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– …

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)?

Multi-unit auctions: Clearing complexity

[Sandholm & Suri IJCAI-01]

In all of the curves together

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)

Screenshot from eMediator[Sandholm AGENTS-00]

Supply/demand curve bids

profit = amounts paid by bidders – amounts paid to sellersCan 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

psell pbuy

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

psell pbuy

One price for each agent ( discriminatory pricing ): greater profit

p1sell

pbuyp2sell

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