1. Combinatorial Auctions By: Shai Roitman e-mail: shairoi@cs.huji.ac.il

2. Auctions One to many mechanism Efficient Allocation of the items. Seller Auctions Buyer Auctions - Reversed Auctions

3. Known Auction Types Open Cry Auctions –English –Dutch Sealed Bid Auctions –First Price –Second Price

4. The equivalence of auctions True Valuations –English –Sealed Bid Second Price Winners Curse –Dutch –Sealed Bid First Price

5. Sealed Bid Auctions advantages Communication efficient The value of the bid can be kept private.

6. Items Value Private Value - An Item has a value to the bidder regardless of the value to the other bidders –Example: Consumer goods Public Value - The item has value in the context of other bidder estimations –Example: Stocks

7. Strategies for the Auctions under private value assumptions English Auction –Small increments until maximum price(true value) reached. Second price Sealed Auction –Submit the evaluated value as the bid

8. First price Sealed Auction & Dutch Auction –Need to evaluate others evaluation (may use some distribution on the values of the other bidders) and use this evaluation for setting the bid. –Winners Curse Complex analysis Strategies for Auctions - continued

9. Multi Item Auctions - Multi Stage Auction Scenario –A set of items has to be sold Naive Solution –Hold auctions for each item or set of items one at a time

10. Multi Item Auctions - Problems How to choose the order of the items to be sold? How to bundle several dependant items? If the items have dependencies multi stage auctions can lead to inefficient allocation

11. Combinatorial auction Items may be grouped as bundles. => Takes into considerations the dependencies between the items. => Greater economic efficiency

12. The Utility function Private - Public Value Super Additive - Supplemental items Sub Additive - Complementary items Monotonic - The more the better Convex - Diversity

13. Uses for combinatorial auctions FCC Radio spectrum Logistics Scheduling Any purchase of dependant multiple items.

14. Logistic explicit use case of combinatorial auctions Logistics.com - OptiBid(TM) –Trucking companies bid on bundles of lanes –Logistics.com - More than $5 billion in transportation contracts been bid to date (January 2000) (Ford, Wal-Mart, K-Mart).

15. Incentive Issues - An example 3 bidders {1,2,3} 2 items {x,y} Bidder 1 values –{x,y}=100 {x}={y}=0 Bidder 2 values –{x,y}=0 {x}={y}=75 Bidder 3 values –{x,y}=0 {x}={y}=40

16. Incentive Issues - An example - continued If bid truthfully - x->2, y->3 (Revenue 115) If Bidder 2 and Bidder 3 belief that the others truthfully bid their values –Bidder 2 can shade his value of {x} and {y} to 65 and still get the same x->2 y->3 (Revenue 105) –Bidder 3 can shade his value of {x} and {y} to 30 and still get the same x->2 y->3 (Revenue 105)

17. If Bidder 2 & Bidder 3 shade their value (65 & 30) then they will lose as {x,y}->1 => Lost of economic efficiency Incentive Issues - An example - continued

18. Threshold Problem a collections of bidders whose combined valuation for distinct portions of a subset of items exceed the bid submitted on that subset by some other bidder. Difficulty in coordination of their bids to outbid the single large bidder on that subset

19. Auction Scheme assumptions Independent private values for bidders values draw from a commonly known distribution risk neutral

20. Auction Design- An optimal mechanism Truth Revelation - revelation principle No Bidder is made worse off by participating Seller Maximum Expected Revenue

21. Efficiency If the allocation of objects to bidders chosen by the seller solves the following equations than the auction is efficient

22. General CAP Formalization

23. Vickrey Clarke Groves (VCG) - part 1

24. Vickrey Clarke Groves (VCG) - part 2

25. Vickrey Clarke Groves (VCG) - part 3 If no agent has a significant effect on the average V is close to V^(-k) thus the revenue is close to the maximum revenue defined in the General CAP.

26. Problems in the VCG mechanism Solving the CAP problem is hard (NP-Hard) Using Approximate solutions => Not incentive compatible Payments in VCG are sensitive to the choice of the solution

27. General CAP Formalization

28. Multiple object in the CAP Formulation

29. The CAP (Combinatorial Auction Problem) Bidders must submit bid for every subset Transmitting the bid sets in a succinct manner

30. Restriction of conditions => solvable solution - an example Restriction –All bidders complement each other –all bidders are symmetric Solution –Auction all the items as one item in an optimal single item auction

31. Cybernomics experiments Performed tests for additive values and valuations with synergies of small, medium or high intensity Results –Combinatorial multi round auctions always superior in efficiency but lower in revenue –Slower convergence (finishing the auctions)

32. The CAP - continued partial solutions –Restriction on the way the bids are transmitted OR / OR* Trees Single mind restriction –Sending an Oracle Problem of deciding the collection of bids to accept

33. The SPP Problem Given a set of M elements collection V of subsets with weights Find the largest weight collection of subsets that are pairwise disjoint.

34. The SPP Formalization

35. SPP Related Problem - Set Partitioning Problem (SPA)

36. SPP Related Problem - Set Covering Problem

37. What is the complexity of SPP? SPP Is a NP-Hard / Complete problem SPP Problem is exponential in |V| (V the number of subsets of M)! No Hope??

38. Effective solution to the CAP Problems Requirements –Number of distinct bids is not large –Underlying SPP problem can be solved reasonable quick.

39. SPP Approximation There is no Polynomial algorithm that can deliver a worst case ration larger than n^(E- 1) for any E>0 There is a worst case ratio of O(n/(log n)^2) algorithm (Polynomial algorithm)

40. Other Approaches Decentralized Methods –Setting up a fictitious market determining an allocation and prices –Choosing an allocation and bidders are required to send improvements

41. Conclusions Combinatorial Auctions can lead to higher economic efficiency Practical Combinatorial Auctions are hard to implement with compliance to the truth revelation principle