Envy-Free Auctions for Digital goods

1. 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

2. Truthful Competitive Envy-free Three desirable properties for auctions

3. Truthful = bid-independent Competitive Envy-free A truthful auction

4. Truthful = bid-independent Competitive = constant fraction of optimal revenue Envy-free A competetive auction

5. Truthful = bid-independent Competitive = constant fraction of optimal revenue Envy-free = no envy among bidders after auction An envy-free auction

6. Truthful = bid-independent Competitive = constant fraction of optimal revenue Envy-free = no envy among bidders after auction Three desirable properties for auctions

7. Truthful = bid-independent Competitive = constant fraction of optimal revenue Envy-free = no envy among bidders after auction Main result No auction can have all three properties

8. Truthful = bid-independent Competitive = constant fraction of optimal revenue Envy-free = no envy among bidders after auction A solution Relax one of the three properties

9. Why • Envy free for consumer acceptance • Truthful for no sabotage • Competitive guarantees profit minimum bound for auctioneer

10. competitive ratio: O(log n) A truthful, envy-free auction

11. Definition 5 • Optimal single price omniscient auction: F(b) = maxkkvk 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)

12. Before continuing… • Two important variables: n = number of bidders m = number of winners in optimal auction

13. Definition 6 • 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: • low constant β(2) and • limm→∞β(m) = 1

14. 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 Sum all revenues that satisfy 2i < m thus i < log m 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

15. 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

16. 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 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

17. 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 auction F(b) = maxkkvk 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

18. Theorem 4 • Truthful auction that is Θ(log n)-competitiveE[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) єΘ(log n) Vector of all submitted bids Optimal auction Number of winners in optimal auction Number of bidders Competitive ratio

19. 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

20. 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

21. CostShare • Truthful • Profit R if R≤ F (or no winners) • Envy-free • Because it cannot guarantee winners, it is not competitive

22. 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

23. Auctions for real

24. Current auction research applied • Frequency auctions • Radio • Mobile phones • Advertisements • Google • MSN • Auction sites • Ebay • Amazon

25. 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

26. Outcomes New Zealand • Extreme outcomes • Not Fraudulent

27. Lessons New Zealand • Vickrey does not work well • With few bidders • When goods are substitutes • Think about details

28. 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

29. Auctioneer strategies • Both English auctions (going, going, gone) • Amazon auction ends after deadline & no bids for 10 minutes • Ebay auction ends after deadline

30. Results for bidders • Nash Amazon = Everybody proxy bidding • Nash Ebay = Everybody proxy or everybody sniping