Exchanges
= markets with many buyers and many sellers
Let’s consider a 1-item 1-unit exchange first
Exchange game in class
• 1 buyer, 1 seller, 1 good
• The agents’ valuations for the good are
drawn uniformly from [0, 100]. This is
common knowledge
• The agents don’t know each others’
valuations
Does a good exchange mechanism exist ?
• E.g: Keith is selling a car to Tuomas
– Both have quasilinear utility functions
– Each party knows his valuation, but not the other’s valuation
– Probability distributions of valuations are common knowledge
• Want a mechanism that is
– Budget balanced: Keith gets what Tuomas pays
– Pareto efficient: Car changes hands if and only if vbuyer > vseller
– Individually rational: Both Keith and Tuomas get higher
expected utility by participating than not
• Thrm. Such a mechanism does not exist (even if
randomized mechanisms are allowed) [Myerson-Satterthwaite]
– This impossibility is at the heart of more general exchange
settings (NYSE, NASDAQ, combinatorial exchanges, …) !
Multi-unit auctions &
exchanges
(multiple indistinguishable units
of one item for sale)
Tuomas Sandholm
Carnegie Mellon University
Markets with multiple
indistinguishable units for sale
• Application 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
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)
Multi-unit auctions & exchanges:
Clearing complexity
[Sandholm & Suri IJCAI-01 & new draft]
Screenshot from
eMediator
[Sandholm AGENTS-00]
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