1. Decentralized Auctions for Uniformly Semimodular Bidders Mahyar Salek Richard Steinberg MSR Cambridge London School of Economics

2. Auctioning Multiple Items Challenge: Values of items don’t add up!

3. Supermodular Valuation Submodular Valuation

4. Outline Mechanisms – PAUSE [Kelly, Steinberg 2000] (high level) – subPAUSE Idea Example Formal Description Revenue Results – General Setting – Multiunit Setting Context Conclusion

5. PAUSE [Kelly, Steinberg 2000] $6 $2 $3 $1$3 $1 1 –Rum Simultaneous Ascending Auction 2 –Allow bids on bundles of size up to 2 -Accept if improving revenue (by minimum increment) Revenue: 16 $9 Revenue: 18... k –Allow bids on bundles of size up to k -Accept if improving revenue.. Finish when no new revenue-improving bid Charge each winning bidder her current bid

6. PAUSE [Kelly, Steinberg 2000] Is NOT truthful Unknown Equilibrium Easy to understand Transparent Decentralized Auctioneer does not compute much Revenue Guarantee under Sincere Bidding new Does PAUSE have an equivalent for the submodular world?

7. subPAUSE Idea 1 –Rum All-item Auction 2 –Allow xor bids on breakups of size up to 2 -Accept if improving revenue Revenue: 15 $15 Revenue: 17... k –Allow xor bids on breakups of size up to k -Accept if improving revenue.. Finish when no new revenue-improving bid $8 $9

8. Bundle /bidder Valuation 88816 24 10 99916171823 ab c c ab c c aba c c b c c ab c c Bidders: 1 23 items:

9. Stage 1: All-Item Auction (Ascending) Private Value Bid 2423 10 23 ab c c Temp. assgn. ab c c

10. Stage 2: Bidders bid to break down the temp assgn into two sets. Private Value Bid 16 10 16 ab Private Value Bid 8 10 98 c c 1 2 3 Bidders 3 wants either {a,b} for 16 or {c} for 8 but not both.

11. Admissibility This bid is admissible. Cheap Disposal Partition Revenue Private Value Bid 16 ab Private Value Bid 98 c c 3

12. Example (cont’) Stage 2: Bidders bid to break down the temp assgn into at most two sets. Private Value Bid 24 Temp. assgn. ab c c 1

13. Auctioneer makes k+1 offers to bidder j: Proposition 1: This is an offer that j can’t refuse (by submodulariy) Auctioneer’s menu

14. Example (cont’) Stage 3: Bidders bid to break down the temp assgn into three sets. Private Value Bid 24 Temp. assgn. ab c c 1 Private Value Bid 10 99 a Private Value Bid 109 99 2 3 Private Value Bid 109 99 b c c

15. Example (cont’) Stage 3: Bidders bid to break down the temp assgn into at most three sets. Private Value Bid 1615 10 Temp. assgn. a b c c 1 2

16. Block Bids Admissible Bids ‘Against’ Bidders Next Stage

17. Example Stage 4: Bidders bid to break down temp assgns into up to four sets. Private Value Bid 98 Private Value Bid 98 3 b Private Value Bid 1615 10 Temp. assgn. a b c c 1 2 c c

18. Example Stage 4: Bidders bid to break down the temp assgn into at most four sets. Private Value Bid 88 10 98 Temp. assgn. a b c c 1 2 3 Terminates

19. Termination Example Stats: – Revenue: 26 – Total available surplus: 28 – Social welfare: 27 – Number of stages: 4 Terminate if there is no more admissible block bid, return the assignment with the highest revenue

20. Formal Description £4 £6£7

21. Stage 1: All-Item Auction (Ascending) £10 £0 (, £6) (, £5)

22. Cheap Disposal Partition Revenue Block Bids Admissible Bids ‘Against’ Bidders Next Stage

23. ‘Against’ Bidders Auctioneer makes k+1 offers to bidder j: Proposition 1: This is an offer that j can’t refuse (by submodulariy) Add Assignment Terminate if there is no more admissible block bid, return the assignment with the highest revenue Proposition 3: The outcome is envy-free

24. Properties Computationally tractable for the auctioneer Transparent to the bidders – A losing bidder can see why he lost Envy-free – No bidder would want to swap his allocation with another bidder Prevents jump bidding (in prices & in packages)

25. Big Picture Abstract Auction Framework PAUSE subPAUSE

26. Summary of the Results PAUSE Abstract Auction Framework subPAUSE Revenue bounds - compared to VCG - compared to SAA and VCG - Improved for bounded curvature - Compared to VCG Ahipasaoglu, Orlin, Steinberg (2011) - Improved for multi-unit - Compared to VCG - Efficiency vs. Revenue

27. δ-identical item-sets

28. Thank you!

29. Example (2)  rejects and therefore A 2,, 12  rejects and therefore A 3,, 13

30. Example (3)  rejects and therefore A 4,, 14 …  rejects and therefore A 4,, 18  rejects and therefore A 4,, 19

31. Example (4)  rejects and therefore A 4,, 20 …  rejects and therefore A 4,, 30 Terminated