On Cheating in
Sealed-Bid Auctions
Ryan Porter
Yoav Shoham
Computer Science Department
Stanford University
Introduction
 Sealed-bid auctions require privacy of the bids
 New security problems online
 How should bidders behave when they are aware of the
possibility of cheating?
 Answer provides insights to auctions without cheating
June 11, 2003
On Cheating in Sealed-Bid Auctions
2
Cheating in Auctions
 After the auction:
 Individual cheating (by seller or winning bidder)
 During the auction:
 Collusion
 Individual cheating
 Seller inserting false bids
 Agents observing competing bids before submitting their own
June 11, 2003
On Cheating in Sealed-Bid Auctions
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Outline
First-Price
Auction
Second-Price
Auction
Seller Cheating Possible
June 11, 2003
Agent Cheating Possible
On Cheating in Sealed-Bid Auctions
4
Outline
First-Price
Auction
No effect on price
Second-Price
Auction
Seller Cheating Possible
June 11, 2003
Agent Cheating Possible
On Cheating in Sealed-Bid Auctions
5
Outline
First-Price
Auction
No effect on price
Truthful bidding a dominant
strategy
Second-Price
Auction
Seller Cheating Possible
June 11, 2003
Agent Cheating Possible
On Cheating in Sealed-Bid Auctions
6
Outline
First-Price
Auction
Second-Price
Auction
No effect on price
Equilibrium
bidding strategy
Continuum of auctions
Seller Cheating Possible
June 11, 2003
Truthful bidding a dominant
strategy
Agent Cheating Possible
On Cheating in Sealed-Bid Auctions
7
Outline
Uniform
First-Price
Auction
Equilibrium
bidding strategy
Cheating as overbidding:
No effect on price
Extension
to first-price
auctions without cheating
Other

Second-Price
Auction
Equilibrium
bidding strategy
Continuum of auctions
Seller Cheating Possible
June 11, 2003
Distribution:
Distributions:
Effects of overbidding
Truthful bidding a dominant
strategy
Agent Cheating Possible
On Cheating in Sealed-Bid Auctions
8
General Formulation
 Single good, owned by a seller
 No reserve price
 N bidders (agents), each characterized by a
privately-known valuation (type) i 2 [0,1]
 Each i is independently drawn from cdf F(i):
 Strictly increasing and differentiable
 Commonly-known
 Let θ = (θ1,…,θN)
 Let θ-i = (θ1,…,θi-1,θi+1,…,θN)
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On Cheating in Sealed-Bid Auctions
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General Formulation
 Bidding strategy: bi: [0,1] ! [0,1]
 Agent utility function:
ui(bi(i),b-i(-i),i) = І(bi(i) > b[1](-i)) ¢ (i – p(bi(i),b-i(-i))
 All agents are assumed to be rational, expected-utility maximizers
 Expected utility: E-i[ui(bi(i),b-i(-i),i)]
 biR(i) is a best response to b-i(-i) if 8 bi'(i):
E-i[ui(biR(i),b-i(-i),i)] ¸ E-i[ui(bi'(i),b-i(-i),i)]
 Solution concept is Bayes-Nash equilibrium (BNE)
 bi*(i) is a symmetric BNE if 8 bi'(i):
E-i[ui(bi*(i),b-i*(-i),i)] ¸ E-i[ui(bi'(i),b-i*(-i),i)]
June 11, 2003
On Cheating in Sealed-Bid Auctions
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Equilibria for Sealed-Bid Auctions
 Sealed-bid auctions without the possibility of cheating:
 First-Price Auction:
 Unspecified F(i):
 F(i) = i (Uniform distribution):
 Second-Price Auction:
June 11, 2003
On Cheating in Sealed-Bid Auctions
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Outline
Uniform
First-Price
Auction
Equilibrium
bidding strategy
Cheating as overbidding:
No effect on price
Extension
to first-price
auctions without cheating
Other

