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On Cheating in Sealed-Bid Auctions Ryan Porter Yoav Shoham Computer Science Department Stanford University

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June 11, 2003 On Cheating in Sealed-Bid Auctions2 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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June 11, 2003 On Cheating in Sealed-Bid Auctions3 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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June 11, 2003 On Cheating in Sealed-Bid Auctions4 Outline First-Price Auction Second-Price Auction Seller Cheating PossibleAgent Cheating Possible

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June 11, 2003 On Cheating in Sealed-Bid Auctions5 Outline First-Price Auction No effect on price Second-Price Auction Seller Cheating PossibleAgent Cheating Possible

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June 11, 2003 On Cheating in Sealed-Bid Auctions6 Outline First-Price Auction No effect on price Second-Price Auction Truthful bidding a dominant strategy Seller Cheating PossibleAgent Cheating Possible

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June 11, 2003 On Cheating in Sealed-Bid Auctions7 Outline First-Price Auction No effect on price Second-Price Auction Equilibrium bidding strategy Continuum of auctions Truthful bidding a dominant strategy Seller Cheating PossibleAgent Cheating Possible

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June 11, 2003 On Cheating in Sealed-Bid Auctions8 Outline First-Price Auction No effect on price Uniform Distribution: Equilibrium bidding strategy Cheating as overbidding: Extension to first-price auctions without cheating Other Distributions: Effects of overbidding Second-Price Auction Equilibrium bidding strategy Continuum of auctions Truthful bidding a dominant strategy Seller Cheating PossibleAgent Cheating Possible

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June 11, 2003 On Cheating in Sealed-Bid Auctions9 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 = (θ 1,…,θ N ) Let θ -i = (θ 1,…,θ i-1,θ i+1,…,θ N )

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June 11, 2003 On Cheating in Sealed-Bid Auctions10 General Formulation Bidding strategy: b i : [0,1] ! [0,1] Agent utility function: u i (b i ( i ),b -i ( -i ), i ) = І ( b i ( i ) > b [1] ( -i ) ) ¢ ( i – p(b i ( i ),b -i ( -i ) ) All agents are assumed to be rational, expected-utility maximizers Expected utility: E -i [u i (b i ( i ),b -i ( -i ), i )] b i R ( i ) is a best response to b -i ( -i ) if 8 b i '( i ): E -i [u i (b i R ( i ),b -i ( -i ), i )] ¸ E -i [u i (b i '( i ),b -i ( -i ), i )] Solution concept is Bayes-Nash equilibrium (BNE) b i * ( i ) is a symmetric BNE if 8 b i '( i ): E -i [u i (b i * ( i ),b -i * ( -i ), i )] ¸ E -i [u i (b i '( i ),b -i * ( -i ), i )]

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June 11, 2003 On Cheating in Sealed-Bid Auctions11 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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June 11, 2003 On Cheating in Sealed-Bid Auctions12 Outline First-Price Auction No effect on price Uniform Distribution: Equilibrium bidding strategy Cheating as overbidding: Extension to first-price auctions without cheating Other Distributions: Effects of overbidding Second-Price Auction Equilibrium bidding strategy Continuum of auctions Truthful bidding a dominant strategy Seller Cheating PossibleAgent Cheating Possible

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June 11, 2003 On Cheating in Sealed-Bid Auctions13 Second-Price Auction, Cheating Seller Payment of highest bidder: second-highest bid if seller does not cheat b i ( i ) if the seller cheats (assumes cheating seller uses full power) P c – 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 P c second-price with probability (1-P c )

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June 11, 2003 On Cheating in Sealed-Bid Auctions14 Equilibrium Unspecified F( i ): F( i ) = i (uniform distribution):

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June 11, 2003 On Cheating in Sealed-Bid Auctions15 Outline First-Price Auction No effect on price Uniform Distribution: Equilibrium bidding strategy Cheating as overbidding: Extension to first-price auctions without cheating Other Distributions: Effects of overbidding Second-Price Auction Equilibrium bidding strategy Continuum of auctions Truthful bidding a dominant strategy Seller Cheating PossibleAgent Cheating Possible

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June 11, 2003 On Cheating in Sealed-Bid Auctions16 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: b i ( i ) > b j ( j ) for all honest agents j i b i ( i ) > j for all cheating agents j Let P a be the probability that an agent cheats commonly-known Discriminatory, probabilistic sealed-bid auction: Payment rule (determined before bidding): second-price with probability P a first-price with probability (1-P a )

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June 11, 2003 On Cheating in Sealed-Bid Auctions17 Equilibrium Cheaters will bid their dominant strategy b i * ( i ) = i What is b i * ( 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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June 11, 2003 On Cheating in Sealed-Bid Auctions18 Robustness of Equilibrium Thm: In a first-price auction in which agents cheat with probability P a, 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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June 11, 2003 On Cheating in Sealed-Bid Auctions19 Effect of Overbidding: Other Distributions Let b i R ( i ) be the best response to b j ( j ) = j, 8 j i For, where k ¸ 1:

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June 11, 2003 On Cheating in Sealed-Bid Auctions20 Effect of Overbidding: Other Distributions

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June 11, 2003 On Cheating in Sealed-Bid Auctions21 Effect of Overbidding: Other Distributions (satisfies F''( i ) = -1)

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June 11, 2003 On Cheating in Sealed-Bid Auctions22 Predicting Direction of Change Direction of change ()'' = – + + –

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June 11, 2003 On Cheating in Sealed-Bid Auctions23 Revenue Loss for Honest Seller Occurs in both settings due to the possibility of cheating b i * ( 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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June 11, 2003 On Cheating in Sealed-Bid Auctions24 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

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On Cheating in Sealed-Bid Auctions Ryan Porter Yoav Shoham Computer Science Department Stanford University

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