COS 444
Internet Auctions:
Theory and Practice
Spring 2009
Ken Steiglitz
ken@cs.princeton.edu
week 12
1
Bidder collusion
• Of course, in general, bidders want to reduce
competition, the seller wants to increase competition
• We’ve seen examples on eBay of hypothetical implicit
bidder collusion (Rasmusen 2006), and likely seller shill
bidding
• Collusive bidding may be easier in multi-item auctions
• P. Cramton & J.A. Schwartz, “Collusive bidding in the
FCC Spectrum Auctions,” J. Regulatory Economics,
1999, describe (highly) probable collusion in
simultaneous ascending price auctions for FCC licenses
(analogous to English for multiple items).
week 12
2
• Code bids: use the trailing digits of the bid (often multimillions of dollars) “to tell other bidders on which licenses
to bid or not bid. … Oftentimes, a bidder (the sender)
would use these code bids as retaliation against another
bidder (the receiver) who was bidding on a license
desired by the sender. The sender would raise the price
on some market the receiver wanted, and use the trailing
digits to tell the receiver on which license to cease
bidding. ”
Cramton & Schwartz 1999
week 12
3
week 12
4
• Here is their disclaimer:
“Disclaimer: For this analysis, we show that several
bidders apparently used signaling to coordinate on
license allocations. This apparent signaling may be
coincidental. The claims we make concerning a
bidders signaling are based on circumstantial evidence,
and though we may attach some meaning to help
explain certain patterns of bidding, this meaning should
be taken as our hypothesis only. We make no
claims concerning the actual intent of the bidders.”
Cramton & Schwartz 1999
week 12
5
From
a first-hand account
of more explicit collusion
week 12
6
“Marks & Co. were kings of the book ring. They were
one of the five leading firms of antiquarian booksellers
who never bid against each other in the auction rooms.
One member of the ring would be allowed to buy a book
for a nominal sum, say £100. As soon as the auction was
over* the five conspirators would hurry to their nearest
safe-house – usually a Lyons tea shop – and conduct a
private auction. If one of them bought the book for £500,
the £400 profit would be divided in cash amongst the
other four†. This process was called a ‘knock-out’, and
Frank Doel once blew an entire operation.
Between Silk and Cyanide, Leo Marks, Harper Collins, London, 1998.
week 12
* a post-auction knock-out
†nota
bene
7
“A famous heart specialist names Evan Bedford
instructed him to bid up to £300 for an edition of
Harvey’s De Motu Cordis, the earliest printed book on
the circulation of the blood, which was coming up for
auction at Hodgson’s. Too busy with his own Hartley
Street salesroom to attend the auction himself, he
telephoned Frank at home late at night demanding to
know why the book had been sold to another dealer for
£200 when he’d authorized Frank to bid three. Frank
confided that it had been sold in the knock-out for £600.
The irate physician immediately undertook to have the
whole question of the book ring raised in the House of
Commons, which caused cardiac arrest amongst its five
participants.
Between Silk and Cyanide, Leo Marks, Harper Collins, London, 1998.
week 12
8
See Cassady 1967 for lots of details about real ring operations
• He describes post-auction knockouts, “…the ring holds a
private sale to liquidate [the goods] and divide them
among ring members.” (p. 180). Notice that the knockout is an example where the utility of a buyer includes
some utility of the seller---since each prospective buyer
has a stake in the seller’s revenue.
• Organizing a ring is often a very complex operation.
• Mentions that in sale of timber rights by U.S. gov’t.
collusion is common; these sales have strong commonvalue features, similar to spectrum auctions.
• Mentions Australian wool trade---the most complex buyer
collusion known to Cassady. (p. 187) One buyer
belonged to thirteen two-member and thirteen threemember rings.
week 12
9
Bidder rings (Graham & Marshall 1987)
