ra = 12, rb = 5; and two advertisers,
x, y, with vx = 5, vy = 4.
(a) Compute the socially optimal allocation and the VCG prices for it.
Socially optimal allocation is {a-x, b-y}
Price paid by x for a:
a
pa,x = VBS−x − VBS−
−x = 48 − 20 = 28
H. K. Chen (SFU)
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Chapter 15 — Ex.5(a)
5. Consider two ad slots a, b, with ra = 12, rb = 5; and two advertisers,
x, y, with vx = 5, vy = 4.
(a) Compute the socially optimal allocation and the VCG prices for it.
Socially optimal allocation is {a-x, b-y}
Price paid by x for a:
a
pa,x = VBS−x − VBS−
−x = 48 − 20 = 28
Price paid by y for b: pb,y = 0
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Chapter 15 — Ex.5(b)
5. Consider two ad slots a, b, with ra = 12, rb = 5; and two advertisers,
x, y, with vx = 5, vy = 4.
(b) Suppose the search engine decides not to sell slot b. Instead, it sells
only slot a using a seal-bid, second-price auction. What bids will the
advertisers submit for slot a, who will win, and what price will they pay?
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Chapter 15 — Ex.5(b)
5. Consider two ad slots a, b, with ra = 12, rb = 5; and two advertisers,
x, y, with vx = 5, vy = 4.
(b) Suppose the search engine decides not to sell slot b. Instead, it sells
only slot a using a seal-bid, second-price auction. What bids will the
advertisers submit for slot a, who will win, and what price will they pay?
Advertisers bid their true values for slot a: ra vj
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Chapter 15 — Ex.5(b)
5. Consider two ad slots a, b, with ra = 12, rb = 5; and two advertisers,
x, y, with vx = 5, vy = 4.
(b) Suppose the search engine decides not to sell slot b. Instead, it sells
only slot a using a seal-bid, second-price auction. What bids will the
advertisers submit for slot a, who will win, and what price will they pay?
Advertisers bid their true values for slot a: ra vj
x wins (since vx > vy )
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Chapter 15 — Ex.5(b)
5. Consider two ad slots a, b, with ra = 12, rb = 5; and two advertisers,
x, y, with vx = 5, vy = 4.
(b) Suppose the search engine decides not to sell slot b. Instead, it sells
only slot a using a seal-bid, second-price auction. What bids will the
advertisers submit for slot a, who will win, and what price will they pay?
Advertisers bid their true values for slot a: ra vj
x wins (since vx > vy )
x pays ra vy = 48
H. K. Chen (SFU)
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Chapter 15 — Ex.5(c)
5. Consider two ad slots a, b, with ra = 12, rb = 5; and two advertisers,
x, y, with vx = 5, vy = 4.
(c) Which of these possible procedures (a) and (b) generate the greater
revenue for the search engine? By how much?
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Chapter 15 — Ex.5(c)
5. Consider two ad slots a, b, with ra = 12, rb = 5; and two advertisers,
x, y, with vx = 5, vy = 4.
(c) Which of these possible procedures (a) and (b) generate the greater
revenue for the search engine? By how much?
Part (b) generates a greater revenue:
R(b) = 48 > 28 = R(a)
By 20
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Chapter 15 — Ex.5(d)
5(d) Now let’s see if the result in part (c) is general. That is, does it
depend on the clickthrough rates and values? Let ra , rb be clickthrough
rates for slots a and b, with ra > rb > 0. Let vx , vy be values per click for
advertisers x and y, with vx > vy > 0. Can you determine which of the
two procedures generates the greater revenue for the search engine?
H. K. Chen (SFU)
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Chapter 15 — Ex.5(d)
5(d) Now let’s see if the result in part (c) is general. That is, does it
depend on the clickthrough rates and values? Let ra , rb be clickthrough
rates for slots a and b, with ra > rb > 0. Let vx , vy be values per click for
advertisers x and y, with vx > vy > 0. Can you determine which of the
two procedures generates the greater revenue for the search engine?
If both slots are sold using VCG mechanism, optimal matching is
{a-x, b-y}
H. K. Chen (SFU)
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Chapter 15 — Ex.5(d)
5(d) Now let’s see if the result in part (c) is general. That is, does it
depend on the clickthrough rates and values? Let ra , rb be clickthrough
rates for slots a and b, with ra > rb > 0. Let vx , vy be values per click for
advertisers x and y, with vx > vy > 0. Can you determine which of the
two procedures generates the greater revenue for the search engine?
If both slots are sold using VCG mechanism, optimal matching is
{a-x, b-y}
Price paid by x is pVCG
a,x = vy (ra − rb )
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Chapter 15 — Ex.5(d)
5(d) Now let’s see if the result in part (c) is general. That is, does it
depend on the clickthrough rates and values? Let ra , rb be clickthrough
rates for slots a and b, with ra > rb > 0. Let vx , vy be values per click for
advertisers x and y, with vx > vy > 0. Can you determine which of the
two procedures generates the greater revenue for the search engine?
If both slots are sold using VCG mechanism, optimal matching is
{a-x, b-y}
Price paid by x is pVCG
a,x = vy (ra − rb )
Price paid by y is pVCG
b,y = 0
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Chapter 15 — Ex.5(d)
5(d) Now let’s see if the result in part (c) is general. That is, does it
depend on the clickthrough rates and values? Let ra , rb be clickthrough
rates for slots a and b, with ra > rb > 0. Let vx , vy be values per click for
advertisers x and y, with vx > vy > 0. Can you determine which of the
two procedures generates the greater revenue for the search engine?
