Issue on the Border of Economics and Computation

Issue on the Border of Economics and Computation

Iterative Combinatorial Auctions

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Outline• Why is VCG rarely used?

• The communication complexity of combinatorial auctions.– Exponential lower bound.

• Ascending-price auctions– Walrasian equilibrium.– Substitutes valuations.– Unit-demand valuations.

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.

– Airport landing slots.– Bus routes.– Internet ads.– Privatization

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.

Combinatorial Auctions• Set M of m indivisible items• Set N of n bidders• Preferences are on subsets S – bundles – of items • Valuation function vi: 2M R

– vi(S) – bidder i’s value for bundle S– monotone: vi(S) not decreasing in S– normalized: vi() = 0

Allocation: mutually-disjoint subsets S1, S2, … Sn

Social welfare of allocation: i vi(Si)

VCG in practice• VCG seems like the ultimate solution: implements

the efficient outcome in dominant strategies.– And bidders simply need to tell the truth!

• It turns out that VCG is rarely used in practice– Except maybe for 2nd-price auctions.

• Reasons:1. Computation Complexity

2. Some economic and behavioral problems.

3. Communication Complexity

Computational complexity• As you saw last week:

– Computing the efficient allocation is NP-hard.– Even hard to approximate (better than m1/2 approximation)– Even for very simple inputs (single-minded bidders)– (We would like mechanisms to be polynomial in n,m.)

• This is a very important issue in large-scale real-life auctions.– Usually solved quite efficiently by linear-programming

solvers.– The computational issue is not considered the biggest

obstacle in combinatorial auctions.

Economic and behavior problems• VCG has some very severe drawbacks:

– Revenue non-monotonicity (adding bidders can decrease revenue).

– Vulnerable to collusion.– Vulnerable to false-name bids.– Is not budget balanced (sometimes the mechanism should

be subsidized).

• Payment structure is non-intuitive – Auction rules need to be simple

Communication Complexity• What if the auctioneer had unlimited communication

power?– Still needs to elicit the preferences of the bidders.

• In the general combinatorial auction problem: bidders have private values of exponential size.– Truthful auctions are not practical even with few dozens of

items.

• Possible solution: design some iterative auctions where bidders transmit only the relevant data.

• Methodological tool: Communication Complexity.

Communication Complexity

• We will now discuss the communication complexity in combinatorial auctions.

Communication Complexity Model

• Goal: compute a function f(x,y)• How many bits of communication are needed to

compute f(x,y) in the worst case?– For a some mechanism (“protocol”) P that computes f,

Cf(P) is the its worst-case communication cost.

– Communication complexity: min P Cf(P)

Alice Bob

Private input: y (k2 bits)

Private input:

x (k1 bits)0011010

11000

01

01001100

Iterative Communication• It is well known that iterative protocols can reduce

communication exponentially:

• Example:

Alice Bob

1

0

0

…

1

0

10010

y (n bits)

x (log(n) bits)

f = y[x]

• It is easy to see that:• Computing f in a simultaneous protocol

requires O(n) bits.• A 2-round protocol takes logn+1 bits.

Communication in CA• So maybe we can carefully design auctions that will

ask bidders for the “relevant” information and compute an efficient allocation with polynomial communication?

• Answer: no!

Every protocol that computes the optimal allocation requires exponential communication for some inputs.– We will show this even for:

• binary valuations (v(S) is either 0 or 1 for every S)• 2 bidders.

• Theorem: every protocol that finds the efficient allocation for every pair of binary valuations, must use at least bits of communication in the worst case.– is exponential in m (by Stirling’s formula).

Communication in CA

• The proof is via a standard approach in communication complexity: construction of a “fooling set”– A collection of pairs of valuations where the protocol must

behave differently for each one.

2/m

m

2/m

m

Comm. Complexity Lower Bound: proofProof:• given a binary valuation v, we define the dual

valuation v* to be v*(S) = 1 - v(SC)

– SC=M\S

• Properties of v*– Monotone.– For every S, v(S) + v*(SC)=1

• The set {v,v*}v is our fooling set, we will show that the protocol behaves differently for every such pair.

Comm. Complexity Lower Bound: proof

• Concluding the theorem from the lemma:– In efficient mechanisms: # of distinct communication

sequences is at least the number of pairs (v,v*)

– There are at least binary valuations:• For example, only count the valuations for which:

v(S)=0 |S|<m/2, v(S)=1 |S|>m/2

• There are sets of size m/2. Each valuation assigns a bit for each such set.

2/2 m

m

2/m

m

The bits transmitted on inputs (v,v*)

The bits transmitted on inputs (u,u*)

Proof (cont.): We will prove the following lemma

Lemma: In an efficient mechanism, for all binary valuations v≠u

Comm. Complexity Lower Bound: proofProof of lemma: (“cut-and-paste argument”)

Assume (v,v*) and (u,u*) have the same comm. sequence in the efficient mechanism.– Consider the pair (v,u*):– Alice behaves exactly as in (v,v*), Bob as in (u,u*).– The same holds for (v*,u).The same allocation for (v,v*),(u,u*) , (v,u*), (v*,u).

u≠v there is T such that v(T) ≠u(T). W.l.o.g. v(T)=1 and u(T)=0. v(T)+u*(TC)=2.

