On Cheating in Sealed-Bid Auctions
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Transcript of On Cheating in Sealed-Bid Auctions
On Cheating in Sealed-Bid Auctions
Ryan Porter Yoav Shoham
Computer Science DepartmentStanford University
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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
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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
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Outline
First-Price Auction
Second-Price Auction
Seller Cheating Possible Agent Cheating Possible
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Outline
First-Price Auction No effect on price
Second-Price Auction
Seller Cheating Possible Agent Cheating Possible
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Outline
First-Price Auction No effect on price
Second-Price Auction
Truthful bidding a dominant strategy
Seller Cheating Possible Agent Cheating Possible
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Outline
First-Price Auction No effect on price
Second-Price Auction
Equilibrium bidding strategyContinuum of auctions
Truthful bidding a dominant strategy
Seller Cheating Possible Agent Cheating Possible
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Outline
First-Price Auction No effect on price
Uniform Distribution:Equilibrium bidding strategyCheating as overbidding:
Extension to first-price auctions without cheating
Other Distributions: Effects of overbidding
Second-Price Auction
Equilibrium bidding strategyContinuum of auctions
Truthful bidding a dominant strategy
Seller Cheating Possible Agent Cheating Possible
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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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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(bi
R(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)]
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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:
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Outline
First-Price Auction No effect on price
Uniform Distribution:Equilibrium bidding strategyCheating as overbidding:
Extension to first-price auctions without cheating
Other Distributions: Effects of overbidding
Second-Price Auction
Equilibrium bidding strategyContinuum of auctions
Truthful bidding a dominant strategy
Seller Cheating Possible Agent Cheating Possible
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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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Equilibrium
Unspecified F(i):
F(i) = i (uniform distribution):
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Outline
First-Price Auction No effect on price
Uniform Distribution:Equilibrium bidding strategyCheating as overbidding:
Extension to first-price auctions without cheating
Other Distributions: Effects of overbidding
Second-Price Auction
Equilibrium bidding strategyContinuum of auctions
Truthful bidding a dominant strategy
Seller Cheating Possible Agent Cheating Possible
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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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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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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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Effect of Overbidding: Other Distributions
Let biR(i) be the best response to bj(j) = j, 8 j i
For , where k ¸ 1:
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Effect of Overbidding: Other Distributions
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Effect of Overbidding: Other Distributions
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Predicting Direction of Change
Direction of change
( )''=–
++
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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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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
On Cheating in Sealed-Bid Auctions
Ryan Porter Yoav Shoham
Computer Science DepartmentStanford University