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 2.2
• 100 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
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•
•
•
•
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. (B2 XOR B3) OR (B1 XOR B3 XOR B4) OR ...
– Our algorithm applies (simply more edges in bid graph )
Winner determination in
combinatorial auction
generalizations
Tuomas Sandholm
Carnegie Mellon University
Computer Science Department
Subhash Suri
University of California
Santa Barbara
Dept of Computer Science
Andrew Gilpin David Levine
CombineNet Inc.
Pittsburgh, PA
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
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•
•
•
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 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):
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
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•
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
Multi-unit exchanges
• CPLEX 7.1 scales very poorly:
#bids/#items = 10
(a1 = .8, a2 = .8)
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:
Conclusions
• Generalizations of combinatorial auctions
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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