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

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”

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.

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?

A multigraph G with values w e on edges e. Goal: assign prices on vertices to maximize total profit, where: APX hard [GHKKKM’05] 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. Algorithm Given a multigraph G with values w e on edges e. Proof simple ! OPT=OPT L +OPT R LR

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 !

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

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]. Based on online auctions for digital goods. See Blum, Kumar, Rudra, Wu, Soda 2003; Blum Hartline, 2005

O(log mn) approx. by picking the best single price [GHKKKM05]. Other known results: Algorithmic Problem  (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]

What about the most general case? 20$ 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. Theorem 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 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 - -

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.

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

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

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

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

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

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