Item Pricing for Revenue Maximization in Combinatorial Auctions

Maria-Florina Balcan,Carnegie Mellon University

Joint with Avrim Blum and Yishay Mansour

Outline of the Talk

• Item Pricing in Unlimited Supply Combinatorial Auctions

• General bidders.

• Item Pricing in Limited Supply Combinatorial Auctions• Bidders with subadditive valuations.

[Balcan-Blum’06]

Revenue Maximization in Combinatorial Auctions

• Single-minded bidders.

[Balcan-Blum-Mansour’07]

[Balcan-Blum-Mansour’07]

Supermarket Pricing Problem

• A supermarket trying to decide on how to price the goods.

Seller’s Goal: set prices to maximize revenue.

• Simple case: customers make separate decisions on each item.

• Or could be even more complex.

• Harder case: customers buy everything or nothing based on sum of prices in list.

““Unlimited supply combinatorial auctionUnlimited supply combinatorial auction with additive / with additive / single-minded /unit-demand/ general bidders”single-minded /unit-demand/ general bidders”

Supermarket Pricing Problem

AlgorithmicAlgorithmic

• Seller knows the market well.

Incentive Compatible AuctionIncentive Compatible Auction

• Must be in customers’ interest (dominant strategy) to report truthfully.

Online PricingOnline Pricing

Various recent results have been focused on single minded and unit demand consumers.

• Customers arrive one at a time, buy what they want at current prices. Seller modifies prices over time.

Algorithmic Problem, Single-minded Bidders [BB’06]

• n item types (coffee, cups, sugar, apples), with unlimited supply of each.

• m customers.

• All marginal costs are 0, and we know all the (Li, wi).

• Customer i has a shopping list Li and will only shop if the total cost of items in Li is at most some amount wi

What prices on the items will make you the most money?What prices on the items will make you the most money?

• Easy if all Li are of size 1.

• What happens if all Li are of size 2?

• A multigraph G with values we on edges e.

• Goal: assign prices on vertices to maximize total profit, where:

• APX hard [GHKKKM’05].

10

40

15

2030

5

10

5

Algorithmic Problem, Single-minded Bidders [BB’06]

Unlimited supply

A Simple 2-Approx. in the Bipartite Case

• Goal: assign prices on vertices to maximize total profit, where:

• Set prices in R to 0 and separately fix prices for each node on L.

• Set prices in L to 0 and separately fix prices for each node on R.

• Take the best of both options.

AlgorithmAlgorithm

• Given a multigraph G with values we on edges e.

ProofProof simple!

OPT=OPTL+OPTR

40

152535

1525

5

L R

A 4-Approx. for Graph Vertex Pricing

• Goal: assign prices on vertices to maximize total profit, where:

• Randomly partition the vertices into two sets L and R.

• Ignore the edges whose endpoints are on the same side and run the alg. for the bipartite case.

AlgorithmAlgorithm

ProofProof In expectation half of OPT’s profit

is from edges with one endpoint in L and one endpoint in R.

• Given a multigraph G with values we on edges e.

simple!

10

40

15

2030

5

10

5

Algorithmic Pricing, Single-minded Bidders,k-hypergraph Problem

What about lists of size · k?

– Put each node in L with probability 1/k, in R with probability 1 – 1/k.

– Let GOOD = set of edges with exactly one endpoint in L. Set prices in R to 0 and optimize L wrt GOOD.

• Let OPTj,e be revenue OPT makes selling item j to customer e. Let Xj,e be indicator RV for j 2 L & e 2 GOOD.

• Our expected profit at least:

AlgorithmAlgorithm

10

15

20

Algorithmic Problem, Single-minded Bidders [BB’06]

Can also apply the [B-B-Hartline-M’05] reductions to obtain good truthful mechanisms.

• 4 approx for graph case.• O(k) approx for k-hypergraph case.

Summary:

• 4 approx for graph case.• O(k) approx for k-hypergraph case.

Can be naturally adapted to the online setting.

Improves the O(k2) approximation [BK’06].

• O(log mn) approx. by picking the best single price [GHKKKM05].

Other known results:

Algorithmic Problem, Single-minded Bidders [BB’06]

(log n) hardness for general case [DFHS06].

