envy-free auctions for digital goods
DESCRIPTION
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. Truthful Competitive Envy-free. Three desirable properties for auctions. Truthful = bid-independent Competitive Envy-free. - PowerPoint PPT PresentationTRANSCRIPT
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