Auctions
Economics 383 - Auction Theory
Instructor: Songzi Du
Simon Fraser University
November 17, 2016
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Auctions
Mechanisms of transaction: bargaining, posted price, auctions
Auction: take bids, allocate resource, and collect payments.
Babylonian wife auction (500 BC)
Auction of the Roman Empire by the Praetorian Guard (who had killed
Emperor Pertinax in 193 AD). The winning bidder Didius Julianus was
crowned Emperor; beheaded 9 weeks later (winner’s curse).
Google AdWords auction (revenue of USD$28 billion in 2010), eBay
Financial auctions (treasury bills, settlement of credit default swap,
stock exchange)
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Second price auction
A single, indivisible good.
Second price auction:
1
2
Every bidder submits a bid, simultaneously (sealed bid).
The highest bidder gets the object and pays the second highest bid;
everyone else does not pay.
Also known as Vickrey auction.
Proxy bidding in eBay: a computer program that automatically and
minimally increases your bid (up to your pre-specified maximum
amount) to ensure that you are the top bidder.
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Second price auction
A single, indivisible good.
Second price auction:
1
2
Every bidder submits a bid, simultaneously (sealed bid).
The highest bidder gets the object and pays the second highest bid;
everyone else does not pay.
Also known as Vickrey auction.
Proxy bidding in eBay: a computer program that automatically and
minimally increases your bid (up to your pre-specified maximum
amount) to ensure that you are the top bidder.
Bidder i has a value vi for the good (his private information), payoff
of vi − Pi if he gets it, 0 if not.
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Ascending bid auction
Also known as English auction.
The auction is carried out interactively in real time.
The auctioneer gradually raises the price, starting from some reserve
price (e.g., zero), bidders drop out until finally only one bidder
remains, and that bidder wins the object at this final price.
Variants of ascending bid auction: bidders shout out prices, or submit
them electronically.
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Ascending bid auction
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Strategy in Second Price Auction
Strategy: a function si (vi ) that maps values to bids.
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Strategy in Second Price Auction
Strategy: a function si (vi ) that maps values to bids.
n bidders
Payoff function:


vi − max(b1 , . . . , bi−1 , bi+1 , . . . , bn )




if bi > max(b1 , . . . , bi−1 , bi+1 , . . . , bn )
Ui (vi , b1 , b2 , . . . , bn ) =

0




otherwise
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Strategy in Second Price Auction
Strategy: a function si (vi ) that maps values to bids.
n bidders
Payoff function:


vi − max(b1 , . . . , bi−1 , bi+1 , . . . , bn )




if bi > max(b1 , . . . , bi−1 , bi+1 , . . . , bn )
Ui (vi , b1 , b2 , . . . , bn ) =

