Download COS 444 Internet Auctions: Theory and Practice

COS 444
Internet Auctions:
Theory and Practice
Spring 2008
Ken Steiglitz
[email protected]
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Multi-unit demand auctions
(Ausubel & Cramton 98, Morgan 01)
•
Examples: FCC spectrum, Treasury debt
securities, Eurosystem: multiple, identical
units
•
Issues: Pay-your-bid (discriminatory) prices
v. uniform-price; efficiency; optimality of
revenue
•
The problem: conventional, uniform-price
auctions provide incentives for demandreduction
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Multi-unit demand auctions
Example 1: (Morgan) 2 units supply
Bidder 1: capacity 2, values $10, $10
Bidder 2: capacity 1, value $8
Suppose bidders bid truthfully; rank bids:
$10 bidder 1
10 bidder 1
8 bidder 2  first rejected bid
If buyers pay this, surplus (1) = $4
revenue = $16
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Multi-unit demand auctions
Example 1: But bidder 1 can do better!
Bidder 1: capacity 2, values $10, $10
Bidder 2: capacity 1, value $8
Suppose bidder 1 shades her demand:
$10 bidder 1 for her first unit
8 bidder 2 for first unit
0 bidder 1 for her 2nd unit  first rej. bid
If buyers pay this, surplus (1) = $10
surplus (2) = $8 inefficient!
revenue
= $0!
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Multi-unit demand auctions
Thus,
uniform price demand reduction inefficiency
The natural generalization of the Vickrey auction
(winners pay first rejected bid) is not incentive
compatible and not efficient
Lots of economists got this wrong!
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Multi-unit demand auctions
Ausubel & Cramton prove, in a simplified model,
that this example is not pathological:
Proposition: There is no efficient equilibrium
strategy in a uniform-price, multi-unit demand
auction.
The appropriate generalization of the Vickrey
auction is the Vickrey-Clark-Groves (VCG)
mechanism…
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The VCG auction for multi-unit demand
Return to example 1: 2 units supply
Bidder 1: capacity 2, values $10, $10
Bidder 2: capacity 1, value $8
Suppose bidders bid truthfully, and order bids:
$10 bidder 1
10 bidder 1
8 bidder 2
Award supply to the highest bidders
… How much does each bidder pay?
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The VCG auction for multi-unit demand
Define:
social welfare = W ( v ) = total value received by
agents, where v is the vector of values
Then the VCG payment of i is
W-i ( 0, x-i ) − W-i ( x )
= welfare to others when i bids 0, minus that when i
bids truthfully
= sum of ki rejected bids (if bidder i gets ki items)
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The VCG auction for multi-unit demand
Example 1: 2 units supply
Bidder 1: capacity 2, values $10, $10
Bidder 2: capacity 1, value $8
If bidder 1 bids 0, welfare = $8,
and is $0 when 1 bids truthfully…
 1 pays $8 for the 2 items
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The VCG auction for multi-unit demand
Example 2: 3 units supply
Bidder 1: capacity 2, values $10, $10
Bidder 2: capacity 1, value $8
Bidder 3: capacity 1, value $6
$10 bidder 1
10 bidder 1  bidder 1 gets 2 items
8 bidder 2  bidder 2 gets 1 item
6 bidder 3
Welfare when 1 bids 0 = $14
Welfare when 1 bids truthfully = $8
 1 pays $6 for the 2 items
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The VCG auction for multi-unit demand
Example 2, con’t
3 units supply
Bidder 1: capacity 2, values $10, $10
Bidder 2: capacity 1, value $8
Bidder 3: capacity 1, value $6
$10 bidder 1  bidder 1 gets 2 items
8 bidder 2  bidder 2 gets 1 item
6 bidder 3
Welfare when 2 bids 0 = $26
Welfare when 2 bids truthfully = $20
 2 pays $6 for the 1 item
(notice that revenue = $12 < $18 =3x$6 in uniformprice case, so not optimal)
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VCG mechanisms
(Krishna 02)
VCG mechanisms are
• efficient
• incentive-compatible
(truthful is weakly dominant)
• individually rational
• max-revenue among all such mechanisms
… but not optimal revenue in general,
and prices are discriminatory, “murky”
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Bilateral trading mechanisms
[Myerson & Satterthwaite 83]
An impossibility result:
The following desirable characteristics of
bilateral trade (not an auction):
1) efficient
2) incentive-compatible
3) individually rational
Cannot all be achieved simultaneously!
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Bilateral trading mechanisms
The setup:
•
one seller, with private value v 1 ,
distributed with density f1 > 0 on [a1 , b1 ]
•
one buyer, with private value v 2 ,
distributed with density f2 > 0 on [a2 , b2 ]
•
risk neutral
… Notice: not an auction in Riley &
Samuelson’s class!
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Bilateral trading mechanisms
Outline of proof: We use a direct
mechanism (p, x ):
where p (v1 , v2 ) = prob. of transfer 12
x (v1 , v2 ) = expected payment 12
b2
p1 (v1 )   p(v1 , t2 ) f 2 (t2 )dt2  prob. selling 1 to 2
a2
b2
x1 (v1 )   x(v1 , t2 ) f 2 (t2 )dt2  E[rev seller 1]
a2
U1(v1)  x1(v1)  v1  p1(v1)  E[profit 1]
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Bilateral trading mechanisms
Main result: If
[a1, b1]  [a2 , b2 ]  
then no incentive-compatible individually rational
trading mechanism can be (ex post) efficient.
Furthermore,
b1
 [1  F (t )]  F (t ) dt
2
1
a2
is the smallest lump-sum subsidy to achieve
efficiency.
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Bilateral trading mechanisms
Examples
1. f i > 0 is necessary: discrete probs.
2. Subsidy for efficiency: v 1 and v 2
both uniform on [0,1]
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Auctions vs. Negotiations
(Bulow & Klemperer 96)
Simple example: IPV, uniform
Case 1) Optimal auction = optimal mechanism
with one buyer. Optimal entry value v* = 0.5;
revenue = 1/4
Case 2) Two buyers, no reserve; revenue = 1/3
>¼
 One more buyer is worth more than setting
reserve optimally!
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Auctions vs. Negotiations, con’t
Bulow & Klemperer 96 generalize to any F,
any number of bidders…
A no-reserve auction with n +1 bidders
is more profitable than an optimal auction
(and hence optimal mechanism) with n
bidders
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Auctions vs. Negotiations, con’t
Optimal reserve, n bidders:
1

