Auction Theory
Class 8 – Multi-unit auctions: part 1
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Outline
• Multi-unit auctions: goals and challenges
• Reminder: Walrasian equilibria
• The simultaneous ascending auction
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Definition
Efficiency
incentives
Spectrum auction examples
• Weaknesses of the Simultaneous Ascending Auction
The art of auction design
• Mechanism design with multiple items is not well
understood:
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Multi-parameter mechanism design
Computational and communication complexity
Complex incentive structure
Bounded rationality of agents
• Still an art, more than science.
• But, based on theoretical and empirical knowledge.
Multiple items
• Goal: design auctions that allow for multiple
kinds of items
– Items are non identical.
– Also known as “Combinatorial Auctions”
• Examples:
– Spectrum licenses.
– Financial assets.
– Transportation of
packages.
– Advertising campaigns
– Industrial equipment.
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–
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Airport landing slots.
Bus routes.
Internet ads.
Privatization
Leading Application: Spectrum Auctions
• Adopted by the FCC in 1994.
– Revolutionized the sale of spectrum.
– Licenses are sold for specific frequencies
in specific geographic areas.
• Has been used since all over the
world.
– Also in UK, Germany, Netherlands,
Canada, The Pacific, India, etc…
• The basis of auctions for complex
resource allocation problems.
– Trigger for research
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Combinatorial preferences
• Why shouldn’t we auction each item separately?
• Auctioning each item alone ignores:
– complements:
v(TV) + v(VCR) < v(TV+VCR)
– Substitutes:
v(TV Toshiba) + v(TV Sony) > v(both TVs)
• Bidding for packages (or bundles) may increase the
efficiency of the auction.
Desired properties
• What would be desirable properties of such auctions?
– Auction finds “market clearing” prices.
– In equilibrium...
– Auction rules are simple.
– Robust to collusion, false-name bids, timing manipulations,
etc.
– Tractable: ends in reasonable time, bidders can compute
their equilibrium strategies.
• Bidders should not be expected to process or to report large amounts
of data.
Challenges
• Sealed-bid auctions?
– Problematic. Bidder preferences may be complex.
• Market-clearing prices may not exist
– Especially when items are complements.
• Bidders do not always know their preference…
– Determining the preferences is often costly.
• Do bidders know the equilibrium strategy in the
complex auction?
Computational complexity
• The complex structure of multi-item auctions
introduces technical problems:
– Complexity of winner determination: hard for computers to
compute the optimal allocation in a reasonable time
• Such computational problems are called NP-hard.
• Prevents, for example, the use of VCG in many settings.
– Communication complexity: Private values to big to
communicate.
– Strategic complexity: bidders need to act quickly. Even if
they are fully rational, computation (computer/ consultants
etc) may be too slow to find the optimal strategy.
• Solutions must take other considerations into account
(incentive, for example)
– Interdisciplinary field.
Outline
• Multi-unit auctions: goals and challenges
• Reminder: Walrasian equilibria
• The simultaneous ascending auction
–
–
–
–
Definition
Efficiency
incentives
Spectrum auction examples
• Weaknesses of the Simultaneous Ascending Auction
Demand
• Bidders observe prices and determine their demand
• Per-item prices are announced
• Bidder decide what is the bundle that maximizes their
surplus under these prices
• (we all compute our demand. For example, when going to
the supermarket.)
• Formally:
For a bidder i, and prices p1,…,pn we say that the
bundle T is a demand of i if for every other bundle S:
vi (T )   p j  vi ( S )   p j
jT
jS
Walrasian Equilibrium
• We would like to reach an outcome where the market clears.
• A Walrasian equilibrium is an allocation S1,…,Sn and
item prices p1,…,pn such that:
– Si is the demand of bidder i under the prices p1,…,pm
– For any item j that is not allocated (not in S1,…,Sn) we have
pj=0
Efficiency of market-clearing prices
• The first welfare theorem:
The allocation in a Walrasian equilibrium is efficient.
