1 Auctions -2 Debasis Mishra QIP Short-Term Course on Electronic Commerce Indian Institute of Science, Bangalore February 17, 2006

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Auctions -2Auctions -2

Debasis MishraDebasis Mishra

QIP Short-Term Course on Electronic QIP Short-Term Course on Electronic CommerceCommerce

Indian Institute of Science, BangaloreIndian Institute of Science, Bangalore

February 17, 2006February 17, 2006

February 17, 2006February 17, 2006QIP Course on E-Commerce : QIP Course on E-Commerce :

Auctions- 2Auctions- 2 22

OutlineOutline

Combinatorial AuctionsCombinatorial Auctions The VCG AuctionThe VCG Auction Iterative VCG AuctionsIterative VCG Auctions

– One-to-one assignmentOne-to-one assignment– General case: iBundle auctionGeneral case: iBundle auction

Winner Determination ProblemWinner Determination Problem



Combinatorial AuctionsCombinatorial Auctions

Sale of multiple items simultaneously.Sale of multiple items simultaneously. Bidders can have non-additive values on Bidders can have non-additive values on

items.items. Allows for bids on bundles of items.Allows for bids on bundles of items. An example: a seller wants to sell two items: An example: a seller wants to sell two items:

– (a) a shopping complex in Goa (a) a shopping complex in Goa – (b) a shopping complex in Mumbai.(b) a shopping complex in Mumbai.

Two buyers with values:Two buyers with values:– 5 for a, 6 for b, and 15 for a+b,5 for a, 6 for b, and 15 for a+b,– 10 for a, 8 for b, and 10 for a+b.10 for a, 8 for b, and 10 for a+b.



Applications in PracticeApplications in Practice

Spectrum wave auctions in different Spectrum wave auctions in different countries: US, European nations, and countries: US, European nations, and India too.India too.

Transportation lane auctions: London Transportation lane auctions: London bus routes, Home Depot in US.bus routes, Home Depot in US.

Airport time-slots by FAA (US).Airport time-slots by FAA (US). Sponsored search auctions by Google, Sponsored search auctions by Google,

Yahoo!, and Microsoft.Yahoo!, and Microsoft.



Efficient Mechanism DesignEfficient Mechanism Design

An economic and algorithmic viewAn economic and algorithmic view Input: “Bids” from buyers, Output: an Input: “Bids” from buyers, Output: an

allocation (assignment of items to allocation (assignment of items to buyers) and payments of buyers.buyers) and payments of buyers.

AllocationAllocation – – efficient allocationefficient allocation: one : one that maximizes the total value of that maximizes the total value of buyers or total payoff/welfare of the buyers or total payoff/welfare of the system.system.

PaymentPayment – to make money AND to – to make money AND to give an give an incentiveincentive to participate. to participate.



Why are Incentives Why are Incentives ImportantImportant

Bidders often indulge in strategizing in Bidders often indulge in strategizing in “badly” designed auctions – a fact from “badly” designed auctions – a fact from spectrum auctions in US and Europe: spectrum auctions in US and Europe: leads to low revenue and loss in efficiency.leads to low revenue and loss in efficiency.

Strategizing is not easy in combinatorial Strategizing is not easy in combinatorial auctions.auctions.

Incentive schemes makes strategies Incentive schemes makes strategies straightforward – straightforward – no need to strategize, no need to strategize, just bid in a just bid in a straightforwardstraightforward manner. manner. Leads to savings for bidders in terms of Leads to savings for bidders in terms of “auction preparation costs”.“auction preparation costs”.



The ModelThe Model

Set of items N={1,…,n}.Set of items N={1,…,n}. Set of buyers M={1,…,m}.Set of buyers M={1,…,m}. Bundles – any subset of items of N.Bundles – any subset of items of N. Buyers have values on bundles: v(i,S)Buyers have values on bundles: v(i,S)

– A1- S is a subset of T means v(i,S) <= A1- S is a subset of T means v(i,S) <= v(i,T)v(i,T)

– A2- v(i,A2- v(i,φφ)=0)=0– A3 - if i pays amount p and gets bundle A3 - if i pays amount p and gets bundle

S, his payoff is v(i,S) - pS, his payoff is v(i,S) - p



The Efficient Allocation (1 The Efficient Allocation (1 of 3)of 3)

An An allocationallocation is X: a partition of items in N is X: a partition of items in N to bundles and assignment of these to bundles and assignment of these bundles to buyers (buyers can get the bundles to buyers (buyers can get the empty bundle) – Xempty bundle) – Xii is bundle of i. is bundle of i.