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
week 12
10
Graham & Marshall’s theoretical model: Secondprice pre- auction knockout (PAKT)
•
•
•
•
IPV, risk neutral
Value distributions F, common knowledge
Identity of winner & price paid common
knowledge
Membership of ring known only to ring
members
week 12
11
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 (sub-winner)
also wins main auction, she pays:
week 12
12
If sub-winner wins main auction, she pays:
•
Main auctioneer P* = SP at main auction
•
Ring center δ = max{ P̃ − P* , 0 }, where P̃
= SP in PAKT
Thus: If the sub-winner wins main auction, she
pays in total the SP among all bids (which
would have happened without the ring)
and… the profit is passed along to the ring
center
week 12
13
week 12
14
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, in expectation
Each ring member receives
P = E[δ | sub-winner wins main auction]/K
week 12
15
Bidder rings
Graham and Marshall prove:
1) Truthful bidding in the PAKT, and following the
recommendation of the ring center is a 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
week 12
16
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
week 12
17
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!
week 12
18
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!
week 12
19
Bilateral trading mechanisms
•
Outline of proof: We use a direct
mechanism (p, x), where
p (v1 , v2 ) = prob. of transfer to buyer
x (v1 , v2 ) = expected payment to seller
b2
p1 (v1 )   p (v1 , t2 ) f 2 (t2 )dt2  prob. selling to buyer
a2
b2
x1 (v1 )   x(v1 , t2 ) f 2 (t2 )dt2  E[payment to seller]
a2
U1 (v1 )  x1(v1 )  v1  p1(v1)  E[profit of seller]
week 12
20
Bilateral trading mechanisms
•
Similarly and symmetrically:
b1
p2 (v2 )   p (t1 , v2 ) f1 (t1 )dt1  prob. transfer to buyer
a1
b1
x2 (v2 )   x(t1 , v2 ) f1 (t1 )dt1  E[payment from buyer ]
a1
U2 (v2 )  v2  p2 (v2 )  x2 (v2 )  E[profit of buyer]
week 12
21
Bilateral trading mechanisms
•
Incentive-compatible means
U1 (v1 )  x1 (vˆ1 )  v1  p1 (vˆ1 )  v1, vˆ1 [a1, b1 ]
U2 (v2 )  v2  p(vˆ2 )  x2 (vˆ2 )  v2 , vˆ2 [a2 , b2 ]
•
Individually rational means
U1 (v1 )  0 v1 [a1, b1 ]
U2 (v2 )  0 v2 [a2 , b2 ]
•
Ex post efficient means
1 if v1  v2
p(v1 , v2 )  
else
0
week 12
22
Bilateral trading mechanisms
•
Incentive-compatible means
U1 (v1 )  x1 (vˆ1 )  v1  p1 (vˆ1 )  v1, vˆ1 [a1, b1 ]
U2 (v2 )  v2  p(vˆ2 )  x2 (vˆ2 )  v2 , vˆ2 [a2 , b2 ]
•
Individually rational means
U1 (v1 )  0 v1 [a1, b1 ]
U2 (v2 )  0 v2 [a2 , b2 ]
•
no incentive
to lie about
v’s
participation does not
entail expected loss
Ex post efficient means
1 if v1  v2
p(v1 , v2 )  
else
0
week 12
object is sold iff buyer
values it more highly
23
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.
week 12
24
Proof steps
•
Part 1: incentive-compatible and individually
rational implies
U1 (b1 )  U2 (a2 )  0
min. E[profit] of seller + min. E[profit] of buyer
•
Part 2: ex post efficient implies
b1
U1 (b1 )  U 2 (a2 )    [1  F2 (t )]F1 (t )dt  0
a2
… contradiction!
week 12
25
Bilateral trading mechanisms
Example: Shows
f i > 0 is necessary:
discrete probs.
buyer
0
3
1 1/ 4 1/ 4
seller
4 1/ 4 1/ 4
Only profitable transaction is 13
week 12
26
Bilateral trading mechanisms
Claim: “sell at price 2 if both are willing, else
no trade” is incentive compatible,
individually rational, and efficient.
•
•
•
week 12
Incentive compatible: truthful reporting is
an equilibrium (check)
Individually rational: E[profit] >0
Efficient: trade occurs only when v1<v2
27
Auctions vs. Negotiations
[ Bulow & Klemperer 96 ]
Simple example: IPV, uniform
Case 1) Optimal auction = optimal mechanism
with one buyer. Optimal entry value v* = 1/2;
revenue = 1/4
Case 2) Two buyers, no reserve; revenue = 1/3
> 1/4
 One more buyer is worth more than setting
reserve optimally!
week 12
28
Auctions vs. Negotiations
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 (IPV)
auction (and hence optimal mechanism)
with n bidders
week 12
29
Auctions vs. Negotiations
Revenue with optimal reserve, n bidders:

