Combinatorial
Combinatorial Auctions Auctions ( ( Bidding and Allocation) Bidding and Allocation)
Adapted from Noam Nisan Adapted 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 Model The 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 trying Auctioneer allocates the items among bidders trying to maximize its revenue:
to maximize its revenue:
Find
Find nn disjoint sets - disjoint sets - SS11,…,,…,SSnn XX s.t. s.t.
vvii((SSii) is maximal.) is maximal.
Issues Issues
• BiddingBidding
– ExpressivenessExpressiveness – SimplicitySimplicity
• AllocationAllocation
– Hardness ResultsHardness Results
– Approximation AlgorithmsApproximation Algorithms
• PaymentPayment
• StrategyStrategy
Bidding Bidding
• 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
Bidding Bidding
• 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 - Examples Bidding - Examples
• The additive valuationThe additive valuation – v v ((S S ) = |) = |S S ||
– No substitutabilities and no complementarities.No substitutabilities and no 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 - Examples Bidding - 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 ) = ) = j=1…|j=1…|S| S| 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 - Examples Bidding - Examples
• Downward sloping symmetric valuationDownward sloping symmetric valuation – A symmetric valuation withA symmetric valuation with
pp11 pp22 … … ppnn 0 0
– Is viewed as the “normal” economic case.Is viewed as the “normal” economic case.
Bidding - Asymmetric Examples Bidding - Asymmetric Examples
• 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 Languages Bidding 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 Languages Basic 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 substitutabilities Can express all bids with no substitutabilities and only them.
and only them.
Basic Bidding Languages Basic 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 language. atoms in OR language.
OR-of-XOR OR-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 -s for the sum of their prices.
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-XOR OR-of-XOR
• TheoremTheorem: The monochromatic valuation requires : The monochromatic valuation requires an 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 red Can’t have a blue atom in one clause and a red atom in another.
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 2
requires XOR of 2mm/2/2 atoms. atoms.
XOR-of-OR XOR-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 valuation -budget valuation requires an
requires an (2(2mm1/41/4) size XOR-of-OR expression.) size XOR-of-OR expression.
OR/XOR Formulae OR/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=, R, RT=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 Items OR Bids with Phantom Items
• OR* bids (Fujishima et al.):OR* bids (Fujishima et al.): Each bidder submits Each bidder submits an OR bid whose atoms (
an OR bid whose atoms (SS,,pp) may introduce new ) may introduce new (phantom) items.
(phantom) items.
• Phantom items are used to express constraints, e.g.:Phantom items are used to express 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 be can be rewritten as an OR* formula of size
rewritten as an OR* formula of size ss and O( and O(s s 22) ) phantom items.
phantom items.
The OR* Language The OR* Language
• Theorem:Theorem: The majority valuation requires at least The majority valuation requires at 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 as /2 real items must appear as an atom (possibly with phantom items).
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 Computability Bidding and Computability
• Definition 1:Definition 1: A bidding language is A bidding language is polynomially polynomially interpretable
interpretable if there exists a polynomial algorithm if there exists a polynomial algorithm receiving a bid
receiving a bid bb in the language and a subset in the language and a subset SS as as input, and outputs
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 verifiable
verifiable if there exists an NP algorithm receiving if there exists an NP algorithm receiving a bid
a bid bb in the language, a subset in the language, a subset SS and a proof and a proof ww of of a lower bound on
a lower bound on b b ((SS ). ).
Allocation Allocation
• 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 and Algorithmically – no difference between real and phantom items.
phantom items.
SPP - Hardness SPP - 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, unless >0, unless NP=ZPP.
NP=ZPP.
Integer Programming Integer Programming
• Formalization of the allocation problem:Formalization of the allocation problem:
Maximize:
Maximize:
ii=1…=1…nnxxiippii
Subject to:
Subject to:
ii||jjSSjjxxii 1, for each 1, for each jj=1…=1…mm xxii{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 Auctions Fractional 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 Prices Single Item Prices
• An allocation An allocation xx11,…,,…,xxnn is is supported by single item supported by single item prices
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= = jjSSiiyyjj
• If every item belongs to some winning bid, it is a If every item belongs to some winning bid, it is a full allocation
full allocation..
Single Item Prices Single Item Prices
• An auction An auction admits single item pricesadmits single item prices if it has a full if it has a full allocation supported by single item prices.
allocation supported by single item prices.
• TheoremTheorem: :
An auction admits single item prices An auction admits single item prices
The linear program produces {0,1} solutions. The linear program produces {0,1} 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 - Example Single 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
Optimal Optimal
• Linear Order BidsLinear Order Bids
The 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 Bids
All 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