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

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

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

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

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)?

General case of efficiency under diminishing values VCG has efficient equilibrium. What about other mechanisms? Model: x i k 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., β i k (x i )= β(x i k ) –Reasoning: efficiency requires x i k > x i r iff β i k (x i ) > β i r (x i ) 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

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

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

In all of the curves together

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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

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)

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

Nondiscriminatory vs. discriminatory pricing Unit price Quantity Supply of agent 1 Aggregate demand Supply of agent 2 One price for sellers, one for buyers ( nondiscriminatory pricing ): profit > 0 p sell p buy One price for each agent ( discriminatory pricing ): greater profit p1 sell p buy p2 sell

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

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

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