Efficient allocation: one which maximizes Efficient allocation: one which maximizes ΣΣi i v(i,Xv(i,Xii).).

An economy consists of buyers and their An economy consists of buyers and their valuations – E(B) is economy with buyers valuations – E(B) is economy with buyers in B (subset of M).in B (subset of M).

E(M) is E(M) is mainmain economy and E(M-i) is a economy and E(M-i) is a marginalmarginal economy. economy.




If X is an efficient allocation of E(B), it If X is an efficient allocation of E(B), it is a disjoint partition of A but assigns is a disjoint partition of A but assigns bundles to buyers in B bundles to buyers in B only.only.

If X is an efficient allocation of E(B), If X is an efficient allocation of E(B), denote V(B)= denote V(B)= ΣΣi i v(i,Xv(i,Xii).).

For an integer programming For an integer programming formulation:formulation:– y(i,S) = 1 if i is assigned bundle S and y(i,S) = 1 if i is assigned bundle S and

zero otherwise.zero otherwise.




Formulation (Formulation (IPIP) for main economy) for main economy

V(M) = max ∑V(M) = max ∑i,S i,S v(i,S)y(i,S)v(i,S)y(i,S)

s.t.s.t.

∑∑S S y(i,S) = 1y(i,S) = 1 for all i for all i in M,in M,

∑∑ii ∑ ∑S: j in S S: j in S y(i,S) = 1y(i,S) = 1 for all j in N, for all j in N,

y(i,S) in {0,1} for all i, y(i,S) in {0,1} for all i, S.S.



The VCG Mechanism (1 of The VCG Mechanism (1 of 2)2)

The Vickrey-Clarke-Groves (VCG) The Vickrey-Clarke-Groves (VCG) mechanism is a sealed-bid auction.mechanism is a sealed-bid auction.

Allocation – Efficient allocation of main Allocation – Efficient allocation of main economy.economy.

Payment - such that payoff of buyer i is Payment - such that payoff of buyer i is V(M)-V(M-i): marginal contribution of iV(M)-V(M-i): marginal contribution of i– Payment is: v(i,XPayment is: v(i,Xii)-[V(M)-V(M-i)], where X is )-[V(M)-V(M-i)], where X is

efficient allocation of main economy.efficient allocation of main economy. Bidding one’s true value as bid is a Bidding one’s true value as bid is a

dominantdominant strategy.strategy.– In many ways, it is a unique mechanism which In many ways, it is a unique mechanism which

is efficient and has a dominant strategy.is efficient and has a dominant strategy.



The VCG Mechanism (2 of The VCG Mechanism (2 of 2)2)

Requires solving (Requires solving (IPIP) (m+1) ) (m+1) times in the worst case.times in the worst case.– ((IPIP) is difficult to solve – NP ) is difficult to solve – NP

Hard (Rothkopf et al., 1998, Hard (Rothkopf et al., 1998, Management Science).Management Science).

Requires every buyer to Requires every buyer to submit an exponential-sized submit an exponential-sized valuation function.valuation function.

Example: 3 buyers, 2 itemsExample: 3 buyers, 2 items V(1,2)=8+8=16; V(1,2)=8+8=16;

V(1)=12,V(2)=14.V(1)=12,V(2)=14.– pp11=8-[16-14]=6; p=8-[16-14]=6; p22=8-[16-=8-[16-

12]=412]=4

aa bb a+a+bb

11 88 99 1122

22 66 88 1144



An Easy Instance: An Easy Instance: The Assignment Problem (1 of 2)The Assignment Problem (1 of 2)

Every buyer is interested in at most one item.Every buyer is interested in at most one item.– Values of buyers can be written as: v(i,j) value of Values of buyers can be written as: v(i,j) value of

buyer i on item j.buyer i on item j. ((IPIP) reduces to following () reduces to following (LPALPA))

V(M) = max ∑V(M) = max ∑i,j i,j v(i,j)y(i,j)v(i,j)y(i,j)

s.t.s.t.

∑∑j j y(i,j) <= 1y(i,j) <= 1 for all i in M, for all i in M,

∑∑ii y(i,j) <= 1 y(i,j) <= 1 for all j in N, for all j in N,

y(i,j) >= 0 for all i, j.y(i,j) >= 0 for all i, j.



