Algorithm for optimal winner determination in combinatorial auctions

1. Algorithm for optimal winner determination in combinatorial auctions Amogh Mahapatra PHD Candidate, CS, UMN I used to save all my rejection slips because I told myself, one day I’m going to autograph these and auction them. And then I lost the box.-James Lee Burke

2. Forum: Journal of AI • Established 1970 • 18/year • Impact factor:3.036(2009)

3. Author: Tuomas W. Sandholm • PHD and MS (Computer Science) from University of Massachusetts, Amherst • BS,MS (Industrial Engineering and Management Sciences) from Helsinki University of Technology, Finland • Professor at Carnegie Mellon University • Research Interests: Market design, game theory, mechanism design, e-commerce, multi agent systems, auctions, voting, coalitions, kidney exchange

4. What are auctions? • Popular way of item distribution, looks autonomous, is its really? • Auctioneer wants highest profit margin, bidders want the best deal/least cost. • Examples: European football league auctions etc. • Rewards are not additive!

5. Existing auction techniques, why combinatorial? • Sequential: Bids come one by one, highest wins. Requires speculating and computing the future moves of other agents. • Parallel: Who bids first? Decide a threshold. Inefficiencies might require exchanges and allowing retraction of bids. Fixable not desirable. • Why combinatorial: One seller, multiple bidders. You can bid on many items at the same times to express complementarity.

6. Winner determination in combinatorial auctions? • Maximize auctioneer’s revenue given all pay their winning bids. • Partition into disjoint sets and decide that highest bid wins. • Number of exhaustive partitions is O(mm) • WOOF!

7. Solution Paradigms: • Dynamic Programming: Enumerates and stores the combinations along the path, reduces complexity to O(3m). • Follows from weighted set packing that it is NP-complete. • Can we approximate: Proof of inapproximability. • Greedy algorithms: Too many restrictions on the mechanism, reduce input/output space.

8. Other solution paradigms: • Bids at most k items • Bids at most share k items • No sharing beyond a threshold • At most 2 bids at a time: O(m3) • Combinations allow a pre-specified data structure • Computational complexity versus economic freedom

9. Optimal Search Algorithm • Why Optimal? • Allows all combinations of bids • No approximations, finds maximum revenue • Completely avoids loops and redundant spaces • Capitalizes on sparseness of bids • Would anyone actually bid 2^64 combinations? • If no, why bother?

10. How to search? • Two stages • Selective generation of children in the search tree(SEARCH2) • Secondary search to find children quickly(SEARCH1) • Heuristics that are optimized for speed.(unless only DFS) • Preprocesses the search space.

11. SEARCH1 • If keeping items increases gain, do so.(dummy bids) • Partition represented in a tree by a path from root to node if children contains items with smallest unused paths and no redundancy • The number of leaves in the tree is a tighter bound than 2n and is equal to (n/m)m • DFS or IDA*

12. SEARCH2 • Bidtree: Binary tree in which the bids are inserted up front as the leaves (only those parts of the tree are generated for which bids are received). • 3 states: BLOCKED(Left), MASKED(Right) , ANY(None) • Complexity O(nm)

13. Preprocessing • Keep only highest bid for a combination • Remove provably non competitive bids • Mark non-competitive tuple of bids • Decompose set of bids into connected components

14. Experimental Set Up • 4 set of experiments • Pick items randomly without replacement price within [0,1] • Pick randomly but weight the price • Draw the same number of items price within [0,1] • Add items using a decay probability and pick the price as weighted

15. Results • Good execution time • Scales up to larger number of bids • Handles the variations of items while fixing the number of bids gracefully • Handles gracefully even if ratio of bids to items is fixed • Can handle incremental quotes calculations as well

16. Other applications • Weighted set packing, weighted independent set, and weighted maximum clique, coalition structure generation. • Not if bids are dense like in some coalition structures • Second, in coalition structure generation the values of coalitions are not known to any party and are only revealed during search

17. Substitutability: • Not always additive, subtractive as well say in case of airport fields • XOR defines that well • But do we always want what we don’t want? • OR of XORs • The same algorithms is tuned to handle the problem here too • Shows the real power of the framework

18. Questions and death threats? • Some good used cases of combinatorial auction algorithms? • Is maximizing revenue the only goal, would an auctioneer not want to give special favors to its best client and be harsh to newbies and still make profit? • If the search space is low , are there competitive mechanisms other than auctions for making assignments? • Should revenue be maximal or satisficing, if yes any ideas how?