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: 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., βi

k(xi)= β(xi

k)

– Reasoning: efficiency requires xik > xi

r iff βik(xi) > βi

r(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

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

∑

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 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 – even

without worrying about incentives