1
v*
M (v) dF (v)
n

1
1/ 2
(2 x  1)dx  1 / 4
Revenue with no reserve, n+1 bidders:

1
0
M (v) dF (v) n1

1
0
week 12
(2 x  1) dx 2  1 / 3
30
Auctions vs. Negotiations
Facts:

1
v*

M (v) dF (v)  E[max{ M (v1 ),..., M (vn ), 0}]
1
0
n
M (v) dF (v) n1  E[max{ M (v1 ) ,..., M (vn1 )}]
E[ M (v)]  0
week 12
Why?
Why?
Why?
31
Auctions vs. Negotiations
Facts:
1

v*
M (v) dF (v) n  E[max{ M (v1 ),..., M (vn ), 0}]
distribution fctn. of max. of n draws, integrate only where M ≥ 0
1

0
M (v) dF (v)n1  E[max{ M (v1 ),..., M (vn1 )}]
distribution fctn. of max. of n+1 draws
E[ M (v)]  0
week 12
expected revenue with only one buyer!
32
Auctions vs. Negotiations
Now compare revenue in a no-reserve
auction with n+1 bidders, and an optimal
auction with n bidders:
  E[no reserve , n  1]  E[optimal , n]
1
  M dF
0
n 1
1
  M dF n
v*
 E[max{ M (v1 ),..., M (vn ), M (vn1 )}]
 E[max{ M (v1 ),..., M (vn ), 0 }]
0
week 12
□
33
“Rational frenzies and crashes,” J. Bulow & P.
Klemperer, J. Political Economy, 102, pp. 1-23, 1994.
• Asset markets are volatile! Common wisdom
attributes to irrational behavior, market
imperfections, market failure
• This paper offers a model of a simple situation in
which completely rational behavior leads to
“frenzies” and “crashes”
• Uses IPV auction theory and the RET in a
dynamic setting
• An elegant economic idealization makes the point
week 12
34
The BK 94 game
• K identical units for sale, one seller, K+L riskneutral potential buyers, each wanting to buy a
single unit
• IPV’s, drawn from F(v) on [0, vmax]
• Buyer derives surplus (v – p) from a purchase at
price p
week 12
35
The simple motivating idea… WTP
• Suppose you’re in a
simple single-item
Vickrey auction with IPV’s
that are uniform on [0,1],
and you have value v.
• You are made a take-itor-leave-it offer at price p.
Should you accept it?
week 12
36
The simple motivating idea… WTP
• Suppose you’re in a
simple single-item
Vickrey auction with IPV’s
that are uniform on [0,1],
and you have value v.
• You are made a take-itor-leave-it offer at price p.
Should you accept it?
• Well, if and only if p ≤
E(second-price | v wins)
= “Willingness To Pay”
week 12
37
Dynamics of BK 94 game
1)
2)
Seller begins offering units at max price vmax and lowers it
until a purchase occurs, at price p
(NEW SALE) When a purchase occurs, every buyer gets
an invitation to purchase 1 unit at price p.
Either:
(a) (FRENZY) all goods are sold at p  game ends
(b) (FRENZY) not all goods are sold at p, no one is left
willing to buy at that price  then go to 1) and continue
lowering price until another NEW SALE takes place
(c) (EXCESS DEMAND) More buyers want to buy at price p
than there are units remaining. Then if there are k+l bidders
offering to buy the remaining k units, go to 1) and restart the
game with these k+l bidders competing for the remaining k
units. All previous sales remain valid.
week 12
38
Solution to game
• With k units and (k+l) bidders remaining: a
symmetric equilibrium strategy is: offer to buy at
price p if and only if p ≤ ω(v), where
 (v)  E[price in a first - rejected price auction | win ]
 E[( k  1) st highest out of (k  l ) values | (k  1) st  v]
 " willingess to pay"
Note that this reduces to Vickrey with one item
and one buyer
week 12
39
• This follows from a straightforward generalization
of the RET: any mechanism selling K identical
items to the bidders with the K highest values in a
unit-demand auction has in equilibrium the same
expected payment conditional on winning, namely
ω(v) (see B&K 94).
• The interesting dynamics are a consequence of the
shape of ω(v). When k goes down, ω(v) goes up,
and this changes the next threshold drastically and
a bunch of buyers may jump in all at once!
week 12
40
Why does ω(v) flatten out dramatically?
week 12
41
week 12
42
Simulation
K=50 , L=100
week 12
43
Term papers due 5pm Tuesday May 12
(Dean’s Date)
 Email me for office hours re term papers
It’s been fun!
Neshmet Bark of Osiris, on a
bronze drachm of M. Aurelius,
Alexandria, Egypt. E. 2160,
174/5 AD.
week 12
44