v*
M (v) dF (v)
n
No reserve, n+1 bidders
1

0
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M (v) dF (v)
n1
20
E[ M (v)]  0
Auctions vs. Negotiations, con’t
Facts:
1

v*
M (v) dF (v) n  E[max{ M (v1 ),..., M (vn ),0}]
1
 M (v) dF (v)
n1
0
E[ M (v)]  0
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 E[max{ M (v1 ),...,M (vn1 )}]
… QED
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Bidder rings (Graham & Marshall 87)
Stylized facts
1) They exist and are stable
2) They eliminate competition among ring
members; yet ensure ring member with
highest value is not undercut
3) Benefits shared by ring members
4) Have open membership
5) Auctioneer responds strategically
6) Try to hide their existence
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Bidder rings
Graham & Marshall’s model: Second-price preauction knockout (PAKT)
i.
IPV, risk neutral
ii.
Value distributions F, common knowledge
iii. Identity of winner & price paid common
knowledge
iv. Membership of ring known only to ring
members
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Bidder rings
Pre-auction knock-out (PAKT):
1) Appoint ring center, who pays P to each
ring member, P to be determined below
2) Each ring member submits a sealed bid to
the ring center
3) Winner is advised to submit her winning
bid at main auction; other ring members
submit only meaningless bids
4) If the winner at the sub-auction (subwinner) also wins main auction, she pays:
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Bidder rings
If sub-winner wins main auction, she
pays:
o
Main auctioneer P* = SP at main
auction
o
Ring center δ = max{ P̃ − P* , 0 },
where P̃ = SP in PAKT
Thus: If the sub-winner wins main
auction, she pays SP among all bids
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Bidder rings
The quantity δ is the amount “stolen” from the
main auctioneer, the “booty”
The ring center receives and distributes
E[δ | sub-winner wins main auction]
 so his budget is balanced
Each ring member receives
P = E[δ | sub-winner wins main auction]/K
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Bidder rings
Graham and Marshall prove:
1)
Truthful bidding in the PAKT, and following the
recommendation of the ring center is SBNE &
weakly dominant strategy (incentive
compatible)
2)
Voluntary participation is advantageous
(individually rational)
3)
Efficient (buyer with highest value gets item)
In fact, the whole thing is equivalent to a Vickrey auction
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Bidder rings
Main auctioneer responds strategically by
increasing reserves or shill-bidding
Graham& Marshall also prove that
1.
Optimal main reserve is an increasing function
of ring size K
2.
Expected surplus of ring member is a
decreasing function of reserve prices
3.
Expected surplus of ring member is an
increasing function of ring size K
So best to be secretive
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Term papers due 5pm Tuesday
May 13 (Dean’s Date)
 Email me for office hours re term papers
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