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

Adapted from Noam NisanAdapted from Noam Nisan

What is a Combinatorial What is a Combinatorial Auction?Auction?

• 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}

The ModelThe Model• mm items for sale: items for sale: XX = { = {xx11,…,,…,xxmm}}

• nn bidders: bidders: vv11,…,,…,vvnn

Every Every vvii is a valuation of subsets of X: is a valuation of subsets of X:

vvii : 2 : 2XX R R++

vvii((S S ) = how much would I pay for ) = how much would I pay for S S XX

• Bids are handed in sealed envelops.Bids are handed in sealed envelops.• Auctioneer allocates the items among bidders Auctioneer allocates the items among bidders

trying to maximize its revenue:trying to maximize its revenue:

Find Find nn disjoint sets - disjoint sets - SS11,…,,…,SSnn XX s.t. s.t.vvii((SSii) is maximal.) is maximal.

IssuesIssues• BiddingBidding

– ExpressivenessExpressiveness– SimplicitySimplicity

• AllocationAllocation– Hardness ResultsHardness Results– Approximation AlgorithmsApproximation Algorithms

• PaymentPayment

• StrategyStrategy

BiddingBidding• No ExternalitiesNo Externalities

v v ((S S ) depends only on ) depends only on SS..

• Free DisposalFree Disposal

S S TT v v ((S S ) ) v v ((T T ))

• NormalizationNormalization

v v (() = 0) = 0

BiddingBidding• Let Let SS and and TT be disjoint item subsets. be disjoint item subsets.

We say We say SS and and TT are: are:

– ComplementaryComplementary, if:, if:

v v ((S S T T ) > ) > v v ((S S ) + ) + v v ((T T ))

– SubstitutesSubstitutes, if:, if:

v v ((S S T T ) < ) < v v ((S S ) + ) + v v ((T T ))

Bidding - ExamplesBidding - Examples• The additive valuationThe additive valuation

– v v ((S S ) = |) = |S S ||– No substitutabilities and no No substitutabilities and no

complementarities.complementarities.• The single item valuationThe single item valuation

– Want to buy just one item.Want to buy just one item.– v v ((S S ) = 1) = 1 (iff (iff SS is not empty) is not empty)– All items are substitutes of each other.All items are substitutes of each other.

• The The KK-budget valuation-budget valuation– Want to buy up to K items.Want to buy up to K items.– v v ((S S ) = min {) = min {KK, |, |S S |}|}

Bidding - ExamplesBidding - Examples• The majority valuationThe majority valuation

– Want to buy most of the items.Want to buy most of the items.– v v ((S S ) =) = 00 for |for |S S | < | < mm / 2 / 2

11 otherwiseotherwise• Symmetric valuationSymmetric valuation

– Let Let pp11,…,,…,ppnn be non-negative numbers. be non-negative numbers.v v ((S S ) = ) = jj=1…|=1…|SS| | ppjj

– Additive: Additive: ppjj=1=1– K-budget: K-budget: ppjj== 00 for for j > Kj > K

11 otherwiseotherwise– Majority: Majority: ppjj== 11 for for j j = = mm/2/2

00 otherwiseotherwise

Bidding - ExamplesBidding - Examples• Downward sloping symmetric valuationDownward sloping symmetric valuation

– A symmetric valuation withA symmetric valuation withpp11 pp22 … … ppnn 0 0

– Is viewed as the “normal” economic case.Is viewed as the “normal” economic case.