•
proof: consider another allocation, T1,…,Tn
vi (Si )   p j  vi (Ti )   p j
• For every bidder:
jSi
jTi
• Summing over all bidders:
n
n
n
n
 v (S )   p   v (T )   p
i 1
i
i
n
j
i 1 jSi
m
i
i 1
n
i
i 1 jTi
n
 v (S )   p   v (T )   p
i 1
i
i
n
i 1
i
i 1
i
i 1 jTi
j
The price of
unallocated items
is 0
n
 v (S )   v (T )
i 1
i
j
i
i
i 1
i
i
S1,…,Sn is efficient…
Outline
• Multi-unit auctions: goals and challenges
• Reminder: Walrasian equilibria
• The simultaneous ascending auction
–
–
–
–
Definition
Efficiency
incentives
Spectrum auction examples
• Weaknesses of the Simultaneous Ascending Auction
Objective
• Seller’s goal: maximize efficiency
– By law for FCC spectrum auctions.
Eliciting the preferences
• How can we run an auction for m items?
• Direct revelation?
– Each bidder has to report vi(S)  exponential size
– More than million values with only 20 items.
• The common solution: iterative auctions.
– Seller can define a set of questions/queries:
• How much are you willing to pay for the bundle S?
• Would you buy the bundle S for $5000?
• Ask the bidders for the demand under the current prices, and then
update the prices. “Ascending-price auctions”.
– One hopes that iterative auctions would require less
communication.
Ascending-price auctions
• Initial prices are announces.
• Prices can only go up.
Revenue considerations:
• It is widely believed that ascending auctions gain more
revenue than other methods
• Rule of thumb, but supported by empirical data and
some theory.
Why ascending-price auctions?
• Simple and intuitive for bidders
– Terminates (relatively) quickly
• “Price discovery” – directs the attention of the bidders to
the relevant items.
– No need to determine the value of the other items/bundles.
– For example:
if they see an aggressive competition on one item, which
become expensive, they may let it go.
– Easier to assemble “packages” of items.
• Decreases the amount of information broadcasted.
– Again, bidders bid only on items that turn out to be relevant.
Simultaneous Ascending Auction
• The prevalent auction for spectrum auctions.
– And other multi-item auctions (Google’s TV,
transportation, etc.)
• Suggested to the FCC for the spectrum auctions in
1994
– By Milgrom, McAfee, and Wilson
• Main ideas:
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Simultaneous
Ascending
Item prices
Stopping rule
Activity rule
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Simultaneous Ascending Auction
Intuitively, the simultaneous ascending
auction works as follows:
1. Start with zero prices.
2. Each bidder reports her favorite bundle (“demand”)
 Provisional winners are announced.
3. Price of over-demanded items is raised by ε.
4. Stop when there are no over-demanded items.
– i.e., Walrasian equilibrium is reached.
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Simultaneous Ascending Auction
For formally proving the properties of the
auction, we will need to define it more
formally.
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Simultaneous Ascending Auction
Before starting:
- pj=0 for every item j.
- Temporary bundle Si is empty.
Repeat the following process:
1. Let Di be a demand of bidder i under these prices:
•
•
pj+ε for items in Si
pj for all other items
2. If for all bidders Si=Di, stop the auction.
3. Otherwise, find a bidder i with Si ≠ Di and update:
•
•
•
For items j in Di but not in Si, set pj=pj+ ε
Si=Di
for every bidder k ≠i, Sk=Sk\Di (that is, remove the items in Di)
Collect
demand
Update
provisional
winners
and raise
prices
The final outcome: Allocation S1,….,Sn and prices p1,…., pn
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Simultaneous Ascending Auction
• Theorem: when all bidders have substitutes
valuations, the simultaneous ascending auction
terminates at a Walrasian equilibrium.
– Therefore, at an efficient outcome.