0




otherwise
Dominant strategy si (vi ) satisfies: for every
(s1 ( · ), . . . , si−1 ( · ), si+1 ( · ), . . . , sn ( · )) and every (v1 , v2 , . . . , vn ),
Ui (vi , s1 (v1 ), . . . , si−1 (vi−1 ), si (vi ), si+1 (vi+1 ), . . . , sn (vn ))
≥Ui (vi , s1 (v1 ), . . . , si−1 (vi−1 ), bi , si+1 (vi+1 ), . . . , sn (vn ))
for every bi ∈ R.
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Dominant Strategy in Second Price Auction
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Why second price? Why not third price?
Third price auction: the highest bidder gets the good and pays the
third highest bid; everyone else do not pay.
Is truthful bidding the dominant strategy?
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Auction of two goods
Auction of two indivisible, identical goods.
Each bidder i wants only one good, has a value vi if he gets a good.
As before, each bidder submits a bid.
Third-price auction: the top two bidders each gets a good, and each
pays the third highest bid; the rest do not pay.
Is truthful bidding the dominant strategy?
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Facts about uniform distribution
Suppose n bidders, with values vi randomly and independently drawn
from the uniform distribution on [0, 1]:
P(vi ≤ x) = x,
P(v1 ≤ x1 , v2 ≤ x2 , v3 ≤ x3 ) = x1 · x2 · x3 ,
for x’s between 0 and 1.
E[max(v1 , v2 , . . . , vn )] =
n−1
n
, E[max2 (v1 , v2 , . . . , vn )] =
,
n+1
n+1
n−2
1
, . . . , E[min(v1 , v2 , . . . , vn )] =
,
n+1
n+1
where max2 means second highest, max3 third highest, etc.
E[max3 (v1 , v2 , . . . , vn )] =
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Reserve price in second price auction
Reserve price (r ): the minimum bid that is considered in the (second
price) auction, announced before the auction.
1
2
The good is sold to the highest bidder if the highest bid is equal or
above r ; otherwise, the good is not sold.
The winning bidder (if any) pays the maximum of the second-place bid
and the reserve price.
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Reserve price in second price auction
Reserve price (r ): the minimum bid that is considered in the (second
price) auction, announced before the auction.
1
2
The good is sold to the highest bidder if the highest bid is equal or
above r ; otherwise, the good is not sold.
The winning bidder (if any) pays the maximum of the second-place bid
and the reserve price.
Why set reserve price? What is the role of reserve price in revenue?
Suppose the seller has no value for the (single, indivisible) good that
he is auctioning. There are n bidders, with values randomly and
independently drawn from the uniform distribution on [0, 1].
What’s the optimal reserve price when n = 1? n = 2?
What happens when the seller uses a posted price?
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Second price auction with reserve price r
Let Rev(r ) be the seller’s revenue given a reserve price r ∈ [0, 1].
If there is only n = 1 bidder:
Rev(r ) = (1 − r ) · r .
Rev0 (r ) = 1 − 2r
Optimal reserve price r = 1/2 (from solving Rev0 (r ) = 0).
If there are n = 2 bidders:
1−r
Rev(r ) = 2r (1 − r ) · r + (1 − r ) · r +
3
2
Rev0 (r ) = 2r (1 − 2r )
Optimal reserve price r = 1/2.
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Second price auction with reserve price r
An observation: the seller’s revenue from the optimal reserve price
and 1 bidder (1/4) is less than his revenue from zero reserve price and
2 bidders (1/3).
This is a general theorem (Bulow and Klemperer).
Setting the optimal reserve price is less profitable than simply
attracting an additional bidder.
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First price auction
A single, indivisible good.
First price auction:
1
2
Every bidder submits a bid, simultaneously (sealed bid).
The highest bidder gets the object and pays his own bid; everyone else
does not pay.
Bidder i has a value vi for the good (his private information), payoff
of vi − Pi if he gets it, 0 if not.
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Descending bid auction
Also known as Dutch auction.
The auction is carried out interactively in real time.
The auctioneer gradually lowers the price from some high initial value
until the first moment when some bidder accepts and pays the current
price.
Flowers have long been sold in the Netherlands using this procedure.
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A Model of First Price Auction
n bidders (n ≥ 2)
Each bidder i (1 ≤ i ≤ n) has a private value vi for the good.
0 ≤ vi ≤ 1.
The distribution of vi is the uniform distribution on [0, 1]. Identical
and independent distribution for every bidder.
Bidding strategy is a function si (vi ) that maps values to bids.
vi is bidder i’s type.
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Strategy in First Price Auction
Payoff function:


vi − bi




if bi > max(b1 , . . . , bi−1 , bi+1 , . . . , bn )
Ui (vi , b1 , b2 , . . . , bn ) =

0




otherwise
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Strategy in First Price Auction
Payoff function:


vi − bi




if bi > max(b1 , . . . , bi−1 , bi+1 , . . . , bn )
Ui (vi , b1 , b2 , . . . , bn ) =

