Hardness of pricing loss leaders

1. Hardness of pricing loss leaders Yi Wu IBM Almaden Research Joint work with PreyasPopat

2. Introduction

3. Example: supermarket pricing Buy coffee and alcohol if under 15$ Buy cereal and milk if under 10$ How to price items to maximize profit? Buy coffee and milk if under 7$

4. Problem Definition • Input: • items. • buyers. each of the buyer is interested in a subset of the items with budget • single minded valuation: buyer buy either all the items in if the total price is less than or buy nothing. • Algorithmic task: price item with profit margin to maximize the overall profit.

5. Special case: -hypergraph pricing • -hypergraph pricing: each buyer is interested in at most of the items. • Graph pricing: each buyer is interested in at most of the items.

7. Special case: highway pricing • Items are aligned on a line and each buyer is interested in buying a path (consecutive items). Driver 2 Driver 3 Driver 1

8. Previous Work For item pricing with items m buyers:  -approximation [Guruswami et al.] hard[Demain et al.] For -hypergraph pricing  O()-approximation [Balcan-Blum]  4-approximaiton for graph pricing (k=2) [Balcan-Blum 06] 17/16-hard [Khandekar-Kimbrel-Makarychev-Sviridenko 09], 2-hard assuming the UGC (Unique Games Conjecture) For highway problem PTAS [Grandoni-Rothvoss-11]  NP-hard[Elbassioni-Raman-Ray-09] All the previous work assumes that the profit margin is positive for every item.

9. Example 30 1 2 10 10 3

10. Optimal Positive Pricing Strategy 30 30 0 1 2 10 10 3 10 Profit is 40.

11. Even better strategy 15 30 15 1 2 10 10 Loss leader 3 -5 Profit is 50.

12. Loss leaders • Definition: Aloss leaderis 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

13. Discount model • Discount Model [Balcan-Blum-Chan-Hajiaghayi-07] The seller assign a profit margin to each item and have profit with the buyer interested in set if the buyer purchase the item. What if the production cost is 0 such as the highway problem?

14. Coupon Model • Coupon Model [Balcan-Blum-Chan-Hajiaghayi-07] The seller assign a profit margin to each item and have profit with the buyer interested in set

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

16. Open Problem [Baclan-Blum 06] • What kind of approximation is achievable for the item pricing problems with prices below cost allowed?

17. Make a guess: • [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.”

18. Main Results

19. Our results: • For 3-hypergraph pricing problem, it is NP-hard to get better than -approximation under either the coupon or discount model. [W-11, Popat-W-11] • For graph vertex pricing (i.e.,) and the highway pricing problem, it is UG-hard to get constantapproximation under the coupon model. [Popat-W-11]

20. Comparison

21. Proof

22. Item pricing: a special Max-CSP • The pricing problem is also a CSP. • Variable: • Constraint: each buyer interested in with valuation is a constraint with the following payoff function: • Discount model: • Coupon Model:

23. Dictator Test for item pricing • A instance of item pricing with items indexed by • A pricing function is a function defined on

24. (c,s)-dictator Test. • Completeness • There exists some function such that for every , the pricing function has a good profit . • Soundness • For non-dictator function, it has profit . [Khot-Kindler-Mossel-O’Donnell-07]:assuming the Unique Games Conjecture, it is NP-hard to get better than -approximation.

25. Dictator Test for 3-hypergraph pricing

26. Hastad’s (1-Dictator Test for • Generate and randomly. • Generate such that each with probability and random from with probability . • Randomly generated a and add a equation

27. Analysis of Hastad’s Test(informal proof) • Completeness: if , this will satisfy fraction of the equations. • Soundness: • Technical Lemma [Austrin-Mossel-09]: non-dictator function can not distinguish the difference between pairwise independent distribution and fully independent distribution on .

28. Equivalent Test for non-dictator (1) • Generate and randomly • Add a equation

29. Equivalent Test for non-dictator (2) • Generate and randomly • Add a equation Passing probability is 1/q.

30. The Dictator Test for 3-hypergraph pricing • Generate and randomly. • Generate such that each with probability and random with probability . • For every Add a buyer interested in )with budget .

31. Completeness • For , we know that with probability we have that and Then for The profit is then at least Completeness c = q log q.

32. Soundness Analysis:Equivalent test for non-dictator (1) • Generate randomly. • Add a buyer interested in with budget for every

33. Equivalent test for non-dictator (2) • Generate randomly. • Add a buyer interested in with budget for every . Then for any , suppose , then the profit is at most Soundness is q.

34. Things not covered • Real valued price function. • NP-hardness reduction • Discount model

35. Dictator Test for graph pricing and highway problem

36. Khot-Kindler-Mossel-O’Donnell’s Dictator Test for • Generate randomly and such that with probability and random in with probability • For every add a equation

37. Informal Proof KKMO (1) • Notation: as the the indicator function of whether . • Let us assume (without justify) that is balanced; i.e., for every • Key Technical Lemma: for any non-dictator , if , then

38. Informal Proof of KKMO(2)

39. A Candidate Test for graph pricing • Generate randomly and such that with probability and random in with probability • For every add a buyer interested in with budget We can not prove the soundness claim for this test.

40. Dictator Test for graph pricing • Generate randomly and such that with probability and random in with probability • For every add a buyer interested in with budget

41. Thing not covered • Unbalanced price function • Real value price function

42. Highway problem • 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.

43. Proof of Lemma1. • Suppose we have n segments of highway with price The constraints are of the form . • If we change the valuable to then the constraint becomes • On bipartite graph for highway problem, we can make the constraint

44. Proof of Lemma 2. • 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 2in expectation.

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

46. Open Problem • Getting better upper and lower bound for hypergraph pricing problem • Can we have a -dictator test for CSP of the form for