On Cheating in Sealed-Bid Auctions

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

2. 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 On Cheating in Sealed-Bid Auctions

3. 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 On Cheating in Sealed-Bid Auctions

4. Outline On Cheating in Sealed-Bid Auctions

5. Outline On Cheating in Sealed-Bid Auctions

6. Outline On Cheating in Sealed-Bid Auctions

7. Outline On Cheating in Sealed-Bid Auctions

8. Outline On Cheating in Sealed-Bid Auctions

9. General Formulation • Single good, owned by a seller • No reserve price • N bidders (agents), each characterized by a privately-known valuation (type) i2 [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) On Cheating in Sealed-Bid Auctions

10. 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)] On Cheating in Sealed-Bid Auctions

11. 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: On Cheating in Sealed-Bid Auctions

12. Outline On Cheating in Sealed-Bid Auctions

13. 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) On Cheating in Sealed-Bid Auctions

14. Equilibrium • Unspecified F(i): • F(i) = i (uniform distribution): On Cheating in Sealed-Bid Auctions

15. Outline On Cheating in Sealed-Bid Auctions

16. 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) On Cheating in Sealed-Bid Auctions

17. 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 On Cheating in Sealed-Bid Auctions

18. 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: On Cheating in Sealed-Bid Auctions

19. Effect of Overbidding: Other Distributions • Let biR(i) be the best response to bj(j) = j, 8 j  i • For , where k ¸ 1: On Cheating in Sealed-Bid Auctions

20. Effect of Overbidding: Other Distributions On Cheating in Sealed-Bid Auctions

21. Effect of Overbidding: Other Distributions (satisfies F''(i) = -1) On Cheating in Sealed-Bid Auctions

22. Predicting Direction of Change ( )'' = – – + + On Cheating in Sealed-Bid Auctions

23. 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” On Cheating in Sealed-Bid Auctions

24. 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

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