1. 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

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

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

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

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

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

7. 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

8. 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

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

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

11. Definition 5 ● Optimal single price omniscient auction: F(b) = max k kv k Vector of all submitted bids i-th component, b i, is bid submitted by bidder i. v i is the i-th largest bid in the vector b (for the max, v k is the final price that each winner pays) Number of winners

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 Optimal auction Number of winners Competitive ratio Desired: low constant β(2) and lim m→∞ β(m) = 1

14. Theorem 4 ● Truthful auction that is Θ(log n) -competitive E[R] = ( v / log n ) Σ i=0 [log m]–1 2 i Expected revenue for worst-case Lowest bid > 0 Average over all log n different auctions Special auction: i picked random from [0,…,[log n]], then run 2 i -Vickrey auction NB revenue for 2 i -Vickrey auction ≈ 2 i v if 2 i < m and 0 otherwise Sum all revenues that satisfy 2 i < m thus i < log m

15. Theorem 4 ● Truthful auction that is Θ(log n) -competitive ( v / log n ) Σ i=0 [log m]–1 2 i = ( v / log n ) 2 [log m] – 1 Math Special auction: i picked random from [0,…,[log n]], then run 2 i -Vickrey auction NB revenue for 2 i -Vickrey auction ≈ 2 i v if 2 i < 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 2 i -Vickrey auction NB revenue for 2 i -Vickrey auction ≈ 2 i v if 2 i < 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) = max k kv k So here F(b) = mv Special auction: i picked random from [0,…,[log n]], then run 2 i -Vickrey auction NB revenue for 2 i -Vickrey auction ≈ 2 i v if 2 i < m and 0 otherwise

18. 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 6 E[A(b)] ≥ F(b) / β(m) we have proven β(m) є Θ(log n) Number of bidders Number of winners in optimal auction Optimal auction Vector of all submitted bids 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 High2nd High NZ $100.000NZ $6 NZ $7.000.000NZ $5.000

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