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Mergers and Collusion in All-Pay
Auctions and Crowdsourcing
Contests Omer Lev, Maria Polukarov, Yoram
Bachrach& Jeffrey S. Rosenschein
AAMAS 2013St. Paul, Minnesota
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All-pay auctions
Bidders bid and pay their bid to the auctioneer
Auction winner is one which submitted the highest bid
Perl
imin
arie
s
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Why all-pay auctions?Explicit all-pay auctions are rare, but implicit ones are extremely common:
Competition for patents between firms
Crowdsourcing competitions (e.g., Netflix challenge, TopCoder, etc.)
Hiring employees
Employee competition (“employee of the month”)
Perl
imin
arie
s
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Auctioneer types
“sum profit”Gets the bids from all bidders – regardless of their winning statusE.g., “emloyee of the month”
“max profit”Gets only the winner’s bid. Other bids are, effectively, “burned”E.g., hiring an emplyee
Perl
imin
arie
s
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All-pay auction equilibriumAll bidders give the object in question a value of 1
A single symmetric equilibrium – for n bidders:
Baye, Kovenock, de Vries
Reg
ular
all-
pay
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All-pay auction equilibrium bidder properties
0Expected utility:
Utility variance:
Expected bid:
Bid variance:
Baye, Kovenock, de Vries
Reg
ular
all-
pay
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All-pay auction equilibrium auctioneer properties
1Sum profit expected profit:Sum profit profit variance:Max profit expected profit:Max profit profit variance: Baye, Kovenock, de Vries
Reg
ular
all-
pay
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Example no collusion case
3 bidders
Bidders’ c.d.f is and the expected bid is ⅓, with variance of . Expected profit is 0 with variance of . Sum profit auctioneer has expected profit of 1 with variance of .
Max profit auctioneer has expected profit of ⅗ with variance of .
Baye, Kovenock, de Vries
Reg
ular
all-
pay
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Mergers
k bidders (out of the total n) collaborate, having a joint strategy. All other bidders are aware of this.
Mer
gers
(c
olla
bora
tion
pub
lic
know
ledg
e)
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Merger propertiesEquilibrium remains the same – but with smaller n
BidderExpected Utility: 0
Utility variance:Expected bid:
Bid variance:
Sum ProfitExpected profit: 1
Profit variance:
Max ProfitExpected profit:Profit variance:
Mer
gers
(c
olla
bora
tion
pub
lic
know
ledg
e)
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Example no collusion case
3 bidders
Bidders’ c.d.f is and the expected bid is ⅓, with variance of . Expected profit is 0 with variance of . Sum profit auctioneer has expected profit of 1 with variance of .
Max profit auctioneer has expected profit of ⅗ with variance of .M
erge
rs
(col
labo
rati
on p
ublic
kn
owle
dge)
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Example merger case
3 bidders, 2 of them merged
Bidders’ c.d.f is uniform, and the expected bid is ½, with variance of . Expected profit is 0 with variance of ⅙.
Sum profit auctioneer has expected profit of 1 with variance of ⅙.
Max profit auctioneer has expected profit of ⅔ with variance of .M
erge
rs
(col
labo
rati
on p
ublic
kn
owle
dge)
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Collusions
k bidders (out of the total n) collaborate, having a joint strategy. Other bidders are not aware of this and continue to pursue their previous strategies.
Col
lusi
on(c
olla
bora
tion
pri
vate
kno
wle
dge)
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Collusion colludersColluders have a pure, optimal strategy
Producing an expected profit of:
Profit variance:
k: n:
k: n:
Colluders’ profit per colluder increases as number of colluders growsC
ollu
sion
(col
labo
rati
on p
riva
te k
now
ledg
e)
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Collusion auctioneers
Sum profit:
Max profit:
k: n:
k: n:
For large enough n exceed non-colluding profitsCol
lusi
on(c
olla
bora
tion
pri
vate
kno
wle
dge)
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Collusion non-colluding bidders
For large enough k (e.g., ) this expression is positive.I.e., non-colluders profit from collusion
Utility for non-colluding bidders is:
If a non-colluder discovers the collusion, best to bid a bit above colludersC
ollu
sion
(col
labo
rati
on p
riva
te k
now
ledg
e)
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Example no collusion case
3 bidders
Bidders’ c.d.f is and the expected bid is ⅓, with variance of . Expected profit is 0 with variance of . Sum profit auctioneer has expected profit of 1 with variance of .
Max profit auctioneer has expected profit of ⅗ with variance of .
Col
lusi
on(c
olla
bora
tion
pri
vate
kno
wle
dge)
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Example merger case
3 bidders, 2 of them merged
Bidders’ c.d.f is uniform, and the expected bid is ½, with variance of . Expected profit is 0 with variance of ⅙.
Sum profit auctioneer has expected profit of 1 with variance of ⅙.
Max profit auctioneer has expected profit of ⅔ with variance of .C
ollu
sion
(col
labo
rati
on p
riva
te k
now
ledg
e)
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Example collusion case
3 bidders, 2 of them collude
One bidder has c.d.f of (expected bid of ⅓), colluders bid ¼. Colluders’ expected profit is ¼, while the non-colluder expected profit is ⅙.
Sum profit auctioneer expected profit only .
Max profit auctioneer has expected profit of .C
ollu
sion
(col
labo
rati
on p
riva
te k
now
ledg
e)
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Future directionsAdding bidders’ skills to model
Detecting collusions by other bidders
Designing crowdsourcing mechanisms less susceptible to collusion
Adding probability to win based on effort
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The End
Thanks for listening!