If both slots are sold using VCG mechanism, optimal matching is
{a-x, b-y}
Price paid by x is pVCG
a,x = vy (ra − rb )
Price paid by y is pVCG
b,y = 0
VCG
Seller’s revenue is RVCG = pVCG
a,x + pb,y = vy (ra − rb )
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Chapter 15 — Ex.5(d)
5(d) Now let’s see if the result in part (c) is general. That is, does it
depend on the clickthrough rates and values? Let ra , rb be clickthrough
rates for slots a and b, with ra > rb > 0. Let vx , vy be values per click for
advertisers x and y, with vx > vy > 0. Can you determine which of the
two procedures generates the greater revenue for the search engine?
If both slots are sold using VCG mechanism, optimal matching is
{a-x, b-y}
Price paid by x is pVCG
a,x = vy (ra − rb )
Price paid by y is pVCG
b,y = 0
VCG
Seller’s revenue is RVCG = pVCG
a,x + pb,y = vy (ra − rb )
If only a is sold using a second price auction, then x gets a
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Chapter 15 — Ex.5(d)
5(d) Now let’s see if the result in part (c) is general. That is, does it
depend on the clickthrough rates and values? Let ra , rb be clickthrough
rates for slots a and b, with ra > rb > 0. Let vx , vy be values per click for
advertisers x and y, with vx > vy > 0. Can you determine which of the
two procedures generates the greater revenue for the search engine?
If both slots are sold using VCG mechanism, optimal matching is
{a-x, b-y}
Price paid by x is pVCG
a,x = vy (ra − rb )
Price paid by y is pVCG
b,y = 0
VCG
Seller’s revenue is RVCG = pVCG
a,x + pb,y = vy (ra − rb )
If only a is sold using a second price auction, then x gets a
Price paid by x is pSPA
= vy ra
x
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Chapter 15 — Ex.5(d)
5(d) Now let’s see if the result in part (c) is general. That is, does it
depend on the clickthrough rates and values? Let ra , rb be clickthrough
rates for slots a and b, with ra > rb > 0. Let vx , vy be values per click for
advertisers x and y, with vx > vy > 0. Can you determine which of the
two procedures generates the greater revenue for the search engine?
If both slots are sold using VCG mechanism, optimal matching is
{a-x, b-y}
Price paid by x is pVCG
a,x = vy (ra − rb )
Price paid by y is pVCG
b,y = 0
VCG
Seller’s revenue is RVCG = pVCG
a,x + pb,y = vy (ra − rb )
If only a is sold using a second price auction, then x gets a
Price paid by x is pSPA
= vy ra
x
SPA
Seller’s revenue is R
= pSPA
= vy ra
x
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Chapter 15 — Ex.5(d)
5(d) Now let’s see if the result in part (c) is general. That is, does it
depend on the clickthrough rates and values? Let ra , rb be clickthrough
rates for slots a and b, with ra > rb > 0. Let vx , vy be values per click for
advertisers x and y, with vx > vy > 0. Can you determine which of the
two procedures generates the greater revenue for the search engine?
If both slots are sold using VCG mechanism, optimal matching is
{a-x, b-y}
Price paid by x is pVCG
a,x = vy (ra − rb )
Price paid by y is pVCG
b,y = 0
VCG
Seller’s revenue is RVCG = pVCG
a,x + pb,y = vy (ra − rb )
If only a is sold using a second price auction, then x gets a
Price paid by x is pSPA
= vy ra
x
SPA
Seller’s revenue is R
= pSPA
= vy ra
x
Clearly, RSPA > RVCG .
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Chapter 15 — Ex.6
6. Suppose a seller has one item, x. There are three buyers, a, b, c. The
values they have for x are 6, 3, 1, respectively.
(a) Suppose that the seller runs a second price auction for the item.
Which buyer will win the auction and how much will this buyer pay?
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Chapter 15 — Ex.6
6. Suppose a seller has one item, x. There are three buyers, a, b, c. The
values they have for x are 6, 3, 1, respectively.
(a) Suppose that the seller runs a second price auction for the item.
Which buyer will win the auction and how much will this buyer pay?
Bidders will bid truthfully, and so buyer a wins the item and pays the
second highest bid, 3.
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Chapter 15 — Ex.6
6. Suppose a seller has one item, x. There are three buyers, a, b, c. The
values they have for x are 6, 3, 1, respectively.
(a) Suppose that the seller runs a second price auction for the item.
Which buyer will win the auction and how much will this buyer pay?
Bidders will bid truthfully, and so buyer a wins the item and pays the
second highest bid, 3.
(b) Suppose the seller uses the VCG procedure to allocate the item. Find
the allocation that results from this procedure. What are the prices
charged to each buyer? Explain why the price that buyer a pays is the harm
that he causes to the remaining bidders by taking the item he is assigned.
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Chapter 15 — Ex.6
6. Suppose a seller has one item, x. There are three buyers, a, b, c. The
values they have for x are 6, 3, 1, respectively.
(a) Suppose that the seller runs a second price auction for the item.
Which buyer will win the auction and how much will this buyer pay?
Bidders will bid truthfully, and so buyer a wins the item and pays the
second highest bid, 3.
(b) Suppose the seller uses the VCG procedure to allocate the item. Find
the allocation that results from this procedure. What are the prices
charged to each buyer? Explain why the price that buyer a pays is the harm
that he causes to the remaining bidders by taking the item he is assigned.
You should be able to do this by now.
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