Multi-Unit Auctions & Exchanges: Strategies and Mechanisms

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. Bidding languages and expressiveness • These bidding languages were introduced for combinatorial auctions, but also apply to multi-unit auctions • OR [default; Sandholm 99] • XOR [Sandholm 99] • OR-of-XORs [Sandholm 99] • XOR-of-ORs [Nisan 00] • OR* [Fujishima et al. 99, Nisan 00] • Recursive logical bidding languages [Boutilier & Hoos 01] • In multi-unit setting, can also use price-quantity curve bids

4. Screenshot from eMediator [Sandholm AGENTS-00, Computational Intelligence 02]

5. Multi-unit auctions: pricing rules • Auctioning multiple indistinguishable units of an item • Naive generalization of the Vickrey auction: uniform price auction • If there are m units for sale, the highest m bids win, and each bid pays the m+1st highest price • Downside with multi-unit demand: Demand reduction lie [Crampton&Ausubel 96]: • m=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 • I.e., if i wins n items, he pays the prices of the n highest losing bids • What about revenue (if market is competitive)?

6. General case of efficiency under diminishing values • VCG has efficient equilibrium. What about other mechanisms? • Model: xik is i’s signal (i.e., value) for his k’th unit. • Signals are drawn iid and support has no gaps • Assume diminishing values • Prop. [13.3 in Krishna book]. An equilibrium of a multi-unit auction where the highest m bids win is efficient iff the bidding strategies are separable across units and bidders, i.e., βik(xi)= β(xik) • Reasoning: efficiency requires xik >xir iff βik(xi) > βir(xi) • So, i’s bid on some unit cannot depend on i’s signal on another unit • And symmetry across bidders needed for same reason as in 1-object case

7. Revenue equivalence theorem (which we proved before) applies to multi-unit auctions • Again assumes that • payoffs are same at some zero type, and • the allocation rule is the same • Here it becomes a powerful tool for comparing expected revenues

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

12. In all of the curves together

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21. 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

22. 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)

23. 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

24. 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

25. 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

26. 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 – even without worrying about incentives