What about the most general case?

20$

30$30$

5$

25$

20$

100$

1$

General Bidders

Can extend [GHKKKM05] and get a log-factor approx for general bidders by an item pricing.

There exists a price a p which gives a log(m) +log

(n) approximation to the total social welfare.

TheoremTheorem

Can we say anything at all??

General Bidders

Can we do this via Item Pricing?

• Can extend [GHKKKM05] and get a log-factor approx for general bidders by an item pricing.

Note: if bundle pricing is allowed, can do it easily.

– Pick a random admission fee from {1,2,4,8,…,h} to charge everyone.

– Once you get in, can get all items for free.

For any bidder, have 1/log chance of getting within factor of 2 of its max valuation.

Unlimited Supply, General BiddersFocus on a single customer. Analyze demand curve.

• Claim 1: # is monotone non-increasing with p.

# items

price

n0

p0=0 p1 p2 pL-1 pL

n1

nL -

-

Unlimited Supply, General BiddersFocus on a single customer. Analyze demand curve.

price

# items n0

p0=0 p1 p2 pL-1 pL

n1

nL -

-

• Claim 2: customer’s max valuation · integral of this curve.

Unlimited Supply, General BiddersFocus on a single customer. Analyze demand curve.

price

n0

p0=0 p1 p2 pL-1 pL

n1

nL -

-

• Claim 2: customer’s max valuation · integral of this curve.

# items

Unlimited Supply, General BiddersFocus on a single customer. Analyze demand curve.

price

n0

p0=0 p1 p2 pL-1 pL

n1

nL -

-

• Claim 2: customer’s max valuation · integral of this curve.

# items

Unlimited Supply, General BiddersFocus on a single customer. Analyze demand curve.

price

n0

p0=0 p1 p2 pL-1 pL

n1

nL -

-

• Claim 2: customer’s max valuation · integral of this curve.

# items

Unlimited Supply, General BiddersFocus on a single customer. Analyze demand curve.

price

n0

p0=0 p1 p2 pL-1 pL

n1

nL -

-

• Claim 2: customer’s max valuation · integral of this curve.

# items

Unlimited Supply, General BiddersFocus on a single customer. Analyze demand curve.

price

n0

0 h/4 h/2 h

n1

nL -

-

• Claim 3: random price in {h, h/2, h/4,…, h/(2n)} gets a log(n)-factor approx.

# items

What about the limited supply setting?

What about Limited Supply?Assume one copy of each item.

Fixed Price (p):

Set R=J. For each bidder i, in some arbitrary order:

• Let Si be the demanded set of bidder i given the following prices: p for each item in R and for each item in J\R.

• Allocate Si to bidder i and set R=R \ Si.

Goal: Profit Maximization

Assume bidders with subadditive valuations.

Limited Supply, Subadditive Valuations

There exists a single price mechanism whose profit is a approximation to the social welfare.

Can show a lower bound, for =1/4.

[DNS’06], [D’07] show that a single price mechanism provides a logarithmic approx. for social welfare in the submodular, subadditive case.

[DNS’06] show a approximation to the total welfare for bidders with general valuations.

welfare & revenue

Other known results:

welfare

A Property of Subadditive ValuationsLemma 1Lemma 1

Let (T1, …, Tm) be feasible allocation. There exists

(L1, …, Lm) and a price p such that :

(2) (L1, …, Lm) is supported at price p.

(1)

Assume vi subadditive.

Li the subset that bidder i buys in a store where he sees only Ti and every item is priced at p.

Subadditive Valuations, Limited Supply

Lemma 1Lemma 1

price p such that :

and (L1, …, Lm) is supported at price p.

Lemma 2Lemma 2

be the allocation produced by FixedPrice (p/2). Then:

Let (T1, …, Tm) be feasible allocation. 9 (L1, …, Lm) and

Assume (L1, …, Lm) is supported at p and let (S1, …, Sm)

• There exists a single price mechanism whose profit is a

TheoremTheorem

approximation to the social welfare.

Conclusions and Open Problems

Summary:

• Item Pricing mechanism for limited supply setting.• Matching upper and lower bounds.

• Better revenue maximizing mechanisms for the limited supply?

Open Problems Open Problems