Envy-Free Auctions for Digital goods A paper by Andrew V. Goldberg and Jason D. Hartline Presented by Bart J. Buter, Paul Koppen and Sjoerd W. Kerkstra.

Envy-Free Auctions for Digital goods

A paper by Andrew V. Goldberg and Jason D. Hartline

Presented by Bart J. Buter , Paul Koppen and Sjoerd W. Kerkstra

Three desirable properties for auctions

Truthful

Competitive

Envy-free

A truthful auction

Truthful = bid-independent

Competitive

Envy-free

A competetive auction

Truthful = bid-independent

Competitive = constant fraction of optimal revenue

Envy-free

An envy-free auction

Truthful = bid-independent

Competitive = constant fraction of optimal revenue

Envy-free = no envy among bidders after auction

Three desirable properties for auctions

Truthful = bid-independent

Competitive = constant fraction of optimal revenue

Envy-free = no envy among bidders after auction

Main result

Truthful = bid-independent

Competitive = constant fraction of optimal revenue

Envy-free = no envy among bidders after auction

No auction can have all three properties

A solution

Truthful = bid-independent

Competitive = constant fraction of optimal revenue

Envy-free = no envy among bidders after auction

Relax one of the three properties

Why

● Envy free for consumer acceptance● Truthful for no sabotage● Competitive guarantees profit minimum bound

for auctioneer

A truthful, envy-free auctioncompetitive ratio: O(log n)

Definition 5

● Optimal single price omniscient auction:

F(b) = maxk kvk

Vector of all submitted bids

i-th component, bi, is bidsubmitted by bidder i.

vi is the i-th largest bidin the vector b(for the max, vk is the finalprice that each winner pays)

Number of winners

Before continuing…

● Two important variables:

n = number of bidders

m = number of winners in optimal auction

Definition 6

● β(m)-competitive for mass-markets

E[A(b)] ≥ F(b) / β(m)

Expectation overrandomized choicesof the auction

Our auction

Optimal auction

Number of winners

Competitive ratioDesired:

• low constant β(2) and• limm→∞ β(m) = 1

Theorem 4

● Truthful auction that is Θ(log n)-competitive

E[R] = ( v / log n ) Σi=0[log m]–1 2i

Expected revenuefor worst-case

Lowest bid > 0

Average over alllog n different auctions

Special auction: i picked random from [0,…,[log n]], then run 2i-Vickrey auction

NB revenue for 2i-Vickrey auction ≈ 2iv if 2i < m and 0 otherwise

Sum all revenues that satisfy2i < mthusi < log m

Theorem 4

● Truthful auction that is Θ(log n)-competitive

( v / log n ) Σi=0[log m]–1 2i = ( v / log n ) 2[log m] – 1

Math

Special auction: i picked random from [0,…,[log n]], then run 2i-Vickrey auction

NB revenue for 2i-Vickrey auction ≈ 2iv if 2i < m and 0 otherwise

Theorem 4

● Truthful auction that is Θ(log n)-competitive

( v / log n ) 2[log m] – 1 ≥ ( v / log n ) ( m – 1 )

Putting a lower bound on the expected revenuefor this specific log-competitive, truthful, envy-free auction

Special auction: i picked random from [0,…,[log n]], then run 2i-Vickrey auction

NB revenue for 2i-Vickrey auction ≈ 2iv if 2i < m and 0 otherwise

Theorem 4

● Truthful auction that is Θ(log n)-competitive

( v / log n ) ( m – 1 ) ≥ F(b) ( m – 1 ) / ( m log n )

Remember the optimal auctionF(b) = maxk kvkSo hereF(b) = mv

Special auction: i picked random from [0,…,[log n]], then run 2i-Vickrey auction

NB revenue for 2i-Vickrey auction ≈ 2iv if 2i < m and 0 otherwise

Theorem 4

● Truthful auction that is Θ(log n)-competitive

E[R] ≥ F(b) ( m – 1 ) / ( m log n ) ≥ F(b) / ( 2 log n )

● By definition 6E[A(b)] ≥ F(b) / β(m)

we have provenβ(m) є Θ(log n)

Number of biddersNumber of winnersin optimal auction

Optimal auction

Vector of allsubmitted bids

Competitive ratio

Theorem 4

● Log n is increasing and competitive ratio shall be non-increasing

● so search for better auction by relaxing envy-free or truthful property

CostShare

● Predefined revenue R● Find largest k such that highest k bidders can

equally share cost R● Price is R/k● No k exists No bidders win

CostShare

● Truthful● Profit R if R ≤ F (or no winners)● Envy-free

● Because it cannot guarantee winners, it is not competitive

CORE

● COnsensus Revenue Estimate

● Price extractor ( = CostShare )● Consensus Estimate

– Defines R to be bid-independent– Bounding variables are introduced to be competitive

again– At the cost of very small chance for no envy-

freeness or (ultimately) no truthfulness

Auctions for real

Current auction research applied

● Frequency auctions – Radio– Mobile phones

● Advertisements– Google– MSN

● Auction sites– Ebay– Amazon

Frequency auctions

• New Zealand Frequency auction– equal lots– simultanious Vickrey

auctions– extreme cases

Milgrom. Putting Auction Theory to Work, Cambridge University Press, 2004. ISBN: 0521536723

Outcomes New Zealand

• Extreme outcomes

• Not FraudulentHigh 2nd High

NZ $100.000 NZ $6

NZ $7.000.000 NZ $5.000

Lessons New Zealand

● Vickrey does not work well– With few bidders– When goods are substitutes

● Think about details

Ebay and Amazon

● Manual bidding● Sniping (placing bid at latest possible time)

– Pseudo collusion● Proxy bidding (place maximum valuation)

Roth, Ockenfels. Late and multiple bidding in second price Internet auctions:

Theory and evidence concerning different rules for ending an auction. Games and Economic Behavior, 55, (2006), 297–320

Auctioneer strategies

● Both English auctions (going, going, gone)● Amazon auction ends after deadline & no bids

for 10 minutes● Ebay auction ends after deadline

Results for bidders

● Nash Amazon = Everybody proxy bidding● Nash Ebay = Everybody proxy or everybody

sniping