An Easy Instance: An Easy Instance: The Assignment Problem (2 of The Assignment Problem (2 of

2)2) Dual of (Dual of (LPALPA) is () is (DPADPA))V(M) = min ∑V(M) = min ∑ii q qi i + ∑+ ∑jj p pjj

s.t.s.t.ppjj + q + qii >= v(i,j) for all i, j, >= v(i,j) for all i, j,ppj j >= 0 for all i, q>= 0 for all i, qjj >= 0 for all j. >= 0 for all j. Leonard (1983) showed that there exists Leonard (1983) showed that there exists

an optimal solution of (an optimal solution of (DPADPA) such that p) such that pjj =VCG payoff of buyer who is assigned =VCG payoff of buyer who is assigned item j in optimal solution of (item j in optimal solution of (LPALPA) .) .– This solution is obtained by maximizing ∑This solution is obtained by maximizing ∑ii q qi i

over all possible solutions of (over all possible solutions of (DPADPA).).



Iterative AuctionsIterative Auctions

Iterative auctions – decentralized Iterative auctions – decentralized implementation of sealed-bid auctions.implementation of sealed-bid auctions.

Better Better preference elicitationpreference elicitation – buyers can – buyers can work on bounds on valuations and no work on bounds on valuations and no need to submit the entire valuation need to submit the entire valuation function.function.

More transparent – shown to generate More transparent – shown to generate more revenue and efficiency (Cramton more revenue and efficiency (Cramton 1998, Eur. Econ. Rev.).1998, Eur. Econ. Rev.).

Popular in practice – English auction more Popular in practice – English auction more popular than the Vickrey auction.popular than the Vickrey auction.



Linear Price Iterative Linear Price Iterative AuctionAuction

For the assignment problem, pFor the assignment problem, pjj denotes price on item denotes price on item j.j.

Price vector p – prices on items.Price vector p – prices on items. Demand set of buyer i at p: D(i,p)={j in N:v(i,j)- pDemand set of buyer i at p: D(i,p)={j in N:v(i,j)- pjj >= >=

v(i,k) – pv(i,k) – pk k for all k in N} – payoff maximizing items.for all k in N} – payoff maximizing items. An iterative auction (Demange et al., 1986):An iterative auction (Demange et al., 1986):

– Start from zero price vector (or a low price). Initially no Start from zero price vector (or a low price). Initially no buyer is assigned.buyer is assigned.

– A buyer bids when he is not assigned and his maximum A buyer bids when he is not assigned and his maximum payoff is more than zero. A buyer bids (truthfully) by payoff is more than zero. A buyer bids (truthfully) by increasing the price of an item in his demand set at the increasing the price of an item in his demand set at the current price by current price by εε..

– The auction stops when there is no bidding.The auction stops when there is no bidding. As As εε approaches zero, this auction implements the approaches zero, this auction implements the

VCG outcome and truthful bidding is a (ex post) Nash VCG outcome and truthful bidding is a (ex post) Nash equilibrium for buyers.equilibrium for buyers.



ExampleExample

Three buyers, two items: Three buyers, two items: v(1,1)=3, v(1,2)=4; v(1,1)=3, v(1,2)=4; v(2,1)=2, v(2,2)=5; v(2,1)=2, v(2,2)=5; v(3,1)=4,v(3,2)=4.v(3,1)=4,v(3,2)=4.

At price (0,0) buyer 1 bids on item 2 At price (0,0) buyer 1 bids on item 2 to make it (0,1). Now, buyer 3 bids on to make it (0,1). Now, buyer 3 bids on item 1 to make it (1,1). Now, buyer 2 item 1 to make it (1,1). Now, buyer 2 bids on item 2 to make it (1,2) … will bids on item 2 to make it (1,2) … will converge approximately to (3,4).converge approximately to (3,4).



Complex Price AuctionsComplex Price Auctions For general combinatorial auction settings, iterative For general combinatorial auction settings, iterative

auctions require complex prices.auctions require complex prices.– Non-linear (every bundle has a price) and non-anonymous Non-linear (every bundle has a price) and non-anonymous

(personalized prices for every buyer): p(i,S).(personalized prices for every buyer): p(i,S).– The underlying theory for such complex prices can be found The underlying theory for such complex prices can be found

in Bikhchandani and Ostroy (2002, J. of Econ. Theory).in Bikhchandani and Ostroy (2002, J. of Econ. Theory). Sometimes, such complex prices can be represented Sometimes, such complex prices can be represented

in a simple way – when items are homogeneous and in a simple way – when items are homogeneous and marginal valuesmarginal values on units/items are non-increasing on units/items are non-increasing Ausubel (2004) designs an iterative auction that Ausubel (2004) designs an iterative auction that maintains a single price but implicitly maintains maintains a single price but implicitly maintains non-linear and non-anonymous prices (Bikhchandani non-linear and non-anonymous prices (Bikhchandani and Ostroy, 2005, Games and Econ. Behavior, and Ostroy, 2005, Games and Econ. Behavior, Forthcoming)Forthcoming)



iBundle Auction (1 of 4)iBundle Auction (1 of 4) First appeared in Parkes (1999, EC’99).First appeared in Parkes (1999, EC’99).

