CombinatorialCombinatorial AuctionsAuctions((Bidding and Allocation)Bidding and Allocation)

Adapted from Noam NisanAdapted from Noam Nisan

The SettingThe Setting

• Set of Products:Set of Products:

• Each customer can bid:Each customer can bid:$700 for { $700 for { ANDAND } }

$1200 for { } $1200 for { } OROR $8 for { } $8 for { }

$6 for { } $6 for { } XORXOR $30 for { } $30 for { }

$3 for {$3 for {ANYANY 3} 3}

“Classic”:–(take-off right) AND (landing right)–(frequency A) XOR (frequency B)

•E-commerce:–chair AND sofa -- of matching colors–(machine A for 2 hours) AND (machine B for 1 hour)– XOR XOR

ExamplesExamples

ModelModel

• We assume:We assume:Each bidder c has a valuation function Each bidder c has a valuation function cc(S), (S), for any set of products S, describing precisely for any set of products S, describing precisely the price c is willing to pay for Sthe price c is willing to pay for S

• c c satisfies:satisfies:–Free disposal: S Free disposal: S T T c c (S) (S) c c (T)(T)

May satisfy additionally:May satisfy additionally:–Complementarity: Complementarity: c c (S(ST) T) c c (S)+ (S)+ c c (T)(T)

–Substitutability: Substitutability: c c (S(ST) T) c c (S)+ (S)+ c c (T)(T)

No externalities:No externalities:c c depends solely on Sdepends solely on S

IssuesIssues

• Consider only Sealed Bid AuctionsConsider only Sealed Bid Auctions• Bidding languages and their expressivenessBidding languages and their expressiveness• Allocation algorithms (maximizing total Allocation algorithms (maximizing total

efficiency)efficiency)

• Not deal with payment rules and bidders’ Not deal with payment rules and bidders’ strategiesstrategies

How Does c Communicates How Does c Communicates c c

• c sends his valuation c sends his valuation c c to auctioneer as:to auctioneer as:

– a vector of numbersa vector of numbers

Problem: Exponential sizeProblem: Exponential size

– a computer program (applet)a computer program (applet)

Problem: requires exponential number Problem: requires exponential number of accesses by any auctioneer algorithmof accesses by any auctioneer algorithm

– Using an Expressive, Efficient Bidding Using an Expressive, Efficient Bidding languagelanguage

Bidding Language:Bidding Language:RequirementsRequirements

• ExpressivenessExpressiveness– Must be expressive enough to represent Must be expressive enough to represent

every possible valuation.every possible valuation.– Representation should not be too longRepresentation should not be too long

• SimplicitySimplicity– Easy for humans to understandEasy for humans to understand– Easy for auctioneer algorithms to handleEasy for auctioneer algorithms to handle

AND, OR, and XOR bidsAND, OR, and XOR bids

• {left-sock, right-sock}:10{left-sock, right-sock}:10

• {blue-shirt}:8 XOR {red-shirt}:7{blue-shirt}:8 XOR {red-shirt}:7

• {stamp-A}:6 OR {stamp-B}:8{stamp-A}:6 OR {stamp-B}:8

General OR bids and XOR General OR bids and XOR bidsbids

• {{a,b}:7 OR {d,e}:8 OR {a,c}:4a,b}:7 OR {d,e}:8 OR {a,c}:4– {a}=0, {a, b}=7, {a, c}=4, {a, b, c}=7, {a, b, d, {a}=0, {a, b}=7, {a, c}=4, {a, b, c}=7, {a, b, d,

e}=15e}=15– Can only express valuations with no Can only express valuations with no

substitutabilities.substitutabilities.

• {a,b}:7 XOR {d,e}:8 XOR {a,c}:4{a,b}:7 XOR {d,e}:8 XOR {a,c}:4– {a}=0, {a, b}=7, {a, c}=4, {a, b, c}=7, {a, b, d, {a}=0, {a, b}=7, {a, c}=4, {a, b, c}=7, {a, b, d,

e}=8e}=8– Can express any valuationCan express any valuation– Requires exponential size to represent Requires exponential size to represent

{a}:1 OR {b}:1 OR … OR {z}:1{a}:1 OR {b}:1 OR … OR {z}:1

OR of XORs exampleOR of XORs example

• {couch}:7 XOR {chair}:5{couch}:7 XOR {chair}:5

OROR

{TV, VCR}:8 XOR {Book}:3{TV, VCR}:8 XOR {Book}:3

OR-of-XORs example 2OR-of-XORs example 2

• Downward sloping symmetric Downward sloping symmetric valuationvaluation::Any first item is valued at 9, the second Any first item is valued at 9, the second at 7, and the third at 5.at 7, and the third at 5.

