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COS 444 Internet Auctions:

Theory and Practice

Spring 2009

Ken Steiglitz ken@cs.princeton.edu

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Multi-unit demand auctions(Ausubel & Cramton 98, Morgan 01)

• Examples: FCC spectrum, Treasury debt securities, Eurosystem: multiple, identical units

• Important questions: Efficiency (do items go to buyers who value them the most?); Pay-your-bid (discriminatory) prices vs. uniform-price; optimality of revenue

• The problem: conventional, uniform-price auctions provide incentives for demand-reduction

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Multi-unit demand auctions

Example 1: (Morgan) 2 units supply Bidder 1: capacity 2, values $10, $10 Bidder 2: capacity 1, value $8

Suppose bidders bid truthfully; rank bids: $10 bidder 1 10 bidder 1 8 bidder 2 first rejected bid, price=$8@If buyers pay this, surplus (1) = $4 revenue = $16

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Multi-unit demand auctions

Example 1: But bidder 1 can do better! Bidder 1: capacity 2, values $10, $10 Bidder 2: capacity 1, value $8

Suppose bidder 1 reduces her demand: $10 bidder 1 for her first unit 8 bidder 2 for first unit 0 bidder 1 for her 2nd unit first rej. bidIf buyers pay this, surplus (1) = $10 (larger)

surplus (2) = $ 8 (larger) revenue = $ 0! … and auction is inefficient

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Multi-unit demand auctions

• Thus,

uniform price demand reduction inefficiency

• The most obvious generalization of the Vickrey auction (winners pay first rejected bid) is not incentive compatible and not efficient

• Lots of economists got this wrong!

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Multi-unit demand auctions

• Ausubel & Cramton 98 prove, in a simplified model, that this example is not pathological:

Proposition:Proposition: There is no efficient equilibrium strategy in a uniform-price, multi-unit demand auction.

• The appropriate generalization of the Vickrey auction is the Vickrey-Clark-Groves (VCG) mechanism… it turns out to be incentive-compatible

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The VCG auction for multi-unit demand

Return to example 1: 2 units supply Bidder 1: capacity 2, values $10, $10 Bidder 2: capacity 1, value $8

Suppose bidders bid truthfully, and order bids: $10 bidder 1 10 bidder 1 8 bidder 2 Award supply to the highest bidders … How much should each bidder pay?

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The VCG auction for multi-unit demand

• Define:Define: social welfare = W (v) = maximum total value

received by agents, where v is the vector of values

• Then the VCG payment of i is defined to be

W( v-i ) − W-i (v) = welfare to others when bidder i drops out (bids 0),

minus welfare to others when i bids truthfully

= sum of highest ki rejected bids (if bidder i gets ki items) --- the “displaced” bids

= her “externality”Notice: this reduces to Vickrey for single item

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The VCG auction for multi-unit demand

Example 1:Example 1: 2 units supply

Bidder 1: capacity 2, values $10, $10

Bidder 2: capacity 1, value $8

If bidder 1 bids 0, welfare to others = $8,

and is $0 when 1 bids truthfully…

1 pays $8 for the 2 items

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The VCG auction for multi-unit demand

Example 2Example 2: 3 units supply Bidder 1: capacity 2, values $10, $10 Bidder 2: capacity 1, value $8 Bidder 3: capacity 1, value $6

$10 bidder 1 10 bidder 1 bidder 1 gets 2 items 8 bidder 2 bidder 2 gets 1 item 6 bidder 3

Welfare to others when 1 bids 0 = $14 Welfare to others when 1 bids truthfully = $8 1 pays $6 for the 2 items

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The VCG auction for multi-unit demandExample 2Example 2, con’t 3 units supply Bidder 1: capacity 2, values $10, $10 Bidder 2: capacity 1, value $8 Bidder 3: capacity 1, value $6

@$10 bidder 1 bidder 1 gets 2 items 8 bidder 2 bidder 2 gets 1 item 6 bidder 3

Welfare to others when 2 bids 0 = $26 Welfare to others when 2 bids truthfully = $20 2 pays $6 for the 1 item

(notice that revenue = $12 < $18 =3x$6 in uniform-price case, so not revenue optimal)

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Summary

VCG mechanisms are• efficient• incentive-compatible (truthful is weakly dominant) …as we’ll see• individually rational (nonnegative E[surplus])• max-revenue among all efficient mechanisms

… but not optimal revenue in general, and prices are discriminatory, “murky”

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Combinatorial auctions

• In the most general kind of single-seller auction, there are multiple copies of multiple items for sale… we thus distinguish between multi-unit and multi-item auctions.

• Typically these come up in important situations like spectrum auctions, where buyers are interested in bundles of items that interact synergistically.

• The textbook example: a seller offers “left shoe”, “right shoe”, “pair”. Clearly, the pair is worth

much more than the sum of values of each.

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Combinatorial auctions

Here’s an example of common practice in real-estate auctions [Cramton, Shoham, Steinberg 06, p.1]:

• Individual lots are auctioned off• Then packages are auctioned off• If price(package) > Σprice(lots in package),

then the package is sold as a unit; else the constituents are sold at their individual prices

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Combinatorial auctions

• The VCG mechanism generalizes to combinatorial auctions very nicely, which we now show. We follow

L.M. Ausubel & P. Milgrom, “The lovely but lonely Vickrey auction,” in Combinatorial Auctions, P. Cramton, Y. Shoham, R.Steinberg (eds.), MIT Press, Cambridge, MA, 2006.

