On Cheating in Sealed-Bid Auctions

On Cheating in Sealed-Bid Auctions

Ryan Porter Yoav Shoham

Computer Science DepartmentStanford University

June 11, 2003 On Cheating in Sealed-Bid Auctions 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


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


Outline

First-Price Auction

Second-Price Auction

Seller Cheating Possible Agent Cheating Possible


Outline

First-Price Auction No effect on price




Outline



Truthful bidding a dominant strategy



Outline



Equilibrium bidding strategyContinuum of auctions




Outline


Uniform Distribution:Equilibrium bidding strategyCheating as overbidding:

Extension to first-price auctions without cheating

Other Distributions: Effects of overbidding






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)


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)]


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:


Outline










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)


Equilibrium

Unspecified F(i):

F(i) = i (uniform distribution):


Outline










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)


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


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:


Effect of Overbidding: Other Distributions

Let biR(i) be the best response to bj(j) = j, 8 j i

For , where k ¸ 1:

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1



0

0.2

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0 0.2 0.4 0.6 0.8 10

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1

0 0.2 0.4 0.6 0.8 1



0

0.2

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0 0.2 0.4 0.6 0.8 10

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0 0.2 0.4 0.6 0.8 1

(satisfies F''(i) = -1)


Predicting Direction of Change

Direction of change

( )''=–

++

–

0

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0 0.2 0.4 0.6 0.8 1

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0 0.2 0.4 0.6 0.8 1

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0 0.2 0.4 0.6 0.8 1


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”


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

On Cheating in Sealed-Bid Auctions

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