Side Constraints and Non-Price Attributes in Markets

1. Side Constraints and Non-Price Attributes in Markets Tuomas Sandholm Subhash Suri Carnegie Mellon University University of California Computer Science Department Santa Barbara Dept of Computer Science

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 LP) if bids can be accepted partially • Proof is by reduction from PARTITION. PARTITION has m integers, and the question is whether they can be divided into two sets so that sum is same in each set. Create two bidders with same m bids (equal to the integers) and same budget constraint k. There is a solution of 2k iff there is a partition. • 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 ! • Proof is by reduction from SET-COVER. SET-COVER has m ground items, and a list of sets of these items, and the question is whether the items can be covered with k sets. Corresponding to each set, generate a bidder who places a $1 bid for every item in the set. Now, there is a set cover of size k iff the auction has a solution with revenue $m and max number of winners k.

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 ! • Proof. Reduce INDEPENDENT-SET to this problem. For each vertex, generate an item and a $1 bid for it. Corresponding to each edge, insert an XOR-constraint between the bids. Now, the auction has a solution of revenue $k iff there is an independent set of size k.

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) • Can bid on combinations of items [Rassenti,Smith & Bulfin 82]... • Bidder’s perspective • Allows bidder to express what she really wants • No need for lookahead / counterspeculationing of items • Auctioneer’s perspective: • Automated optimal bundling • Binary winner determination problem: • Label bids as winning or losing so as to maximize sum of bid prices (= revenue) • 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

10. Combinatorial reverse auction[Sandholm, Suri, Gilpin & Levine AGENTS-01 workshop on Agents for B2B] • Example: procurement in supply chains • Auctioneer wants to buy a set of items (has to get all) as cheaply as possible • Sellers place bids on how cheaply they are willing to sell bundles of items • Thrm. Binary clearing is NP-complete • Thrm. Binary clearing is approximable • k = 1 + log( largest #items that any bid contains ) • Thrm. Even finding a feasible solution is NP-complete with XORs • If seller(s) cannot keep items and buyer(s) cannot take extras, the set of feasible solutions becomes same for combinatorial auctions & reverse auctions • Thrm. Even finding a feasible solution is NP-complete • Fractional clearing is polytime using LP

11. Combinatorial exchange • Each bid can buy some items, sell other items, and pay or request a payment • Maximize surplus = sum of accepted buy bid prices – sum of accepted sell bid prices • NP-complete and inapproximable in the binary case • Polytime solvable via LP in the fractional case • Our results hold for combinatorial markets (auctions, reverse auctions & exchanges) • We prove the negative results for auctions & positive results for exchanges

12. 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

13. 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

14. Need for XORs: substitutability [Sandholm ICE-98, IJCAI-99, AIJ-01] • 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 • Risk free scenario bidding • Clarke-Groves pricing mechanism can be applied to make truthful bidding a dominant strategy  “generalized Vickrey auction” • Worst case: Need to bid on all 2#items-1 combinations • OR-of-XORs bids maintain full expressiveness & are more concise • E.g. (B2XOR B3) OR (B1XOR B3XOR B4) OR ... • Note: coding XORs using dummy items [Fujishima, Leyton-Brown, Shoham IJCAI-99] does not work in the fractional case

15. 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

16. Non-price attributes in markets • Combinatorial markets exist (logistics.com, Bondconnect, FCC, …) 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 post-processor in exchanges • Buyers care which sellers goods come from & vice versa

17. 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