Item Pricing for Revenue Maximization in Combinatorial Auctions

1. Item Pricing for Revenue Maximization in Combinatorial Auctions Maria-Florina Balcan

2. 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. • Harder case: customers buy everything or nothing based on • sum of prices in list. • Or could be even more complex. “Unlimited supply combinatorial auction with additive / single-minded /unit-demand/ general bidders”

3. Supermarket Pricing Problem Algorithmic • Seller knows the market well. Incentive Compatible Auction • Must be in customers’ interest (dominant strategy) to report truthfully. Online Pricing • Customers arrive one at a time, buy what they want at current prices. Seller modifies prices over time.

4. Algorithmic Problem, Single-minded Bidders [BB’06] • n item types (coffee, cups, sugar, apples), with unlimited supply of each. • m customers. • Customer i has a shopping list Liand will only shop if the total cost of items in Li is at most some amount wi • All marginal costs are 0, and we know all the (Li, wi). 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?

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

6. L R 15 25 35 15 25 5 40 A Simple 2-Approx. in the Bipartite Case • Given a multigraph G with values we on edges e. • Goal: assign prices on vertices to maximize total profit, where: Algorithm • 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. simple! Proof OPT=OPTL+OPTR

7. 5 15 10 10 40 20 30 5 A 4-Approx. for Graph Vertex Pricing • Given a multigraph G with values we on edges e. • Goal: assign prices on vertices to maximize total profit, where: Algorithm • 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. Proof simple! In expectation half of OPT’s profit is from edges with one endpoint in L and one endpoint in R.

8. 15 10 20 Algorithmic Pricing, Single-minded Bidders,k-hypergraph Problem What about lists of size · k? Algorithm • 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:

9. Algorithmic Problem, Single-minded Bidders [BB’06] Summary: • 4 approx for graph case. • O(k) approx for k-hypergraph case. • 4 approx for graph case. • O(k) approx for k-hypergraph case. • Improves the O(k2) approximation [BK’06]. Can also apply the[B-B-Hartline-M’05]reductions to obtain good truthful mechanisms. • Can be naturally adapted to the online setting. • Based on online auctions for digital goods. • See Blum, Kumar, Rudra, Wu, Soda 2003; Blum Hartline, 2005

10. Algorithmic Problem Other known results: • O(log mn) approx. by picking the best single price [GHKKKM05]. • (log n) hardness for general case [DFHS06]. • Other interesting problems: • the highway problem: a log approx [BB06], a PTAS [Grandoni, Rothvoss, SODA 2011] • pricing below cost [BBCH, WINE 2007] [Wu, ICS 2011]

11. 100$ 20$ 5$ 30$ 25$ 30$ 1$ 20$ What about the most general case?

12. General Bidders Can we say anything at all?? There exists a price a p which gives a log(m) +log (n) approximation to the total social welfare. Can extend[GHKKKM05]and get a log-factor approx for general bidders by anitem pricing. Theorem

13. General Bidders • Can extend[GHKKKM05]and get a log-factor approx for general bidders by anitem pricing. Can we do this via 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.

14. # items n0 n1 - nL - pL-1 pL p0=0 p1 p2 price Unlimited Supply, General Bidders Focus on a single customer. Analyze demand curve. • Claim 1: # is monotone non-increasing with p.

15. Unlimited Supply, General Bidders Focus on a single customer. Analyze demand curve. # items n0 n1 - nL - pL-1 pL p0=0 p1 p2 price • Claim 2: customer’s max valuation · integral of this curve.

16. Unlimited Supply, General Bidders Focus on a single customer. Analyze demand curve. # items n0 n1 - nL - pL-1 pL p0=0 p1 p2 price • Claim 2: customer’s max valuation · integral of this curve.

17. Unlimited Supply, General Bidders Focus on a single customer. Analyze demand curve. # items n0 n1 - nL - pL-1 pL p0=0 p1 p2 price • Claim 2: customer’s max valuation · integral of this curve.

18. Unlimited Supply, General Bidders Focus on a single customer. Analyze demand curve. # items n0 n1 - nL - pL-1 pL p0=0 p1 p2 price • Claim 2: customer’s max valuation · integral of this curve.

19. Unlimited Supply, General Bidders Focus on a single customer. Analyze demand curve. # items n0 n1 - nL - pL-1 pL p0=0 p1 p2 price • Claim 2: customer’s max valuation · integral of this curve.

20. Unlimited Supply, General Bidders Focus on a single customer. Analyze demand curve. # items n0 n1 - nL - 0 h/2 h/4 h price • Claim 3: random price in {h, h/2, h/4,…, h/(2n)} gets a • log(n)-factor approx.

21. Unlimited Supply, General Bidders Focus on a single customer. Analyze demand curve. # items n0 n1 - nL - 0 h/2 h/4 h price • Claim 3: random price in {h, h/2, h/4,…, h/(2n)} gets a • log(n)-factor approx.