Auction Theory
Class 3 – optimal auctions
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Optimal auctions
• Usually the term optimal auctions stands for revenue
maximization.
• What is maximal revenue?
– We can always charge the winner his value.
• Maximal revenue: optimal expected revenue in
equilibrium.
– Assuming a probability distribution on the values.
– Over all the possible mechanisms.
– Under individual-rationality constraints (later).
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Next: Can we get better revenue?
• Can we achieve better revenue than the 2nd-price/1st
price?
• If so, we must sacrifice efficiency.
– All efficient auction have the same revenue….
• How?
– Think about the New-Zealand case.
3
4
5
Vickrey with Reserve Price
•
Seller publishes a minimum (“reserve”) price R.
•
Each bidder writes his bid in a sealed envelope.
•
The seller:
–
–
•
Collects bids
Open envelopes.
Winner:
Bidder with the highest bid, if bid is above R.
Otherwise, no one wins.
Payment: winner pays max{ 2nd highest bid, R}
Still Truthful?
Yes. For bidders, exactly like an
extra bidder bidding R.
6
Can we get better revenue?
• Let’s have another look at 2nd price auctions:
1
2 wins
v2
1 wins
x
1 wins and pays x
(his lowest winning bid)
0
0
x
v1
1
7
Can we get better revenue?
• I will show that some reserve price improve revenue.
1
Revenue
increased
2 wins
v2
1 wins
R
Revenue
increased
0
0
1
When
comparing
R to the 2nd-price auction with no reserve
1 (efficiency loss too)
price: Revenue loss here
v
8
Can
we
get
better
revenue?
1
v2
We will be
here with
probability
R(1-R)
2 wins
We will be
here with
probability
R2
1 wins
Average
loss is R/2
Loss is
always at
most R
0
0
v1
1
• Gain is at least 2R(1-R) R/2 = R2-R3  When R2-2R3>0, reserve
price of R is beneficial.
• Loss is at most R2 R = R3
(for example, R=1/4)
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Reservation price
Let’s see another example:
How do you sell one item to one bidder?
– Assume his value is drawn uniformly from [0,1].
• Optimal way: reserve price.
– Take-it-or-leave-it-offer.
Probability that
the buyer will
accept the price
The payment for
the seller
• Let’s find the optimal reserve price:
E[revenue] = ( 1-F(R) ) × R = (1-R) ×R
 (1  R) R
 1  2R  0
R
 R=1/2
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Back to New Zealand
• Recall:
Vickrey auction.
Highest bid: $100000.
Revenue: $6.
• Two things to learn:
– Seller can never get the whole pie.
• “information rent” for the buyers.
– Reserve price can help.
• But what if R=$50000 and highest bid was $45000?
• Of the unattractive properties of Vickrey Auctions:
– Low revenue despite high bids.
– 1st-price may earn same revenue, but no explanation needed…
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Optimal auctions: questions.
• Is indeed Vickrey auctions with reserve price achieve
the highest possible revenue?
• If so, what is the optimal reserve price?
• How the reserve price depends on the number of
bidders?
– Recall:
for the uniform distribution with 1 bidder the optimal
reserve price is ½.
What is the optimal reserve price for 10 players?
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Optimal auctions
• So auctions with the same allocation has the same
revenue.
• But what is the mechanism that obtains the highest
expected revenue?
13
Virtual valuations
• Consider the following transformation on the value
of each bidder: v~(v)  v  1  F (v)
f (v )
– This is called the virtual valuation.
~ (v) when a
– Like bidders’ values: The virtual valuation is v
player wins and zero otherwise.
• Example: the uniform distribution on [0,1]
– Recall: f(v)=1, F(v)=v for every v
1 v
~
v (v )  v 
 2v  1
1
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Optimal auctions
• Why are we interested in virtual valuations?
A key insight (Myerson 81’):
In equilibrium,
E[ revenue ] = E[ virtual valuation ]
• Meaning: for maximizing revenue we will need to
maximize virtual values.
– Allocate the item to the bidder with the highest virtual value.
• Like maximizing efficiency, just when considering
virtual values.
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Optimal auctions
• An optimal auction allocates the item to the bidder
with the highest virtual value.
– Can we do this in equilibrium?
• Is the bidder with the highest value is the bidder with
the highest virtual value?
– Yes, when the virtual valuation is monotone nondecreasing.
– And when values are distributed according to the same F
– Therefore, Vickrey with a reserve price is optimal.
• Will see soon what is the optimal reserve price.
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Optimal auctions
• Bottom line:
The optimal auction allocates the item to the
bidder with the highest virtual value, and this
1  F (v )
~
v
(
v
)