Bidding - Asymmetric Bidding - Asymmetric ExamplesExamples

• The monochromatic valuationThe monochromatic valuation– mm/2 red items, /2 red items, mm/2 blue items./2 blue items.– Want to buy items of just one color.Want to buy items of just one color.– For For SS having having kk reds and reds and ll blues, blues,

v v ((S S ) = max{) = max{kk,,l l }}

• One-of-each-kind valuationOne-of-each-kind valuation– mm/2 pairs of items./2 pairs of items.– Want to buy just one item from each pair.Want to buy just one item from each pair.– For For SS having having kk pairs and pairs and ll singletons (| singletons (|S S ||

=2=2kk++ll),),v v ((S S ) = ) = k k ++ll

Bidding LanguagesBidding Languages• A simple language:A simple language:

– Specify Specify vv explicitly as a 2 explicitly as a 2mm vector. vector.– Impractical - bids are too big.Impractical - bids are too big.

• Language must allow:Language must allow:– To express any “reasonable” valuation with To express any “reasonable” valuation with

polynomial (in polynomial (in mm) size expressions.) size expressions.– To be computationally easy: given To be computationally easy: given vv and and SS, ,

compute compute v v ((S S ) in polynomial time.) in polynomial time.• The applet-language:The applet-language:

– Specify Specify vv as a computer program. as a computer program.– Doesn’t allow efficient allocation algorithms.Doesn’t allow efficient allocation algorithms.

Basic Bidding LanguagesBasic Bidding Languages• Atomic bidsAtomic bids

– vv = ( = (SS,,pp))v v ((T T ) =) = pp if if T T SS

00 otherwiseotherwise– Can’t represent the additive valuation.Can’t represent the additive valuation.

• OR bidsOR bids

– vv = ( = (SS11,,pp11) ) OROR ( (SS22,,pp22) ) OROR … … OROR ( (SSkk,,ppkk))

– If If SSii and and SSjj are disjoint, are disjoint, v v ((SSii SSjj) = ) = ppii + + ppjj

– Can express all bids with no Can express all bids with no substitutabilities and only them.substitutabilities and only them.

Basic Bidding LanguagesBasic Bidding Languages• XOR bidsXOR bids

– vv = ( = (SS11,,pp11) ) XORXOR ( (SS22,,pp22) ) XORXOR … … XORXOR ( (SSkk,,ppkk))

– v v ((S S ) = max ) = max ppii s.t. s.t. SS SSii

– Can express all bids.Can express all bids.– The additive valuation requires 2The additive valuation requires 2mm atoms in atoms in

XOR language but only XOR language but only mm atoms in OR atoms in OR language.language.

OR-of-XOROR-of-XOR• vv = = uu11 OROR uu22 OROR … … OROR uukk

each each uuii is a XOR bid. is a XOR bid.

• The bidder is willing to obtain any number of The bidder is willing to obtain any number of uu-s for the sum of their prices.-s for the sum of their prices.

• Downward sloping bid:Downward sloping bid:[({[({xx11},},pp11) XOR … XOR ({) XOR … XOR ({xxmm},},pp11)] OR)] OR[({[({xx11},},pp22) XOR … XOR ({) XOR … XOR ({xxmm},},pp22)] OR)] OR……[({[({xx11},},ppmm) XOR … XOR ({) XOR … XOR ({xxmm},},ppmm)])]

OR-of-XOROR-of-XOR• TheoremTheorem: The monochromatic valuation : The monochromatic valuation

requires an requires an (2(2mm/2/2) size OR-of-XOR expression.) size OR-of-XOR expression.• Proof:Proof:

– W.L.O.G. every (W.L.O.G. every (SS,,pp) is monochromatic.) is monochromatic.– pp = | = |S S ||– Can’t have a blue atom in one clause and a Can’t have a blue atom in one clause and a

red atom in another.red atom in another.– All atoms must be in one XOR clause.All atoms must be in one XOR clause.– The additive valuation on The additive valuation on mm/2 red items /2 red items

requires XOR of 2requires XOR of 2mm/2/2 atoms. atoms.

XOR-of-ORXOR-of-OR• vv = = uu11 XORXOR uu22 XORXOR … … XORXOR uukk

each each uuii is an OR bid. is an OR bid.

• The bidder is willing to obtain the maximal The bidder is willing to obtain the maximal uu..

