Download Agents in Combinatorial Markets

Side Constraints and
Non-Price Attributes
in Markets
Tuomas Sandholm
Carnegie Mellon University
Computer Science Department
[Paper by Sandholm & Suri 2001]
Side constraints in markets
• Traditionally, markets (auctions, reverse auctions,
exchanges) have been designed to optimize unconstrained
economic value (Pareto efficiency/revenue)
• Side constraints are required in many practical markets
(especially in B2B) to encode legal, contractual and
business constraints
• Side constraints could be imposed by any party
–
–
–
–
–
Sellers
Buyers
Auctioneer
Market maker
…
• Side constraint have significant implications on the
complexity of clearing the market
Outline
• Side constraints in non-combinatorial markets
• Side constraints in combinatorial markets
– Constraints under which the winner determination
problem stays polynomial time solvable (if bids can be
accepted partially)
– Constraints under which the winner determination
problem is NP-complete even if bids can be accepted
partially
– Constraints under which the winner determination
problem is polynomial-time solvable even if bids have
to be accepted entirely or not at all
Noncombinatorial auctions
• There are m items for sale
• Each bidder can submit any number of bids
– Each bid is for one item
• Without side constraints, winners can be
determined in polynomial time by selecting the
highest bid for each item separately
Budget constraints in
noncombinatorial auctions
• Thrm. If bidders can have budget constraints,
revenue-maximizing winner determination is
NP-complete
– Polynomial time (using linear programming = LP)
if bids can be accepted partially
• Max number of items per bidder =>
polynomial time ! [Tennenholtz AAAI-00]
Max winners constraint in
noncombinatorial auctions
• Thrm. If there can be at most k winners,
revenue-maximizing winner determination is
NP-complete
– This holds even if bids can be accepted partially !
XOR constraints in
noncombinatorial auctions
• In some auctions, bidders may want to submit
XOR constraints between bids
– E.g. “I want a Sony TV XOR an RCA TV”
– “Scenario bids” (e.g., for restricted capacity settings)
• Under XOR-constraints , revenue-maximizing
winner determination is NP-complete
– This holds even if bids can be accepted partially !
Notes about generality
• The results from above hold whether or not
the auctioneer has to sell all items
• They also hold if prices are restricted to be
integers
Combinatorial auction (CA)
• Auctioneer’s perspective:
– Binary winner determination problem:
• Label bids as winning or losing so as to maximize sum of
bid prices
– Each item can be allocated to at most one bid
• NP-complete [Rothkopf et al 98, Karp 72]
• Inapproximable [Sandholm IJCAI-99 using Hastad 99]
– Fractional winner determination problem: Bids can be
accepted partially
• Polynomial time using LP
• The results that we will discuss apply to
combinatorial auctions, combinatorial reverse
auctions & combinatorial exchanges
Side constraints in combinatorial
markets
• Thrm. Practical side constraint classes under which the
fractional case remains polytime solvable and the binary
case remains NP-complete
– Cost constraints, e.g. mutual business, trading volume,
minorities, long-term competitiveness via monopoly avoidance,
risk hedging by requiring that at least k bidders get certain
volume
– Unit constraints
– Absolute or % compared to some group
– >, <, or =
– Gross or net in exchanges
Side constraints in combinatorial
markets…
• Thrm. Practical side constraint classes under
which both the fractional and the binary case
are NP-complete
– Counting constraints
• E.g. max winners
• => there is no way to construct a counting gadget in LP
– XOR-constraints between bids
• Needed for full expressiveness => inherent tradeoff
between expressiveness and clearing complexity
Side constraints in combinatorial
markets…
• Thrm. Theoretical side constraint under
which even the binary clearing problem
becomes polytime solvable (the fractional
case remains polytime solvable)
– Extreme equality: each bid has to be accepted
to the same extent
Non-price attributes in markets
• Combinatorial markets exist (logistics.com, Bondconnect, FCC,
CombineNet, …) and multi-attribute markets exist (Frictionless,
Perfect, …), but have not been hybridized
• Here we propose a way to hybridize them
• Attribute types
– Attributes from outside sources, e.g., reputation databases
– Attributes that bidders fill into the partial item description
• Handling attributes in combinatorial auctions & reverse auctions
– Attribute vector b
– Reweight bids, so p’ = f(p, b)
– Side constraints could be specified on p or p’
• Same complexity results on side constraints hold
• Attributes cannot be handled as a preprocessor in exchanges
– Buyers care which sellers goods come from & vice versa
– Have to handle attributes as part of the main winner determination
optimization problem
Conclusions
• Combinatorial markets are important & now feasible
– Market types differ in clearing complexity & approximability
– Expressive bidding language removes guesswork & sets
correct incentives
– Side constraints extend usability of dynamic pricing
• Allow the advantages of dynamic pricing while keeping the advantages
of long-term contracts
• Different side constraints lead to different clearing complexity
– Can make problem harder or easier
– Even non-combinatorial markets become NP-complete to clear under
natural side constraints
» Complexity is not an argument against (only) combinatorial markets