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Winner determination in combinatorial auction generalizations

Winner determination in combinatorial auction generalizations. Tuomas Sandholm Subhash Suri Andrew Gilpin David Levine Carnegie Mellon University University of California CombineNet Inc. Computer Science Department Santa Barbara Pittsburgh, PA

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Winner determination in combinatorial auction generalizations

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  1. Winner determination in combinatorial auction generalizations Tuomas Sandholm Subhash Suri Andrew Gilpin David Levine Carnegie Mellon University University of California CombineNet Inc. Computer Science Department Santa Barbara Pittsburgh, PA Dept of Computer Science CS 15-892 Foundations of Electronic Marketplaces

  2. Search algorithms for winner determination in combinatorial auctions • Our first optimal search algorithm [Sandholm ICE-98, IJCAI-99] • Capitalizes on sparsely populated space of bids • Generates only populated parts of space of allocations • Has become a hot topic in CS since 1999 • Andersson, Aspnes, Boutilier, Conen, DasGupta, de Vries, Fujishima, Gilpin, Gonen, Grosz, Hoos, Hunsberger, Kamei, Kao, Kutanoglu, Lehmann, Levine, Leyton-Brown, Matsubara, Monderer, Muller, Nisan, Parkes, Pearson, Penn, Sakurai, Schulz, Shoham, Suri, Tenhunen, Tenneholtz, Ungar, Vohra, Walsh, Wellman, Wurman, Ygge, Yokoo, … • CABOB: Our newest optimal anytime algorithm[Sandholm & Suri AAAI-00, IJCAI-01] • Literature has focused mainly on combinatorial auctions and a bit on multi-unit combinatorial auctions

  3. New generalizations of combinatorial auctions • No free disposal (sellers cannot keep items, buyers cannot take extras) • Single- or multi-unit • Combinatorial reverse auctions • Single- or multi-unit • Combinatorial exchanges (= many-to-many auctions) • Single- or multi-unit

  4. Combinatorial reverse auction • Example: procurement in supply chains • Auctioneer wants to buy a set of items (has to get all) • Can take extras if there is free disposal • Sellers place bids on how cheaply they are willing to sell bundles of items • Thrm. Winner determination is NP-complete even in single-unit case with free disposal • Thrm. Single unit case with free disposal is approximable • k = 1 + log m (m = largest number of items that any bid contains) • Greedy algorithm: Keep choosing bid with lowest price / #items

  5. No free disposal • Free disposal: seller can keep items, buyers can take extras • Free disposal has been assumed in the combinatorial auction literature so far • In practice, freeness of disposal can vary across items & bidders • Without free disposal, the set of feasible solutions is same for combinatorial auctions & reverse auctions • Thrm. Even finding a feasible solution is NP-complete

  6. What fraction of solutions is feasible ? • Consider a combinatorial auction or reverse auction • m items, n bids • Each bid has k randomly chosen items • p = ( mk) possible bids • i = ( pn) problem instances • m! Feasible instances • 1st bid’s first item can be any of m • 1st bid’s second item can be any of m-1 … • Prob(randomly selected instance feasible) = m! / i

  7. Combinatorial exchange • Example bid: (buy 20 tons of water, sell 10 cubic meters of hydrogen, sell 5 cubic meters of oxygen, ask $500) • Example application: manufacturing where a participant bids for inputs & outputs of a production plan simultaneously • Label bids as winning or losing so as to maximize (revealed) surplus: sum of amounts paid by bidders minus sum of amounts paid to bidders • On each item, sell quantity  buy quantity • Equality if there is no free disposal • Thrm. NP-complete even in the single-unit case • Thrm. Inapproximable even in the single-unit case • Could also maximize trading volume • Thrm. Without free disposal, even finding a feasible solution is NP-complete (even in the single-unit case)

  8. Experiments on generalizations • 933 MHz Pentium III, 512M RAM • CPLEX 7.0 • Each plot point is mean over 50 instances • Significantly slower to find optimal solution than to prove infeasibility => we plot times on feasible instances • With free disposal, all instances are feasible • On distributions where CPLEX finds optimum with no search, it also tends to prove infeasibility with no search • On distributions where CPLEX needs search to find optimum, it also tends to need search to prove infeasibility

  9. Lack of free disposal makes problem much harder Complexity is polynomial in bids (even in worst case) Reverse auctions with free disposal seldom require search on these distributions Auctions require more search & more often (as inapproximability suggests) Single unit auctions & reverse auctions

  10. Single unit auctions & reverse auctions…

  11. Multi-unit auctions & reverse auctions • Decay-decay distribution: • Number of demanded units for each item chosen with decay probability .99 • For each bid • Number of items chosen with decay probability a1 • For each item, #units chosen with decay probability a2 • All instances were easy. E.g., at a1 = .6, a2 = .9, LP solved • 74% of reverse auctions with free disposal • 52% of auctions with free disposal • 50% of auctions without free disposal • 22% of reverse auctions without free disposal • Hardest setting (a1 = .8, a2 = .8):

  12. Multi-unit auctions & reverse auctions… • CATS multipaths: • Almost all reverse auctions (with/without free disposal) & auctions without free disposal were infeasible • CPLEX could not scale to 2,000 bids on auctions with free disposal

  13. Exchanges • Exchange decay-decay distribution: For each bid • Number of items chosen with decay probability a1 • For each item, #units chosen with decay probability a2 • Sign is negated w.p. .5 • Price is random number between 0 and 1, multiplied by total #units (negative half the time) • Single unit case comes from a2 = 0 • Single-unit of each item • Scales well • Free disposal case slightly harder

  14. Multi-unit exchanges • CPLEX 7.1 scales very poorly: #bids/#items = 10 (a1 = .8, a2 = .8)

  15. Multi-unit exchanges • Complexity increases super-exponentially in a2 and a1

  16. Multi-unit exchanges • 50 items, 500 bids (a1 = .8, a2 = .8) • CPLEX 7.1 never finished • Without free disposal, did not even find a feasible solution • CPLEX 7.1 had very poor anytime performance:

  17. Conclusions • Generalizations of combinatorial auctions • No free disposal • Reverse auctions • Exchanges • Single- and multi-unit settings • Theoretical results • All these generalizations are NP-complete • With free disposal • Auction and exchanges are inapproximable • Reverse auctions are approximable (to a logarithmic ratio bound) • But even finding a feasible solution is NP-complete if XORs are allowed • Without free disposal, even finding a feasible solution is NP-complete • Experimental results • Search does well on auctions & at times even better on reverse auctions • Search does well on single-unit exchanges, poorly on multi-unit exchanges • Better algorithms needed • Lack of free disposal makes the problem much harder

  18. Hot off the press[Kothari, Suri & Sandholm ACM-EC-03] • Q: How many bids have to be accepted fractionally (in worst case) so as to obtain maximum surplus in a multi-item multi-unit combinatorial exchange / combinatorial auction? • Trivial answer: #bids • A: #items (this is independent of #units) • Q: How many bids have to be accepted fractionally (in worst case) so as to maximize liquidity in a multi-item multi-unit combinatorial exchange? • Trivial answer: #bids • A: #items + 1 (this is independent of #units) • Q: How complex is it to find such a solution? • A: Can be found using any LP algorithm that terminates in an optimal vertex of the LP polytope

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