• Monochromatic bid:Monochromatic bid:((OROR over all reds) over all reds) XORXOR ( (OROR over all blues) over all blues)

• Theorem:Theorem: Fix Fix KK = = mm/2. The /2. The KK-budget -budget valuation requires an valuation requires an (2(2mm1/41/4) size XOR-of-OR ) size XOR-of-OR expression.expression.

OR/XOR FormulaeOR/XOR Formulae• Definition:Definition: Let Let vv and and uu be valuations. Then, be valuations. Then,

– (v (v XORXOR u)(S) = max{v(S),u(S)} u)(S) = max{v(S),u(S)}– (v (v OROR u)(S) = max{v(R)+u(T)|R u)(S) = max{v(R)+u(T)|RT=T=, ,

RRT=S}T=S}

• Stronger than OR-of-XOR Stronger than OR-of-XOR XOR-of-OR. E.g.: XOR-of-OR. E.g.:

vv = = (monochromatic on (monochromatic on mm/2 items) OR/2 items) OR((KK-budget on -budget on mm/2 items)/2 items)

OR Bids with Phantom ItemsOR Bids with Phantom Items• OR* bids (Fujishima et al.):OR* bids (Fujishima et al.): Each bidder Each bidder

submits an OR bid whose atoms (submits an OR bid whose atoms (SS,,pp) may ) may introduce new (phantom) items.introduce new (phantom) items.

• Phantom items are used to express Phantom items are used to express constraints, e.g.:constraints, e.g.:

((SS11,,pp11) ) XORXOR ( (SS22,,pp22) =) =((SS11{{gg},},pp11) ) OROR ( (SS22{{gg},},pp22))

• Theorem:Theorem: Any OR/XOR formula of size Any OR/XOR formula of size ss can can be rewritten as an OR* formula of size be rewritten as an OR* formula of size ss and and O(O(s s 22) phantom items.) phantom items.

The OR* LanguageThe OR* Language• Theorem:Theorem: The majority valuation requires at The majority valuation requires at

least (least (mmmm/2/2) atoms in the OR* language.) atoms in the OR* language.

• Proof:Proof:– ((SS,,pp) with ) with p p > 0 must have at least > 0 must have at least mm/2 real /2 real

items.items.– Every subset of Every subset of mm/2 real items must appear /2 real items must appear

as an atom (possibly with phantom items).as an atom (possibly with phantom items).

• Open problem:Open problem: Is OR* strictly stronger than Is OR* strictly stronger than OR/XOR?OR/XOR?

• OR* can express externalities.OR* can express externalities.

Bidding and ComputabilityBidding and Computability• Definition 1:Definition 1: A bidding language is A bidding language is polynomially polynomially

interpretableinterpretable if there exists a polynomial if there exists a polynomial algorithm receiving a bid algorithm receiving a bid bb in the language and in the language and a subset a subset SS as input, and outputs as input, and outputs b b ((SS ). ).

• Only Only AtomicAtomic and and XORXOR are polynomially are polynomially interpretable.interpretable.

• Definition 2:Definition 2: A bidding language is A bidding language is polynomially polynomially verifiableverifiable if there exists an NP algorithm if there exists an NP algorithm receiving a bid receiving a bid bb in the language, a subset in the language, a subset SS and and a proof a proof ww of a lower bound on of a lower bound on b b ((SS ). ).

AllocationAllocation

• Bids are given in OR*.Bids are given in OR*.

• Auctioneer can treat them as one OR* bid:Auctioneer can treat them as one OR* bid:{{BBii}}ii=1..=1..nn , where , where BBii=(=(SSii,,ppii) is an atomic bid.) is an atomic bid.

• Algorithmically – no difference between real Algorithmically – no difference between real and phantom items.and phantom items.

SPP - HardnessSPP - Hardness

• SPP – Set Packing ProblemSPP – Set Packing Problem

• Is equivalent to Max-Clique and Max-Is equivalent to Max-Clique and Max-Independent-Set with weighted vertices.Independent-Set with weighted vertices.