– Up to an ε
Based on literature by Kelso and Crawford (1982), Demange, Gale and
Sotomayor (1986), Gul and Stacchetti (1999), Milgrom (2000)
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Substitutes
• The economic intuition: if a,b are substitutes and the
price of a increases, then the bidder still demands b.
• Formal definition:
Consider prices p1,…,pm and prices q1,…,qm where some
of the prices are increased.
For every demand Ai of bidder i under prices p1,…,pm,
bidder i demands a bundle Di under the prices q1,…,qn
that contains all the items in Ai that their prices were not
changed.
– (Think about the case where the demand is a unique bundle.
Quantifiers in the definition handle the case of multiple demand)
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Substitutes: examples
$3 $8
$7
$1
$10
The following valuations satisfy the substitutes
condition:
• Additive: there is a value attached to each item, and the
value of a bundle is the sum of the values of the item in it.
V(
) = 3 + 1 + 10 = 14
• Unit demand: bidder have value per each item, but want
at most one item
– The value of a bundle is the maximal value of an item
in it.
– V(
) = max{ 3, 1, 10 } = 10
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Proof
The theorem follows from the following lemma:
• Lemma: At every stage of the auction, and for every bidder i,
Si  Di
(*)
• Proof: By induction. (Clearly holds at the beginning.)
– Assume (*) holds, and we will show it still holds after the update.
– Consider bidder i that is selected at stage 2:
• After the updating, we have Si=Di.
• Consider bidder k ≠ i:
– Two changes can occur:
• If items were taken from Sk by i, Sk became smaller and (*)
still holds.
• If prices of items not in Sk were increased, due to the
substitutes property the only items that can be removed now
are items whose prices were increased. (But those were
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taken to i.)
Simultaneous Ascending Auction
• Theorem: when all bidders have substitutes
valuations, the simultaneous ascending auction
terminates at a Walrasian equilibrium.
– Therefore, at an efficient outcome.
– Up to an ε
We will conclude the theorem from the lemma:
• Every item with non-zero price was allocated to one of the bidders.
• From this point on, it was demanded by at least one of the bidders.
• At the end, all demanded items are allocated – Walrasian equilibrium.
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Walrasian equilibrium
• We actually proved: with substitutes valuations,
Walrasian equilibrium always exists.
– Constructive proof: it is reached by the ascendingprice auctions.
• However, Walrasian equilibria do not always exist.
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Walrasian eq. does not always exist
• Consider the following valuations:
a
b
{a,b}
Alice
2
2
2
Bob
0
0
3
Claim: there is no Walrasian equilibrium for these
preferences.
Why?
• What is the efficient allocation?
– Bob gets both items.
 Alice should demand nothing in any Walrasian equilibrium.
 Price of both items should be at least 2
 But then Bob will not demand {a,b}.
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Outline
• Multi-unit auctions: goals and challenges
• Reminder: Walrasian equilibria
• The simultaneous ascending auction
–
–
–
–
Definition
Efficiency
incentives
Spectrum auction examples
• Weaknesses of the Simultaneous Ascending Auction
Incentives
• In the analysis we assumed that bidders bid
“straightforward” bidding:
bid their actual demand at each stage.
• Will they? Is the SSA truthful?
– For substitutes preferences.
• Answer: no.
Reason: Bidders can benefit from “demand reduction”.
• Nevertheless: variants of this auction are used
successfully in many environments.
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Profitable Demand reduction
• Consider the following substitutes preferences:
a
b
{a,b}
Alice
4
4
4
Bob
5
5
10
• SSA auction terminates with a Walrasian
equilibrium: pa=4, pb=4 and Bob wins {a,b}.
– Bob’s payoff: 2
• If Bob strategically reduces his demand to a only:
auction stops at pa=pb=0. Bob wins a, Alice wins b.
– Bob’s payoff: 5 !
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Unit-demand preferences
• However, the simultaneous ascending auction is
incentive compatible for a sub-family of substitutes
preferences: unit-demand valuations.