Setup: Setup:

nnn

nnn

xselectionofvalueactualxv

xselectionofvaluereportedxv

availablegoodsofvectorx

)(

)(ˆ

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Combinatorial auctions

• To illustrate notation: Suppose there are 4 bidders. The supply vector and a feasible assignment are:

= 1 3 4 2 (supply vector)

x1 = 0 2 1 0 bidder 1 gets two of item 2, etc.

x2 = 0 0 1 1 x3 = 0 0 0 1 x4 = 1 0 1 0

x

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Combinatorial auctions

• Define the maximizing assignment x* by

• Each buyer n pays:

• αn is the max. possible welfare of others with bidder n absent, doesn’t depend on what bidder n reports

)(..)(ˆmax SCstraintsupply conxxtsxvn

nnn

n

)(ˆ)(ˆmax *m

SCnm

mm

SCnm

mn xvxvp

)(ˆ *m

SCnm

mn xv

in general relative to the reporting function of others

…thus efficient if truthful

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Combinatorial auctions

• So in VCG (truthful bidding) the bidder pays the difference between the max. possible welfare of others with n absent (αn), minus the welfare of others that results from the maximization with n present. You can think of this as the externality of bidder n, or, perhaps, the social cost of n’s presence.

)(ˆ *m

SCnm

mnn xvp

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Combinatorial auctions

• Theorem:Theorem: In a VCG mechanism truthful bidding is dominant, and truthful bidding is efficient (maximizes total welfare).

• Proof:Proof: Fix everyone else’s report, (not necessarily truthful). Denote by x* and p* the allocation and payment when bidder n reports truthfully; and by and the allocation and payment when bidder n reports .

nmmv }ˆ{

x̂ p̂

nv̂

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Combinatorial auctions• Bidder n’s surplus when she reports is

which is n’s surplus when reporting truthfully. □

nv̂

nmnm

mnnnnn xvxvpxvsurplus

)ˆ(ˆ)ˆ(ˆ)ˆ(

nnm

mmnnSC

xvxv

)(ˆ)(max

nnm

mmnn xvxv

)(ˆ)( **

**)( nnn pxv

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Combinatorial auctions• Bidder n’s surplus when she reports is

which is n’s surplus when reporting truthfully. □

nv̂

nmnm

mnnnnn xvxvpxvsurplus

)ˆ(ˆ)ˆ(ˆ)ˆ(

nnm

mmnnSC

xvxv

)(ˆ)(max

nnm

mmnn xvxv

)(ˆ)( **

**)( nnn pxv

true value

by def. of x*

by def. of p*

by def. of x̂

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Combinatorial auctions

Ausubel & Milgrom 2006 discuss the virtues of the VCG mechanism (for general combinatorial auctions) :

• Very general (constraints easily incorporated)• Truthful is a dominant strategy• Efficient• Maximum revenue among efficient mechanisms

…Sounds good, but as we’ve seen, revenue can be disastrously low!

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Combinatorial auctions

Weaknesses of VCG (for general combinatorial auctions):• Low (or even zero) revenues• Non-monotonicity of revenues as functions of no. of

bidders and amounts bid• Vulnerability to collusion of losing bidders• Vulnerability to use of multiple bidding identities by a

single bidder• Prices are discriminatory• Loses dominant-strategy property when values not

private• In general case bid expression, winner determination,

payoff calculations become computationally intractable

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NP-complete in a nutshell (1)

• We think of problems as language-acceptance questions: a problem is a set of strings that describe YES-instances, and if there is a Turing machine that accepts that string in polynomial time, the problem is “easy”, that is, in P.

• Example: is node a connected to node b in a given graph?• The class of problems NP are those problems whose YES-

instances can be checked in polynomial time.• Example: Does a given graph have a Hamilton circuit?

Easy to check, hard to find!• Cook’s theorem: All problems in NP reduce in poly. time to

Boolean Satisfiability: does a given logical expression have a satisfying truth assignment? That is, if there is a fast algorithm for recognizing satisfiable expressions, then P=NP. Any problem in NP with this property is called NP-complete.

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NP-complete in a nutshell (2)

• See the list of NP-complete problems in wikipedia for

> 3000 such problems. It is widely believed that NP-complete problems are intractable in the sense of having no poly.-time algorithms.

• The usual way to show that a problem is NP-complete is to reduce a known NP-complete problem to it. By transitivity of reduction such a problem is as hard as any in NP.

• Example: SUBSET SUM: given a finite set of integers Α

and a positive integer B, is there a subset of Α whose sum is precisely B ?

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complexity classes

P and NP may all be one class!(but everyone thinks P≠NP)

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Payment calculation in VCG is NP-complete

)(..)(ˆmax SCstraintsupply conxxtsxvn

nnn

n

Consider the step in VCG where the maximizing assignment is calculated:

Formulate this as the recognition problem: Is there an assignment that achieves a given total value B ? SUBSET SUM clearly reduced to this; just make the values equal to the set members of Α and set the target value to B .

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An example of a (real) intractable auction Here’s a real example of an auction in which winner

determination is intractable. It’s suggested by a comment on Frank Robinson’s mail-bid sale: “You can bid with a budget limit, or with alternate choices.”

A coin dealer conducts an auction by mail as follows. She sends out an illustrated catalog describing n items k customers. Each customer then returns a list of integer bids for the items; that is, the maximum amount she is willing to spend for each item. (Bids may be 0.) In addition, the customer sends an integer limit, which is a limit on the total amount of money she is willing to spend on all the items she purchases.

She wants to award items in such a way as to respect the bids, limits, and at the same time maximize the total revenue

realized. (Proof of NP-completeness of winner determination is left as an exercise.)

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Single-seller auctions