Second-Price
Auction
Equilibrium
bidding strategy
Continuum of auctions
Seller Cheating Possible
June 11, 2003
Distribution:
Distributions:
Effects of overbidding
Truthful bidding a dominant
strategy
Agent Cheating Possible
On Cheating in Sealed-Bid Auctions
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Second-Price Auction, Cheating Seller
 Payment of highest bidder:
 second-highest bid if seller does not cheat
 bi(i) if the seller cheats
(assumes cheating seller uses full power)
 Pc – probability with which the seller will cheat
 commonly-known
 Interpretation as a probabilistic sealed-bid auction:
 payment rule (determined when auction clears):
 first-price with probability Pc
 second-price with probability (1-Pc)
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On Cheating in Sealed-Bid Auctions
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Equilibrium
 Unspecified F(i):
 F(i) = i (uniform distribution):
June 11, 2003
On Cheating in Sealed-Bid Auctions
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Outline
Uniform
First-Price
Auction
Equilibrium
bidding strategy
Cheating as overbidding:
No effect on price
Extension
to first-price
auctions without cheating
Other

Second-Price
Auction
Equilibrium
bidding strategy
Continuum of auctions
Seller Cheating Possible
June 11, 2003
Distribution:
Distributions:
Effects of overbidding
Truthful bidding a dominant
strategy
Agent Cheating Possible
On Cheating in Sealed-Bid Auctions
15
Revised Formulation
 Single cheating agent j will bid up to j
 Several cheating agents:
 One possibility is an English auction among the cheaters
 Suffices to know that, from an honest agent’s point of view,
in order to win:
 bi(i) > bj(j) for all honest agents j  i
 bi(i) > j for all cheating agents j
 Let Pa be the probability that an agent cheats
 commonly-known
 Discriminatory, probabilistic sealed-bid auction:
 Payment rule (determined before bidding):
 second-price with probability Pa
 first-price with probability (1-Pa)
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On Cheating in Sealed-Bid Auctions
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Equilibrium
 Cheaters will bid their dominant strategy bi*(i) = i
 What is bi*(i) for the honest agents?
 Unspecified F(i): fixed point equation
 F(i) = i (uniform distribution):
 For a first-price auction without cheating,
is the optimal tradeoff between increasing probability of
winning and increasing profit conditional on winning
 Cheating agents decrease probability of winning
 Natural to expect that an honest should compensate by
increasing his bid
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On Cheating in Sealed-Bid Auctions
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Robustness of Equilibrium
 Thm: In a first-price auction in which agents cheat with
probability Pa, and F(i) = i, the BNE bidding strategy for
honest agents is:
 Thm: In a first-price auction without cheating where F(i) =
i in which each agent j  i bids according to:
best response is:
 Support for Bayes-Nash equilibrium
 However, if 9 j j < 0, then:
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On Cheating in Sealed-Bid Auctions
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Effect of Overbidding: Other Distributions
 Let biR(i) be the best response to bj(j) = j, 8 j  i
 For
, where k ¸ 1:
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0
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0.2
0.4
0.6
0.8
1
0
0.2
0.4
On Cheating in Sealed-Bid Auctions
0.6
0.8
1
19
Effect of Overbidding: Other Distributions
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0
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0.2
0.4
0.6
0.8
1
0
0.2
On Cheating in Sealed-Bid Auctions
0.4
0.6
0.8
1
20
Effect of Overbidding: Other Distributions
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
(satisfies F''(i) = -1)
June 11, 2003
On Cheating in Sealed-Bid Auctions
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Predicting Direction of Change
(
)''
Direction of
change
1
=
–
–
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
1
0.8
0.6
0.4
0.2
0
1
0.8
0.6
0.4
0.2
0
1
+
0.8
0.6
0.4
0.2
0
1
+
0.8
0.6
0.4
0.2
0
June 11, 2003
On Cheating in Sealed-Bid Auctions
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Revenue Loss for Honest Seller
 Occurs in both settings due to the possibility of cheating
 bi*(i) allows us to quantify the expected loss
 This analysis could be applied to more general settings:
 Seller could pay to improve security
 Multiple sellers and multiple markets
 Relates to “market for lemons”
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On Cheating in Sealed-Bid Auctions
23
Conclusion
 We considered two settings in which cheating may occur in a
sealed-bid auction due to a lack bid privacy:
 In both cases, we presented equilibrium bidding strategies
 Second-price auction, cheating seller:
 Related first and second-price auctions without cheating (and their
equilibria) as endpoints of a continuum
 First-price auction, cheating agents:
 Counterintuitive results on the effects of overbidding
 Preliminary results on characterizing the direction of the effect
June 11, 2003
On Cheating in Sealed-Bid Auctions
24
On Cheating in
Sealed-Bid Auctions
Ryan Porter
Yoav Shoham
Computer Science Department
Stanford University