Transcript Envy-Free Auctions for Digital goods

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
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Envy free for consumer acceptance
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Truthful for no sabotage
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Competitive guarantees profit minimum bound
for auctioneer
A truthful, envy-free auction
competitive ratio: O(log n)
Definition 5
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Optimal single price omniscient auction:
F(b) = maxk kvk
Vector of all submitted bids
i-th component, bi, is bid
submitted by bidder i.
Number of winners
vi is the i-th largest bid
in the vector b
(for the max, vk is the final
price that each winner pays)
Before continuing…
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Two important variables:
n = number of bidders
m = number of winners in optimal auction
Definition 6
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β(m)-competitive for mass-markets
E[A(b)] ≥ F(b) / β(m)
Expectation over
randomized choices
of the auction
Our auction
Number of winners
Optimal auction
Competitive ratio
Desired:
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low constant β(2) and
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limm→∞ β(m) = 1
Theorem 4
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Truthful auction that is Θ(log n)-competitive
E[R] = ( v / log n ) Σi=0[log m]–1 2i
Expected revenue
for worst-case
Lowest bid > 0
Average over all
log n different auctions
Sum all revenues that satisfy
2i < m
thus
i < log m
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
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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
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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 revenue
for 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
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Truthful auction that is Θ(log n)-competitive
( v / log n ) ( m – 1 ) ≥ F(b) ( m – 1 ) / ( m log n )
Remember the optimal auction
F(b) = maxk kvk
So here
F(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
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Truthful auction that is Θ(log n)-competitive
E[R] ≥ F(b) ( m – 1 ) / ( m log n ) ≥ F(b) / ( 2 log n )
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By definition 6
E[A(b)] ≥ F(b) / β(m)
we have proven
β(m) є Θ(log n)
Competitive ratio
Vector of all
submitted bids
Optimal auction
Number of winners
in optimal auction
Number of bidders
Theorem 4
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Log n is increasing and competitive ratio shall
be non-increasing
so search for better auction by relaxing envyfree or truthful property
CostShare
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Predefined revenue R
Find largest k such that highest k bidders can
equally share cost R
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Price is R/k
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No k exists  No bidders win
CostShare
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Truthful
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Profit R if R ≤ F (or no winners)
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Envy-free
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Because it cannot guarantee winners, it is not
competitive
CORE
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COnsensus Revenue Estimate
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Price extractor ( = CostShare )
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Consensus Estimate
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Defines R to be bid-independent
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Bounding variables are introduced to be competitive
again
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At the cost of very small chance for no envyfreeness or (ultimately) no truthfulness
Auctions for real
Current auction research applied
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Frequency auctions
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Advertisements
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Radio
Mobile phones
Google
MSN
Auction sites
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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 Fraudulent
High
2nd High
NZ $100.000
NZ $6
NZ $7.000.000
NZ $5.000
Lessons New Zealand
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Vickrey does not work well
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With few bidders
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When goods are substitutes
Think about details
Ebay and Amazon
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Manual bidding
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Sniping (placing bid at latest possible time)
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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
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Both English auctions (going, going, gone)
Amazon auction ends after deadline & no bids
for 10 minutes
Ebay auction ends after deadline
Results for bidders
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Nash Amazon = Everybody proxy bidding
Nash Ebay = Everybody proxy or everybody
sniping