v

is a truthful mechanism when
f (v )
is non-decreasing.
– Vickrey auction with a reserve price.
• Remark: distribution for which the virtual valuation
is non-decreasing are called Myerson-regular.
– Example: for the uniform distribution v~ (v)  2v  1
is Myerson-regular.
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Optimal auctions: proof
A key insight (Myerson 81’):
In equilibrium,
E[revenue] = E[virtual valuation]
1  F (v )
~
where the virtual valuations is: v (v)  v 
f (v )
(Note: this theorem does not require that the virtual valuation
is Myerson-monotone.)
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Calculus reminder:
Integration by parts
h( x) g ( x)'  h' ( x) g ( x)  h( x) g ' ( x)
h( x) g ' ( x)  h( x) g ( x)'h' ( x) g ( x)
Integrating:
 h( x) g ' ( x)dx  h( x) g ( x)   h' ( x) g ( x)dx
And for definite integral )‫(אינטגרל מסויים‬:
b


 b


 g ( x)dx   h' ( x) g ( x)dx dy

h
(
x
)
g
(
x
)
dx

h
(
x
)
a

 a




 a


b
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Optimal auctions: proof
•
We saw:
consider a truthful mechanism where the probability of a
player that bids v’ to win is Qi(v).
Then, bidder i’s expected payment must be:
v'
pi (v' )  v' Qi (v' )   Qi (v)dv
a
• The expected payment of bidder i is the average over all his
possible values:
v'


E[ pi (v' )]   pi (v' ) f (v' )dv'    v' Qi (v' )   Qi (v)dv  f (v' )dv'
a
a

a
b
b
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Optimal auctions: proof
v'


E[ pi (v' )]    v' Qi (v' )   Qi (v)dv  f (v' )dv'
a
a

b
 v'

  v' Qi (v' ) f (v' )dv'    Qi (v)dv  f (v' )dv'
a
aa

b
b
Let’s simplify this term….
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Optimal auctions: proof
b
Formula of integration b


 b





a h( x) g ( x)dx h( x)  g ( x)dx   a h' ( x)  g ( x)dx dy
by parts:

where


a  a Qi (v)dv  f ( x)dx 
b

x
 a


x
h( x)   Qi (v)dv
a
g ( x)  f ( x)
b

 b

  Qi (v)dv   F ( x)   Qi ( x)F ( x)dx 
 a
 a a

x
b
b

  Qi (v)dv  1  0   Qi ( x)F ( x)dx 


a
a


Recall that:
x
F ( x)   f ( x)dx
a
b
 1  F ( x)Q ( x)dx
i
a
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Optimal auctions: proof
x


E[ pi ( x)]    xQi ( x)   Qi (v)dv  f ( x)dx
a
a

b
x

  xQi ( x) f ( x)dx     Qi (v)dv  f ( x)dx
a
aa

b
b
b
b
  xQi ( x) f ( x)dx   1  F ( x) Qi ( x)dx
a
Let’s simplify this term….
a
Taking out a
factor of
Qi(x)f(x)


1  F ( x)  
dx
  Qi ( x) f ( x) x 
f ( x) 

a
b
 E[virtual  valuation  of  player  i ]
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Optimal auctions: proof
Ev1 ,..., vn [ pi (vi )]  Ev1 ,..., vn [v~i ]
Expected
payment of
bidder I
Expected virtual
valuation of
player i
n

 n ~
Ev1 ,..., vn  pi (vi )  Ev1 ,..., vn  vi 
 i 1

 i 1 
Expected revenue
Expected virtual
valuation
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Optimal auctions
• Bottom line:
The optimal auction allocates the item to the bidder
with the highest virtual value, and this is a truthful
mechanism when v~ (v)  v  1  F (v)
f (v )
is non-decreasing.
• The auction will not sell the item if the maximal
virtual valuation is negative.
– No allocation  0 virtual valuation.
• The optimal auction is Vickrey with reserve price p
such that v~ ( p )  0
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Optimal auctions: uniform dist.
• The virtual valuation: v~ (v)  2v  1
1
~
• The optimal reserve price is ½: v ( )  0
2
• The optimal auction is the Vickrey auction with a
reserve price of ½.
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Remarks
• Reservation price is independent of the number of
bidders
– With uniform distribution, R=1/2 for every n.
• With non-identical distributions (but still statistically
independent), the same analysis works
– Optimal auction still allocate the item to the bidder with
the highest virtual valuation.
– However, Vickrey+reserve-price is not necessarily the
optimal auction in this case.
• (it is not true anymore that the bidder with the highest value is the bidder
with the highest virtual value)
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Summary: Efficiency vs. revenue
Positive or negative correlation ?
• Always: Revenue ≤ efficiency
– Due to Individual rationality.
More efficiency makes the pie larger!
• However, for optimal revenue one needs to sacrifice
some efficiency.
• Consider two competing sellers: one optimizing
revenue the other optimizing efficiency.
– Who will have a higher market share?
– In the longer terms, two objectives are combined.
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