Side Constraints and Non-Price Attributes in Markets

1. Side Constraints and Non-Price Attributes in Markets Tuomas Sandholm Carnegie Mellon University Computer Science Department [Paper by Sandholm & Suri 2001]

2. 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

3. 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

4. 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

5. 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]

6. 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 !

7. 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 !

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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