1. Item Pricing for Revenue Maximization in Combinatorial Auctions Maria-Florina Balcan, Carnegie Mellon University Joint with Avrim Blum and Yishay Mansour

2. 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]

3. 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 auction with additive / single-minded /unit-demand/ general bidders”

4. Supermarket Pricing Problem Algorithmic Seller knows the market well. Incentive Compatible Auction Must be in customers’ interest (dominant strategy) to report truthfully. Online 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.

5. 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 (L i, w i ). Customer i has a shopping list L i and will only shop if the total cost of items in L i is at most some amount w i What prices on the items will make you the most money? Easy if all L i are of size 1. What happens if all L i are of size 2?

6. A multigraph G with values w e on edges e. Goal: assign prices on vertices to maximize total profit, where: APX hard [GHKKKM’05]. 10 40 15 20 30 5 10 5 Algorithmic Problem, Single-minded Bidders [BB’06] Unlimited supply

7. 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. Algorithm Given a multigraph G with values w e on edges e. Proof simple ! OPT=OPT L +OPT R 40 15 25 35 15 25 5 LR

8. 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. Algorithm Proof 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 w e on edges e. simple ! 10 40 15 20 30 5 10 5

9. 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 OPT j,e be revenue OPT makes selling item j to customer e. Let X j,e be indicator RV for j 2 L & e 2 GOOD. Our expected profit at least: Algorithm 10 15 20

10. 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(k 2 ) approximation [BK’06].

11. 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].

12. What about the most general case? 20$ 30$ 5$ 25$ 20$ 100$ 1$

13. 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. Theorem Can we say anything at all??

14. 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.

15. Unlimited Supply, General Bidders Focus on a single customer. Analyze demand curve. Claim 1: # is monotone non-increasing with p. # items price n0n0 p 0 =0 p1p1 p2p2 p L-1 pLpL n1n1 nLnL - -

16. Unlimited Supply, General Bidders Focus on a single customer. Analyze demand curve. price # items n0n0 p 0 =0 p1p1 p2p2 p L-1 pLpL n1n1 nLnL - - Claim 2: customer’s max valuation · integral of this curve.

17. Unlimited Supply, General Bidders Focus on a single customer. Analyze demand curve. price n0n0 p 0 =0 p1p1 p2p2 p L-1 pLpL n1n1 nLnL - - Claim 2: customer’s max valuation · integral of this curve. # items

18. Unlimited Supply, General Bidders Focus on a single customer. Analyze demand curve. price n0n0 p 0 =0 p1p1 p2p2 p L-1 pLpL n1n1 nLnL - - Claim 2: customer’s max valuation · integral of this curve. # items

19. Unlimited Supply, General Bidders Focus on a single customer. Analyze demand curve. price n0n0 p 0 =0 p1p1 p2p2 p L-1 pLpL n1n1 nLnL - - Claim 2: customer’s max valuation · integral of this curve. # items

20. Unlimited Supply, General Bidders Focus on a single customer. Analyze demand curve. price n0n0 p 0 =0 p1p1 p2p2 p L-1 pLpL n1n1 nLnL - - Claim 2: customer’s max valuation · integral of this curve. # items

21. Unlimited Supply, General Bidders Focus on a single customer. Analyze demand curve. price n0n0 0 h/4 h/2 h n1n1 nLnL - - Claim 3: random price in {h, h/2, h/4,…, h/(2n)} gets a log(n)-factor approx. # items

22. Unlimited Supply, General Bidders Focus on a single customer. Analyze demand curve. price n0n0 0 h/4 h/2 h n1n1 nLnL - - Claim 3: random price in {h, h/2, h/4,…, h/(2n)} gets a log(n)-factor approx. # items

23. What about the limited supply setting?

24. 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 S i 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 S i to bidder i and set R=R \ S i. Goal: Profit Maximization Assume bidders with subadditive valuations.

25. 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

26. A Property of Subadditive Valuations Lemma 1 Let (T 1, …, T m ) be feasible allocation. There exists (L 1, …, L m ) and a price p such that : (2) (L 1, …, L m ) is supported at price p. (1) Assume v i subadditive. L i the subset that bidder i buys in a store where he sees only T i and every item is priced at p.

27. Subadditive Valuations, Limited Supply Lemma 1 price p such that : and (L 1, …, L m ) is supported at price p. Lemma 2 be the allocation produced by FixedPrice (p/2). Then: Let (T 1, …, T m ) be feasible allocation. 9 (L 1, …, L m ) and Assume (L 1, …, L m ) is supported at p and let (S 1, …, S m ) There exists a single price mechanism whose profit is a Theorem approximation to the social welfare.

28. 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