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1. Bayesian Combinatorial Auctions • Giorgos Christodoulou, Annamaria Kovacs, Michael Schapira האוניברסיטה העברית בירושלים The Hebrew University of Jerusalem

2. Combinatorial Auctions

3. Combinatorial Auctions bidders items valuations (normalized) (monotone) Objective: Find a partition of the items that maximizes the social welfare

4. Valuations • Submodular (SM) The marginal value of the item decreases as the number of items increases. • Fractionally-subadditive (FS) additive

5. FS Valuations items add. valuations

6. Combinatorial Auctions - Challenges • Strategic • We want bidders to be truthful. • VCG implements the opt. (exp. time) • Computational • approximation algorithms (not truthful)

7. Unknown Valuations

8. Huge Gaps • Submodular (SM) 1-1/e-e[Feige-Vondrak] • Fractionally-subadditive (FS) 1-1/e[Dobzinski-Schapira] O(log(m) log log(m))[Dobzinski]

9. Solution? • We do not know whether reasonable truthful and polynomial-time approximation algorithms exist. • How can we overcome this problem? • An old/new approach.

10. Partial Information is drawn from D

11. Complete Information

12. Player i will bid • Strategy Profile • Algorithm = allocation + payments Utility of player i Auction Setting

13. Bayesian Combinatorial Auctions Question:Can we design an auction for which anyBayesian Nash Equilibrium provides good approximation to the social welfare?

14. ( ) (Pure) Bayesian Nash [Harsanyi] • Bidding function • Informal: In a Bayesian Nash (B1,…,Bn), given a probability distribution D, Bi(vi) maximizes the expected utility of player i (for all vi).

15. Expected Social of a B.N.E. Bayesian PoA Optimal Social Welfare for fixed v Bayesian PoA = biggest ratio between SW(OPT) and SW(B) (over all D, B)

16. Price of Anarchy Bayesian PoA [Gairing, Monien, Tiemann, Vetta]

17. Utility of player i Second Price • Player i will bid • Strategy Profile • Algorithm: • Give item j to the player i with the highest bid. • Charge I the second highest bid.

18. Second Price Social Welfare = 1

19. Second Price Social Welfare =

20. PoA=1/ Second Price Social Welfare =

21. Supporting Bids • Bidders have only partial info (beliefs) • They want to avoid risks. (ex-post IR) Supporting Bids:(for all S)

22. Lower Bound opt=2

23. Lower Bound Nash=1 PoA=2

24. Our Results • Bayesian setting: • The Bayesian PoA for FS valuations (supporting bids, mixed) is 2. • Complete-information setting: • FS Valuations:Existence of pure N.E. • Myopic procedure for finding one. • PoS=1. • SM Valuations: Algorithm for computing N.E. in poly time.

25. Valuations • Submodular (SM) The marginal value of the item decreases as the number of items increases. • Fractionally-subadditive (FS) additive

26. Upper Bound(full-info case) Lemma. For any set of items S, where is the maximizing additive valuation for the set S.

27. Upper Bound Let be a fixed valuation profile

28. Upper Bound Let be a fixed valuation profile optimum partition: Nash partition:

29. maximum additive valuation wrt Upper Bound Let be a fixed valuation profile optimum partition: Nash partition: Since b is a N.E

30. maximum additive valuation wrt Upper Bound Let be a fixed valuation profile optimum partition: Nash partition: Since b is a N.E

31. Since b is a N.E Upper Bound and so

32. Since b is a N.E Upper Bound and so using lemma we get

33. Since b is a N.E Upper Bound and so using lemma we get and so

34. Upper Bound summing up

35. But… • Open Question: Does a (pure) BN with supporting bids always exist? • Open Question:Can we find a (mixed) BN in polynomial time? • We consider the full-information setting.

36. The Potential Procedure • Start with item prices 0,…,0. • Go over the bidders in some order 1,…,n. • In each step, let one bidder i choose his most demanded bundle S of items. Update the prices of items in S according to i’s maximizing additive valuation for S. • Once no one (strictly) wishes to switch bundle, output the allocation+bids.

37. The Potential Procedure • Theorem: If all bidders have fractionally-subadditive valuation functions then the Potential Procedure always converges to a pure Nash (with supporting bids). • Proof: The total social welfare is a potential function.

38. The Potential Procedure • Theorem: After n steps the solution is a 2-approximation to the optimal social welfare (but not necessarily a pure Nash). [Dobzinski-Nisan-Schapira] • Theorem: The Potential Procedure might require exponentially many steps to converge to a Pure Nash.

39. The Potential Procedure • Open Question: Can we find a pure Nash in polynomial time? • Open Question: Does the Potential Procedure converge in polynomial time for submodular valuations?

40. The Marginal-Value Procedure • Start with bid-vectors bi=(0,…,0). • Go over the items in some order 1,…,m. • In each step, allocate item j to the bidder i with the highest marginal value for j. Set bij to be the second highest marginal value.

41. The Marginal-Value Procedure • Theorem: The Marginal-Value Procedure always outputs an allocation that is a 2-approximation to the optimal social-welfare. [Lehmann-Lehmann-Nisan] • Proposition: The bids the Marginal-Value Procedure outputs are supporting bids and are a pure Nash equilibrium.

42. Open Questions Design an auction that minimizes the PoA for B.N.E. • Can a (mixed) Bayesian Nash Equilibrium be computed in poly-time? • Algorithm that computes N.E. in poly time for FS valuations. Second Price

43. Thank you!