– Each bidder wants at most one item (but items are still nonidentical)
• Non identical items: a, b, c, d, e,
• Each bidder has a value for each item
vi(a),vi(b),bi(c),..
vi (S )  maxjS vi ( j)
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Example
a
b
c
Bidder 1 10
7
4
Bidder 2 0
8
5
Bidder 3 5
5
1
• Let’s run the SSA auction with the following
valuations.
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VCG and Walrasian equilibrium
• Theorem: for unit-demand valuations
1. The VCG outcome is a Walrasian equilibrium
2. The SSA terminates at VCG prices
• Actually it is known that VCG prices are the lowest
Walrasian equilibrium for such preferences.
– That is, least preferable to the seller.
• Conclusion:
Truthful behavior is an equilibrium in the auction.
(Any deviation leads to a non-optimal outcome)
Example
a
b
c
Bidder 1 10
7
4
Bidder 2 0
8
5
Bidder 3 5
5
1
• What is the VCG outcome?
1. What is the efficient allocation?
2. VCG payment for Bidder 1?
– Without Bidder 1, others gain 13.
– With him, they gain 10.
p1=3
Similarly, p2=0, p3=3.
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Outline
• Multi-unit auctions: goals and challenges
• Reminder: Walrasian equilibria
• The simultaneous ascending auction
–
–
–
–
Definition
Efficiency
incentives
Spectrum auction examples
• Weaknesses of the Simultaneous Ascending Auction
Auction examples
• Let’s consider some positive and negative examples
for spectrum auctions.
• We will see some consideration that do not appear in
our formal model.
The UK 2000 spectrum auction
• In 2000, the UK government ran an auction for
selling spectrum licenses for 3G wireless services.
• Government declared objectives:
– Maximize efficiency
– Promote competition
– Increases value for customers and tax-payers.
• First time an auction is use to allocate spectrum in
the UK.
• Details from one of the economists involved in this auction:
Peter Cramton http://www.cramton.umd.edu/papers2000-2004/01naocramton-report-on-uk-3g-auction.pdf
The UK 2000 spectrum auction
• The first important decision by the UK government:
what to sell?
– 4 or 5 licenses?
• A license: grants right to use the spectrum +
obligation to build a network with sufficient
coverage.
• 4 incumbent 2G wireless companies in the UK at that
time
– Vodafone, Orange, 1-2-1, , British Telecom.
– (But Vodafone was trying to take over Orange.)
– Incumbents have much higher values than new entrants
(e.g., current infrastructure, 2G revenue loss is incurred if
failed to secure 3G services)
The UK 2000 spectrum auction
• Challenge: design an auction to attract other bidders
to promote competition.
• Selling 4 licenses:
– With four big incumbents, other bidders may stay out.
– Without more bidders, prices will be very low
• Selling 5 licenses:
– Is it reasonable to build 5 separate networks?
• May hurt efficiency.
– May earn less revenue: the marginal value of the 6th
bidder may be significantly lower.
• Main uncertainties:
– Who is the strongest new entrant?
– Who is the strongest incumbent (Vodafone or BT)?
The UK 2000 spectrum auction
• Government decides to sell 5 licenses:
– Bandwidths: A (35) ,B (30) ,C (25) ,D (25) ,E (25)
– License A was reserved to a new entrant.
• Auction is simpler than the US auctions.
– Bidders can win at most one license.
• SSA is efficient and incentive compatible for unit-demand bidders.
– No geographic complexity (all licenses are nationwide)
The UK 2000 spectrum auction
• The auction:
– A simultaneous ascending auction.
• All licenses are simultaneously on sale
• Bidding in rounds.
• Full transparency: After each round, all bids and bidders were
identified.
• Minimum bids: percentage of previous bids (5% at the beginning,
then 1.5%)
• Activity rule:
– bidders must either be current winners, or place bids.
– Each bidder has 3 waivers to spend.
– Bidders can also call for a recess and stop all bidding until next day, up to
twice per each bidder).
• No bid withdrawal.