– Maintains non-linear and non-anonymous prices.Maintains non-linear and non-anonymous prices. Given such a price vector p, demand set of buyer i Given such a price vector p, demand set of buyer i

is D(i,p) = {all bundles S: v(i,S)-p(i,S) >= v(i,T) – is D(i,p) = {all bundles S: v(i,S)-p(i,S) >= v(i,T) – p(i,T) for all bundles T}p(i,T) for all bundles T}– payoff maximizing bundles at price p.payoff maximizing bundles at price p.

Supply set of seller at price p as L(p) = {all Supply set of seller at price p as L(p) = {all allocations X: ∑allocations X: ∑i i p(i,Xp(i,Xii) >= ∑) >= ∑ii p(i,Y p(i,Yii) for all ) for all allocations Y}allocations Y}– revenue maximizing allocations.revenue maximizing allocations.

Define L*(p)={all allocations X: X in L(p) and XDefine L*(p)={all allocations X: X in L(p) and Xii in in D(i,p) or Xi = D(i,p) or Xi = φφ}: buyer compatible allocations in }: buyer compatible allocations in supply set.supply set.



iBundle Auction (2 of 4)iBundle Auction (2 of 4) Start from zero prices (or low prices).Start from zero prices (or low prices). At every iteration with price p:At every iteration with price p:

– Collect demand sets of buyers at p.Collect demand sets of buyers at p.– Find an allocation X (provisional allocation) from Find an allocation X (provisional allocation) from

L*(p)L*(p) The auction ensures that L*(p) is not empty.The auction ensures that L*(p) is not empty.

– Define losers in X as: O(X,p)={i: XDefine losers in X as: O(X,p)={i: Xii notin D(i,p)}. notin D(i,p)}.– If O(X,p) is empty go to last step. Else for every i If O(X,p) is empty go to last step. Else for every i

in O(X,p) and S in D(i,p) set p(i,S):=p(i,S) + 1 (this in O(X,p) and S in D(i,p) set p(i,S):=p(i,S) + 1 (this bid increment is for convenience) and repeat.bid increment is for convenience) and repeat.

The final allocation is final provisional The final allocation is final provisional allocation X and payment of buyer i is p(i, Xallocation X and payment of buyer i is p(i, Xii).).



iBundle Auction (3 of 4)iBundle Auction (3 of 4) An example: An example:

– v(1,a)= 4, v(1,b)=3, v(1,a+b)=6; v(1,a)= 4, v(1,b)=3, v(1,a+b)=6; – v(2,a)=2, v(2,b) = 5, v(2,a+b)=8.v(2,a)=2, v(2,b) = 5, v(2,a+b)=8.

At price (0,0,0;0,0,0): At price (0,0,0;0,0,0): – D(1,p)=D(2,p)= {a+b}. L*(p)={1 gets a+b, 2 gets D(1,p)=D(2,p)= {a+b}. L*(p)={1 gets a+b, 2 gets φφ}. }.

Next price (0,0,0;0,0,1) where D( ) are unchanged. Next price (0,0,0;0,0,1) where D( ) are unchanged. – L*(p)={2 gets a+b, 1 gets L*(p)={2 gets a+b, 1 gets φφ}. Next price (0,0,1;0,0,1). This goes }. Next price (0,0,1;0,0,1). This goes

on …on … Price reaches (0,0,2;0,0,2): Price reaches (0,0,2;0,0,2):

– D(1,p)={a,a+b}, D(2,p)={a+b}, L*(p)={1 gets a+b}. D(1,p)={a,a+b}, D(2,p)={a+b}, L*(p)={1 gets a+b}. Next price (0,0,2;0,0,3}: Next price (0,0,2;0,0,3}:

– D(1,p)={a,a+b}, D(2,p)={b,a+b}, L*(p)={2 gets a+b}D(1,p)={a,a+b}, D(2,p)={b,a+b}, L*(p)={2 gets a+b}– Next price (1,0,3;0,0,3} … Next price (1,0,3;0,0,3} …

Finally, price reaches (3,2,5;0,2,5): Finally, price reaches (3,2,5;0,2,5): – D(1,p)={a,b,a+b}, D(2,p)={b,a+b}, L*(p)={1 gets a, 2 gets b}. D(1,p)={a,b,a+b}, D(2,p)={b,a+b}, L*(p)={1 gets a, 2 gets b}. – Auction ends with 1 paying 3 and 2 paying 2.Auction ends with 1 paying 3 and 2 paying 2.– Note that these are VCG payments.Note that these are VCG payments.



iBundle Auction (4 of 4)iBundle Auction (4 of 4) The fact that final payments in the iBundle auction The fact that final payments in the iBundle auction

is VCG payment is no coincidence in the example.is VCG payment is no coincidence in the example. Ausubel and Milgrom (2002, Frontiers of Theo. Ausubel and Milgrom (2002, Frontiers of Theo.

