Multi-unit auctions & exchanges (multiple indistinguishable units of one item for sale)

1. Multi-unit auctions & exchanges (multiple indistinguishable units of one item for sale) Tuomas Sandholm Computer Science Department Carnegie Mellon University

2. Auctions with multiple indistinguishable units for sale • Examples • IBM stocks • Barrels of oil • Pork bellies • Trans-Atlantic backbone bandwidth from NYC to Paris • …

3. Multi-unit auctions: pricing rules • Auctioning multiple indistinguishable units of an item • Naive generalization of the Vickrey auction: uniform price auction • If there are k units for sale, the highest k bids win, and each bid pays the k+1st highest price • Demand reduction lie [Crampton&Ausubel 96]: • k=5 • Agent 1 values getting her first unit at $9, and getting a second unit is worth $7 to her • Others have placed bids $2, $6, $8, $10, and $14 • If agent 1 submits one bid at $9 and one at $7, she gets both items, and pays 2 x $6 = $12. Her utility is $9 + $7 - $12 = $4 • If agent 1 only submits one bid for $9, she will get one item, and pay $2. Her utility is $9-$2=$7 • Incentive compatible mechanism that is Pareto efficient and ex post individually rational • Clarke tax. Agent i pays a-b • b is the others’ sum of winning bids • a is the others’ sum of winning bids had i not participated • What about revenue (if market is competitive)?

4. Multi-unit auctions: Clearing complexity[Sandholm & Suri IJCAI-01]

8. In all of the curves together

17. Multi-unit reverse auctions with supply curves • Same complexity results apply as in auctions • O(#pieces log #pieces) in nondiscriminatory case with piecewise linear supply curves • NP-complete in discriminatory case with piecewise linear supply curves • O(#agents log #agents) in discriminatory case with linear supply curves

18. Multi-unit exchanges • Multiple buyers, multiple sellers, multiple units for sale • By Myerson-Satterthwaite thrm, even in 1-unit case cannot obtain all of • Pareto efficiency • Budget balance • Individual rationality (participation)

19. Screenshot from eMediator [Sandholm AGENTS-00]

20. profit psell pbuy • One price for everyone (“classic partial equilibrium”): • profit = 0 • One price for sellers, one for buyers ( nondiscriminatory pricing ): profit > 0 Supply/demand curve bids profit = amounts paid by bidders – amounts paid to sellers Can be divided between buyers, sellers & market maker Quantity Aggregate supply Aggregate demand Unit price

21. pbuy p2sell p1sell One price for each agent ( discriminatory pricing ): greater profit Nondiscriminatory vs. discriminatory pricing Aggregate demand Quantity Supply of agent 1 Supply of agent 2 Unit price psell pbuy One price for sellers, one for buyers ( nondiscriminatory pricing ): profit > 0

22. Shape of supply/demand curves • Piecewise linear curve can approximate any curve • Assume • Each buyer’s demand curve is downward sloping • Each seller’s supply curve is upward sloping • Otherwise absurd result can occur • Aggregate curves might not be monotonic • Even individuals’ curves might not be continuous

23. Pricing scheme has implications on time complexity of clearing • Piecewise linear curves (not necessarily continuous) can approximate any curve • Clearing objective: maximize profit • Thrm. Nondiscriminatory clearing with piecewise linear supply/demand: O(p log p) • p = total number of pieces in the curves • Thrm. Discriminatory clearing with piecewise linear supply/demand: NP-complete • Thrm. Discriminatory clearing with linear supply/demand: O(a log a) • a = number of agents • These results apply to auctions, reverse auctions, and exchanges • So, there is an inherent tradeoff between profit and computational complexity