Experiments on combinatorial auctions Only some of the techniques of CABOB deployed in these results

Experimental setup Comparison between CABOB and CPLEX 7.0 CPLEX 7.0 was the fastest earlier algorithm for the problem System: 933 MHz Pentium III, 512 MB RAM, Linux instances for each data point All distributions produced distinct bids

Experiments: Random Distribution Random: Choose a random number of items without replacement. Pick price from [0,1]. [Sandholm IJCAI-99] CABOB does well primarily because of column dominance test. CPLEX is able to solve these without searching 47% of the time.

Experiments: Weighted Random Distribution Weighted Random: Choose a random number of items without replacement. Pick the price from between 0 and the number of items in the bid. [Sandholm IJCAI-99] CPLEX only searches 5% of the time. CABOB searches 88% of the time.

Experiments: CATS Distributions CATS: Combinatorial Auction Test Suite [Leyton-Brown, Pearson, Shoham, 2000] Random bid distributions Modeled after realistic bidder preferences Distributions have many parameters; we varied # bids and used default parameters for rest

Experiments: CATS PATHS CATS PATHS Simulates bids on paths in a 2-D space. Examples: truck routes, bandwidth allocation Neither algorithm searched. CABOB’s simple preprocessing techniques are faster

Experiments: CATS MATCHING CATS MATCHING Simulates bid where complementarity is based on a temporal aspect of items Example: airport take-off and landing slots CPLEX never searched, CABOB rarely searched

Experiment: Uniform Distribution Each bid has same number of items Prices chosen at random from [0,1] Cases with few items per bid were hardest for earlier algorithms [Sandholm IJCAI-99, Fujishima et al IJCAI-99]

Experiments: Bounded Distribution… Bounded: Choose number of items between a lower and upper bound. Pick price between 0 and number of items in bid Similar to Uniform distribution [Sandholm IJCAI-99], but more realistic CABOB performs better here mainly because of the complete bid graph test

Experiments: Components Distribution A number of independent components from the uniform distribution where each bid has same #items CABOB’s decomposition techniques capitalize on this structure, CPLEX does not

Anytime Performance Feasible solution found quickly Solution improves rapidly over time Optimal algorithms might be the best approximation algorithms too !

Generalizations of combinatorial auctions Free disposal Substitutability Multiple units of each item Combinatorial exchanges (= many-to-many auctions) Reservation prices –On items –On combinations –With substitutability Combinatorial reverse auctions Combinations of these generalizations

Generalization: substitutability [Sandholm IJCAI-99] What if agent 1 bids –$7 for {1,2} –$4 for {1} –$5 for {2} ? Bids joined with XOR –Allows bidders to express general preferences –Groves-Clarke pricing mechanism can be applied to make truthful bidding a dominant strategy –Worst case: Need to bid on all 2 #items -1 combinations OR-of-XORs bids maintain full expressiveness & are more concise –E.g. (B 2 XOR B 3 ) OR (B 1 XOR B 3 XOR B 4 ) OR... –Our algorithm applies (simply more edges in bid graph )

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

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

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

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

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)

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

Single unit auctions & reverse auctions 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… On decay distribution, even with free disposal, reverse auctions take longer than auctions (unlike approximability would suggest)

Multi-unit auctions & reverse auctions Decay-decay: Number of units for each item chosen with decay probability.99 –For each bid Number of items chosen with decay probability  1 For each item, #units chosen with decay probability  2 –All instances were easy. E.g., at  1 =.6,  2 =.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 (  1 =.8,  2 =.8):

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

Exchanges Exchange decay-decay distribution: For each bid –Number of items chosen with decay probability  1 –For each item, #units chosen with decay probability  2 –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  2 = 0 Single-unit of each item –Scales well –Free disposal case slightly harder

Multi-unit exchanges CPLEX 7.1 scales very poorly: #bids/#items = 10 (  1 =.8,  2 =.8)

Multi-unit exchanges 50 items, 500 bids (  1 =.8,  2 =.8) CPLEX 7.1 never finished –Without free disposal, did not even find a feasible solution CPLEX 7.1 had very poor anytime performance:

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 –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

Hot off the press [Kothari, Suri & Sandholm 2002] 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: Polynomial time = fast