Econ.) show that under a Econ.) show that under a submodularitysubmodularity condition condition on V( ), satisfied in the example, final payments in on V( ), satisfied in the example, final payments in iBundle will always be VCG payments.iBundle will always be VCG payments.

For valuations that do not satisfy the submodular For valuations that do not satisfy the submodular condition, we have to apply iBundle for marginal condition, we have to apply iBundle for marginal economies too and give economies too and give discountsdiscounts to buyers at the to buyers at the end (Mishra and Parkes, 2005, J. of Econ. Theory, end (Mishra and Parkes, 2005, J. of Econ. Theory, Forthcoming).Forthcoming).– These discounts are marginal contribution of a buyer to These discounts are marginal contribution of a buyer to

the revenue of the seller.the revenue of the seller.



Winner Determination Problem (1 Winner Determination Problem (1 of 2)of 2)

Computing L*(p) is called the winner Computing L*(p) is called the winner determination problem (determination problem (WDPWDP).).

max ∑max ∑i i ∑∑S in D(i,p) or S in D(i,p) or φφ p(i,S)y(i,S)p(i,S)y(i,S)

s.t.s.t.

∑∑S in D(i,p) or S in D(i,p) or φφ y(i,S) = 1y(i,S) = 1 for all i in M, for all i in M,

∑∑S: j in S S: j in S ∑∑i:S in D(i,p) i:S in D(i,p) y(i,S) = 1y(i,S) = 1 for all j in N, for all j in N,

y(i,S) in {0,1} for all i, S in D(i,p) or y(i,S) in {0,1} for all i, S in D(i,p) or φφ..



Winner Determination Problem (2 Winner Determination Problem (2 of 2)of 2)

((WDPWDP) is NP-Hard) is NP-Hard– Lot of research on quickly solving WDP Lot of research on quickly solving WDP

(Rothkopf et al. 1998, Tuomas (Rothkopf et al. 1998, Tuomas Sandholm’s papers).Sandholm’s papers).

Observe that (Observe that (WDPWDP) is a smaller ) is a smaller version of (version of (IPIP).).– In iterative auctions, we solve smaller In iterative auctions, we solve smaller

instances of multiple (instances of multiple (IPIP) instead of ) instead of solving one huge instance of (solving one huge instance of (IPIP).).



Linear Programs and Linear Programs and Iterative AuctionsIterative Auctions

Close connection exists.Close connection exists. Almost all iterative auctions can be Almost all iterative auctions can be

interpreted as a linear programming interpreted as a linear programming algorithm to solve an appropriate algorithm to solve an appropriate linear program (de Vries et al., 2005, linear program (de Vries et al., 2005, J. of Econ. Theory, Forthcoming).J. of Econ. Theory, Forthcoming).

In particular, (almost) every auction In particular, (almost) every auction in literature is either a primal-dual in literature is either a primal-dual algorithm or a subgradient algorithm.algorithm or a subgradient algorithm.



Concluding ThoughtsConcluding Thoughts Similar to ascending auctions, possible to design Similar to ascending auctions, possible to design

descending auctions that implement the VCG descending auctions that implement the VCG outcome (Mishra and Parkes, 2004a, 2004b).outcome (Mishra and Parkes, 2004a, 2004b).– Descending auctions have better preference elicitation Descending auctions have better preference elicitation

properties than ascending ones.properties than ascending ones. Combinatorial auction design is difficult due to Combinatorial auction design is difficult due to

complexity of the input.complexity of the input. Carefully handling the WDP stage should help Carefully handling the WDP stage should help

implement practical combinatorial auctions.implement practical combinatorial auctions. Besides incentives, stress should be given on Besides incentives, stress should be given on

simplicitysimplicity (unfortunately, simplicity and (unfortunately, simplicity and incentives are not compatible) – simplicity in incentives are not compatible) – simplicity in terms of prices, bidding terms of prices, bidding languages languages (how to (how to submit bids).submit bids).

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1 Auctions -2 Debasis Mishra QIP Short-Term Course on Electronic Commerce Indian Institute of Science, Bangalore February 17, 2006