The UK 2000 spectrum auction
• Outcome:
– 13 bidders in the auction (4 incumbents + 9 others,
including big international companies)
– 150 rounds.
– Winners: 1 new entrant, 4 incumbents
– Auction raised $39 Billion Dollars (!!)
• “Largest auction in History”
• Prices exceeded everyone’s expectations.
– High prices:
• mainly set by BT (losing the big license) and NTL (the strongest
losing bidder)
The UK 2000 spectrum auction
The UK 2000 spectrum auction
• Why high prices?
• Mainly set by BT (losing the big license) and NTL (the
strongest losing bidder)
• The decision to sell 5 licenses attracted new entrants
that raised the bidding levels.
– Setting aside a license for a new bidder added to this
incentive.
– Winner’s curse on license A: 9 new entrants competing for it.
• Ascending auction format:
– Not raising is a confession of inferiority. “If the license is worth
a lot for my competitor, why not to me?”
– Auction took seven weeks: enough time to go to the board
and ask for more money…
• Specific environement:
– Auction held at the peak of the internet bubble.
– Auction was a first of a series of auctions: all large companies
participated.
The UK 2000 spectrum auction
• Strategies:
– Vodafone was bidding exclusively on the B license.
• The only large license available for incumbents.
• Often use “jump bids” to express its focus on the B
license.
– Orange was bidding exclusively on E
• After B became too expensive.
The Netherlands spectrum auction
• Let’s compare the UK 3G spectrum auction outcome
to a one held in the Netherlands.
• Following the auction in the UK, the government
decided to sell 5 licenses.
• Incumbents in the Netherlands:
– Five 2G operators
– Bidding partnership is allowed.
• Outcome?
The Netherlands spectrum auction
• Outcome:
– Before the auction started, all major international
firms had a partnership with a local incumbent.
– Only one potential new entrant was left
(“Versatel”)
– On the first day, one big (joint) player sent Versatel
a letter: “you can’t win, you should drop out
immediately”.
– Soon after, Versatel drops out.
– Revenue: 3 Billion Euros. (under UK prices,
estimated revenue should have been 10 Billion).
• A conclusion: sell more licenses than the number of
incumbents.
The Netherlands spectrum auction
• One conclusion: sell more licenses than the number
of incumbents.
Outline
• Multi-unit auctions: goals and challenges
• Reminder: Walrasian equilibria
• The simultaneous ascending auction
–
–
–
–
Definition
Efficiency
incentives
Spectrum auction examples
• Weaknesses of the Simultaneous Ascending Auction
Complementarities
• This auction is problematic in the presence of
complementarities.
• Among the reasons:
– Market clearing price do not exist.
– Exposure problems.
• What should one do with complementarities?
– A problem. Not well understood.
– Auction uses bundle prices. Or combinations of item and
bundle prices.
– More complex….
The exposure problem
• bidders may have to make committing bids early in
the auction, when they are uncertain about the
eventual cost of the complete package.
• Think about a new entrant to the spectrum market
– Needs a certain minimal mass of spectrum to begin.
– might end up with few very expensive spectrum, but not
enough licenses.
– Resell is not always possible
• (opponents might have exhausted their budgets, legal problem etc.)
– Bidders fear of being “exposed” to losses, and this lead to
conservative bidding and inefficiencies.
SSA: weaknesses
• Weaknesses of the simultaneous ascending
auction:
– The exposure problem
• In general, limited expressiveness in case of complementarities.
– Incentives for demand reduction
– Complex strategies for bidders
– Tacit collusion
– Parking
• We will see some alternative auctions in next class
Summary
• Variants of the simultaneous ascending auction
have been widely in use in the past 20 years.
– Mainly in spectrum auctions, also in advertising allocation
• When the bidders have substitute preferences:
ends up with market clearing prices ( efficient
allocation)
– In the more restricted case of unit demand: also end up
with VCG payments.
• More complex solutions are needed in the presence
of complementarities.
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