0




otherwise
Bayesian Nash Equilibrium: strategy profile
(s1 (v1 ), s2 (v2 ), . . . , sn (vn )) such that for every bidder i and every vi ,
E[Ui (vi , s1 (v1 ), . . . , si−1 (vi−1 ), si (vi ), si+1 (vi+1 ), . . . , sn (vn ))]
≥E[Ui (vi , s1 (v1 ), . . . , si−1 (vi−1 ), bi , si+1 (vi+1 ), . . . , sn (vn ))]
for every bi ∈ R.
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Solving for equilibrium (first price auction)
We focus on symmetric equilibrium: s1 = s2 = · · · = sn = s.
What is bidder i’s profit from bidding s(vi ), given that others also bid
according to s?
Ui (vi ) = (vi )n−1 · (vi − b(vi ))
Bidder i of type vi maximizes (by bidding s(x)):
max x n−1 (vi − s(x))
0≤x≤1
FOC:
(n − 1)x
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n−2
vi − (n − 1)x
n−2
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s(x) − x
s (x)
n−1 0
=0
x=vi
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Solving for equilibrium (first price auction)
FOC:
(n − 1)(vi )n−2 vi − (n − 1)(vi )n−2 s(vi ) − (vi )n−1 s 0 (vi ) = 0
Rearrange:
s(vi ) = vi −
vi 0
s (vi )
n−1
Guess: s(vi ) = A(vi )k
⇒ A(vi )k = vi −
Clearly k = 1. Then A = 1 −
Equilibrium bidding strategy:
A
n−1 ,
s(vi ) =
vi
Ak(vi )k−1 .
n−1
i.e., A =
n−1
n .
n−1
vi
n
s(vi ) < vi . This is called bid shading.
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All Pay Auction
All pay auction: the highest bidder gets the good, everyone pays
his/her bid. Everything else as before (a single good, simultaneous
bids, private values, etc.)
Example (bribery): in 2008, Governor Rod Blagojevich of Illinois tried
to sell Barack Obama’s senate seat to the highest bidder.
Other examples: war of attrition, political campaign, Olympic game,
etc.
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Solving for equilibrium (all pay auction)
We focus on symmetric equilibrium: s1 = s2 = · · · = sn = s.
What is bidder i’s profit from bidding s(vi ), given that others also bid
according to s?
Ui (vi ) = (vi )n−1 vi − s(vi )
Bidder i of type vi maximizes (by bidding s(x)):
max x n−1 vi − s(x)
0≤x≤1
FOC:
(n − 1)x
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n−2
vi − s (x)
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0
=0
x=vi
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Solving for equilibrium (all pay auction)
FOC:
(n − 1)(vi )n−1 = s 0 (vi ).
Guess: s(vi ) = A(vi )k
⇒ (n − 1)(vi )n−1 = Ak(vi )k−1 .
k = n and n − 1 = Ak, i.e., A =
n−1
n .
Equilibrium bidding strategy:
s(vi ) =
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n−1
(vi )n
n
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Average Price Auction
Average Price Auction: the highest bidder gets the good, pays the
average of all bids; everyone else does not pay. Everything else as
before (a single good, simultaneous bids, private values, etc.)
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Solving for equilibrium (average price auction)
We focus on symmetric equilibrium: s1 = s2 = · · · = sn = s.
What is bidder i’s profit from bidding s(vi ), given that the other also
bids according to s?
1
Ui (vi ) = (vi )n−1 · vi − (s(vi ) + (n − 1)E[s(vj ) | vj ≤ vi ]) ,
n
j 6= i
Bidder i of type vi maximizes (by bidding s(x)):
1
n−1
max x
· vi − · (s(x) + (n − 1)E[s(vj ) | vj ≤ x])
0≤x≤1
n
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Solving for equilibrium (average price auction)
Guess: s(vi ) = Avi
⇒ max (x)
0≤x≤1
n−1
1
Ax
· vi − · Ax + (n − 1)
n
2
FOC:
(n−1)(vi )
A=
n−1
n+1
n+1
n−1 −
n·Ax
= (n−1)(vi )n−1 −
A(vi )n−1 = 0
2n
2
x=vi
2(n−1)
n+1 .
Equilibrium bidding strategy:
s(vi ) =
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2(n − 1)
vi
n+1
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Revelation Principle
Bidder i’s equilibrium strategy si (vi ) is his “agent.”
Bidder i tells the “agent” his true value, the “agent” bids on his
behalf.
No incentive to deviate from the strategy si is equivalent to an
incentive to report the true value to the “agent.” This is known as
the revelation principle.
Bidder i is not necessarily bidding truthfully with si (vi ) (i.e., si (vi )
needs not be vi ).
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Comparing payments from auctions
First price auction: sfp (vi ) =
n−1
n vi .
Second pay auction: ssp (vi ) = vi .
n−1
n
n (vi ) .
save (vi ) = 2(n−1)
n+1 vi .
All pay auction: sall (vi ) =
Average price auction:
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Comparing payments from auctions
First price auction: sfp (vi ) =
n−1
n vi .
Second pay auction: ssp (vi ) = vi .
n−1
n
n (vi ) .
save (vi ) = 2(n−1)
n+1 vi .
All pay auction: sall (vi ) =
Average price auction:
In all of these auctions, the expected payment of a bidder i with value
n
vi is n−1
n (vi ) .
Same payment, i.e., revenue equivalence!
Bidders respond strategically to the change in auction rule, un-do the
intended change.
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