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