Yi Wu IBM Almaden Research Joint work with Preyas Popat

Buy cereal and milk if under 10$ Buy coffee and milk if under 7$ Buy coffee and alcohol if under 15$ How to price items to maximize profit?

 Items are aligned on a line and each buyer is interested in buying a path (consecutive items). Driver 1 Driver 2 Driver 3

Profit is 40.

Profit is 50. Loss leader

 Definition: A loss leader is a product sold at a low price (at cost or below cost) to stimulate other profitable sales.  Example of loss leader ◦ Printer and ink ◦ E-book reader and E-book ◦ Movie ticket and popcorn and drink

What if the production cost is 0 such as the highway problem?

[Balcan-Blum 06]: The maximum profit can be log n-times more when loss leaders are allowed (under either coupon or discount model).

 What kind of approximation is achievable for the item pricing problems with prices below cost allowed?

 [Balcan-Blum-Chan-Hajiaghayi-07]: “Obtaining constant factor appropriation algorithms in the coupon model for general graph vertex pricing problem and the highway problem with arbitrary valuations seems believable but very challenging.”

Positive profit pricesLoss leaders Item pricing 3-hyper graph vertex pricing 8.1-approxmiatoin APX-hard, 2-UGhard Graph vertex pricing 4-approximation 2-UGhardSuper-constant UG- hardness Highway pricingPTAS NP-hardSuper-constant UG- hardness

Passing probability is 1/q.

Completeness c = q log q.

Soundness is q.

 Real valued price function.  NP-hardness reduction  Discount model

We can not prove the soundness claim for this test.

 Unbalanced price function  Real value price function

 Lemma 1: The approximability of bipartite graph pricing is equivalent to highway problem on bipartite graph.  Lemma 2: Super-constant hardness of graph pricing also implies super-constant hardness of bipartite graph pricing.

 Given a non-bipartite instance G, we can randomly partition the graph into two parts G’ and only consider the bipartite sub-graph.  We know that for any price function, the profit change by a factor of 2 in expectation.

 Pricing loss leaders is hard even for the those tractable cases under the positive profit prices model.