• Is approximable within O(Is approximable within O(nn/log/log22nn))

• Not approximable within Not approximable within nn1/2-1/2- for any for any >0.>0.• Not approximable within Not approximable within nn1-1- for any for any >0, >0,

unless NP=ZPP.unless NP=ZPP.

Integer ProgrammingInteger Programming

• Formalization of the allocation problem:Formalization of the allocation problem:Maximize:Maximize:

ii=1…=1…nn xxiippii

Subject to:Subject to:ii||jjSSjj xxii 1, for each 1, for each jj=1…=1…mmxxii{0,1} , for each {0,1} , for each ii=1…=1…nn

• Relaxation to linear programming:Relaxation to linear programming:xxii 0, for each 0, for each ii=1…=1…nn

Fractional AuctionsFractional Auctions

• Example: communication lines for saleExample: communication lines for sale

BB11 = ({TA-Paris, Paris-NY, P}, 10) = ({TA-Paris, Paris-NY, P}, 10)BB22 = ({TA-London, London-NY, P}, 10) = ({TA-London, London-NY, P}, 10)

BB11 is 1/3 winning and is 1/3 winning and BB22 is 2/3 winning. is 2/3 winning.

Can use Can use BB11 for a 1/3 of its bandwidth, and for a 1/3 of its bandwidth, and BB22 – for 2/3 of its bandwidth. – for 2/3 of its bandwidth.

Single Item PricesSingle Item Prices

• An allocation An allocation xx11,…,,…,xxnn is is supported by single supported by single item pricesitem prices yy11,…,,…,yynn if: if:– For every winning bid (For every winning bid (xxii=1), =1), ppii jjSSii yyjj

– For every losing bid (For every losing bid (xxii=0), =0), ppii jjSSii yyjj

• The allocation is The allocation is exactly supportedexactly supported if for every if for every winning bid, winning bid, ppii = = jjSSii yyjj

• If every item belongs to some winning bid, it is If every item belongs to some winning bid, it is a a full allocationfull allocation..

Single Item PricesSingle Item Prices

• An auction An auction admits single item pricesadmits single item prices if it has a if it has a full allocation supported by single item prices.full allocation supported by single item prices.

• TheoremTheorem: : An auction admits single item pricesAn auction admits single item prices The linear program produces {0,1} The linear program produces {0,1} solutions.solutions.

Then the supporting prices are the solutions to Then the supporting prices are the solutions to the dual linear program.the dual linear program.

Single Item Prices - ExampleSingle Item Prices - Example

• Bidder #1:Bidder #1:

• Bidder #2:Bidder #2:

({A},5) XOR ({B},6)({A,P},5) OR ({B,P},6)

({B},3)

• Bidder #1 wins Bidder #1 wins AA for 5$. for 5$.Bidder #2 wins Bidder #2 wins BB for 3$. for 3$.

• Supporting prices: Supporting prices: AA = 2$, = 2$, BB = 3$, = 3$, PP = 3$ = 3$

Cases Where LP Relaxation Is Cases Where LP Relaxation Is OptimalOptimal

• Linear Order BidsLinear Order BidsThe items can be linearly ordered, The items can be linearly ordered, GG = { = {gg11,,…,…,ggmm}, such that all bids are for sub-ranges, }, such that all bids are for sub-ranges, SS = = {{ggkk,…,,…,ggll}.}.

– Hierarchical BidsHierarchical BidsAll sets form a nested hierarchy.All sets form a nested hierarchy. Every two bids Every two bids SS, , TT are either disjoint or are either disjoint or one contains the other.one contains the other.

• OR-of-XORs of Singleton BidsOR-of-XORs of Singleton Bids– Single Item BidsSingle Item Bids– Downward Sloping Symmetric BidsDownward Sloping Symmetric Bids