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

2. 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

3. 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

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

5. 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

6. 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

7. 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

8. 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

9. 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 )

10. 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 )]

11. 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:

12. 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

13. 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 )

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

15. 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

16. 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 )

17. 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

18. 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:

19. 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:

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

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

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

23. 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”

24. 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

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