{a}:9 XOR {b}:9 XOR {c}:9 XOR {d}:9{a}:9 XOR {b}:9 XOR {c}:9 XOR {d}:9

OROR

{a}:7 XOR {b}:7 XOR {c}:7 XOR {d}:7 {a}:7 XOR {b}:7 XOR {c}:7 XOR {d}:7

OROR

{a}:5 XOR {b}:5 XOR {c}:5 XOR {d}:5{a}:5 XOR {b}:5 XOR {c}:5 XOR {d}:5

XOR of ORs exampleXOR of ORs example

• The Monochromatic valuationThe Monochromatic valuation::Even numbered items are red, and odd ones Even numbered items are red, and odd ones blue. Bidder wants to stick to one color, and blue. Bidder wants to stick to one color, and values each item of that color at 1.values each item of that color at 1.

{a}:1 OR {c}:1 OR {e}:1 OR {g}:1{a}:1 OR {c}:1 OR {e}:1 OR {g}:1

XORXOR

{b}:1 OR {d}:1 OR {f}:1 OR {h}:1{b}:1 OR {d}:1 OR {f}:1 OR {h}:1

Bidding Language:Bidding Language:LimitationsLimitations

Theorem:Theorem: The downward sloping symmetric The downward sloping symmetric valuation with valuation with nn items requires exponential items requires exponential size XOR-of-OR bids.size XOR-of-OR bids.

Theorem:Theorem: The monochromatic valuation with The monochromatic valuation with nn items requires exponential size OR-of-XOR items requires exponential size OR-of-XOR bids.bids.

OR* Bidding LanguageOR* Bidding Language (Fujishima et (Fujishima et

al)al)

• Allow each bidder to introduce phantom items, Allow each bidder to introduce phantom items, and incorporate them in an OR bid.and incorporate them in an OR bid.

Example:Example: {a,z}:7 OR {b,z}:8 {a,z}:7 OR {b,z}:8 (z phantom)(z phantom)

– equivalent to (7 for a) XOR (8 for b)equivalent to (7 for a) XOR (8 for b)

Lemma:Lemma: OR* can simulate OR-of-XORs OR* can simulate OR-of-XORs

Lemma:Lemma: OR* can simulate XOR-of-ORs OR* can simulate XOR-of-ORs

AllocationAllocation

• A computational problem:A computational problem:– Input: bids Input: bids – Outputs: allocation of items to biddersOutputs: allocation of items to bidders– Difficult computational problem (NP-Difficult computational problem (NP-

complete)complete)• Existing approaches:Existing approaches:

– Very restricted bidding languages Very restricted bidding languages (Rothkopf (Rothkopf et al)et al)

– Search over allocation space Search over allocation space (Fujishima etal, (Fujishima etal, Sandholm)Sandholm)

– Fast heuristics Fast heuristics (Fujishima etal, Lehman (Fujishima etal, Lehman et al)et al)

Integer-Programming Integer-Programming FormalizationFormalization

• n items: m atomic bids:n items: m atomic bids:

• Goal:Goal:– Maximize social efficiency Maximize social efficiency

– subject to constraintssubject to constraints

m

jjj px

1

}1,0{, jxj 1,,

jSij

jxi

m1j PjSj,

ni ..1

Relaxation:produces “fractional” allocations: xj specifies fraction of bid j obtainedIf we’re lucky, the solution is 0,1

0

The Dual Linear ProblemThe Dual Linear Problem

• n items: m atomic bids:n items: m atomic bids:

• Goal:Goal:– Minimize Implicit Prices Minimize Implicit Prices

– subject to constraintssubject to constraints

n

iiy

1

0, iyi jSi

i pyjj

,

m1j PjSj,

ni ..1

The meaning of the dualThe meaning of the dual

Intuition: Intuition: yyi i is the implicit price for item is the implicit price for item ii

Definition:Definition: Allocation {xAllocation {xjj} is supported } is supported by prices {yby prices {yii} if} if

Theorem:Theorem: There exists an allocation that There exists an allocation that is supported by prices iff the LP solution is supported by prices iff the LP solution is 0,1is 0,1

When do we get 0,1 When do we get 0,1 solutions?solutions?

TheoremTheorem: in each one of the cases below, : in each one of the cases below, the LP will produce optimal 0,1 results:the LP will produce optimal 0,1 results:– Hierarchical valuationsHierarchical valuations– 1-dimensional valuations1-dimensional valuations– Downward sloping symmetric Downward sloping symmetric

valuationvaluation– OR of XORs of singletonsOR of XORs of singletons– ““independent” problems with 0,1 independent” problems with 0,1

solutionssolutions– problem with 0,1 solution + low bidsproblem with 0,1 solution + low bids