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