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
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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
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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
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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., 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
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 1unit case cannot obtain all of
• Pareto efficiency
• Budget balance
• Individual rationality (participation)
Supply/demand curve bids
Quantity
Aggregate supply
Aggregate demand
profit
psell
pbuy
Unit price
profit = amounts paid by bidders – amounts paid to sellers
Can be divided between buyers, sellers & market maker
One price for everyone (“classic partial equilibrium”):
profit = 0
One price for sellers, one for buyers ( nondiscriminatory pricing ):
profit > 0
Nondiscriminatory vs.
discriminatory pricing
Aggregate demand
Quantity
Supply of agent 1
Supply of agent 2
psell pbuy
Unit price
p2sell
pbuy
p1sell
One price for sellers, one for buyers
( nondiscriminatory pricing ):
profit > 0
One price for each agent
( discriminatory pricing ):
greater profit
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