Let S,SC be the efficient allocation for v, u*: v(S)+u*(SC)≥2.

But we know that v(S)+v*(SC)+ u(S)+u*(SC) = 1 + 1 = 2u(S)+v*(SC)≤0 (S,SC) is not optimal for (u,v*)

Contradiction.

No hope?• This lower bound can be strengthened:

exponential communication is needed even for achieving better than m1/2 approximation to the social welfare.

• So how should we design large scale combinatorial auctions?

• Key insights:– Auctions will usually be iterative.– Auctions will be tailored to specific classes of valuations.




Demand

• 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:

Sj

jiTj

ji pSvpTv )()(

• 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.)

Walrasian Equilibrium• We would like to reach an outcome where the market clears.

– Bidder receives their demand– No excess demand or excess supply for goods.

• 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

Walrasian Equilibrium and welfare• Intuitively: In a steady state, there is no excess

demand or excess supply and prices are thus stable. Market are in competitive/Walrasian equilibrium.– How efficient are these equilibria?

• The two fundamental welfare theorems say: (1) these equilibria are efficient, (2) (roughly) they are always feasible.

– Adam Smith’s invisible hand, etc…

• Our model is for divisible goods, different than the classic models.– Welfare theorems holds nevertheless.

Efficiency of market-clearing prices• The first welfare theorem:

The allocation in a Walrasian equilibrium is efficient.

• proof: consider another allocation, T1,…,Tn

• For every bidder:

• Summing over all bidders:

n

i Tjj

n

iii

n

i Sjj

n

iii

ii

pTvpSv1111

)()(

ii Tj

jiiSj

jii pTvpSv )()(

n

i Tjj

n

iii

m

ii

n

iii

i

pTvpSv1111

)()( The price of unallocated items is 0

n

iii

n

iii TvSv

11

)()( S1,…,Sn is efficient…



• Ascending-price auctions– Walrasian equilibrium.– Simultaneous ascending auctions.– Substitutes valuations.– Unit-demand valuations.

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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Objective

• Seller’s goal: maximize efficiency– By law for FCC spectrum auctions.

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 communication volume.– Again, bidders bid only on items that turn out to be relevant.

Simultaneous Ascending Auction

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• 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:– Simultaneous– Ascending– Item prices– Stopping rule– Activity rule


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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Intuitively, the simultaneous ascending auction works as follows: (“tâtonnement ”)


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For formally proving the properties of the auction, we will need to define it more formally.

Simultaneous Ascending AuctionBefore 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 for items in Si

• pj+ε 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)

The final outcome: Allocation S1,….,Sn and prices p1,…., pn 32

Collect demand

Update provisional winners and raise prices


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• 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)

Substitutes (or “gross substitutes”)

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• The economic intuition: if a,b are substitutes and the price of a increases, then the bidder still demands b.

• Formal definition: Substitutes valuations satisfy the following condition: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)

Substitutes: examples

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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 = maximal value of an item in it.– V( ) = max{ 3, 1, 10 } = 10

• Identical goods with downward sloping values.

$3 $8 $7 $1 $10

Substitutes: examples

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Substitute valuations rule out any form of complementarities:

For example, if I am willing to pay 10 for the bundle {a,b},then I will demand both items at prices {3,3},I will demand nothing if the price of a is raised to 8.

$3 $8 $7 $1 $10


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Based on literature by Kelso and Crawford (1982), Demange, Gale and Sotomayor (1986), Gul and Stacchetti (1999), Milgrom (2000)

Proof

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The theorem follows from the following lemma:• Lemma: At every stage of the auction, and for every bidder i,

• 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. (needs the substitutes property)

• 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 taken to i.)

ii DS (*)


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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.


Walrasian equilibrium

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• We actually proved: with substitutes valuations, Walrasian equilibrium always exists.– Constructive proof: it is reached by the ascending-price

auctions.– The ascending auction can be viewed as a primal-dual

algorithm.– The auction terminates in at most steps.

• However, Walrasian equilibria do not always exist.

maxvm

Walrasian eq. does not always exist

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• Consider the following valuations:

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 2But then Bob will not demand {a,b}.

a b {a,b}Alice 2 2 2

Bob 0 0 3




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

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a b {a,b}Alice 4 4 4Bob 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 !

• Consider the following substitutes preferences:

Unit-demand preferences• However, the simultaneous ascending auction is

incentive compatible for a sub-family of substitutes preferences: unit-demand valuations.

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– Each bidder wants at most one item (but items are still non-identical)

• Non identical items: a, b, c, d, e,

• Each bidder has a value for each itemvi(a),vi(b),bi(c),..

)(max)( jvSv iSji

VCG and Walrasian equilibrium

• Actually it is known that the VCG price vector is 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)

• Theorem: for unit-demand valuations

1. The VCG outcome is a Walrasian equilibrium

2. The SSA terminates at VCG prices

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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Issue on the Border of Economics and Computation