Hardness of pricing loss leaders

Yi WuIBM Almaden Research

Joint work with Preyas Popat

Introduction

Example: supermarket pricing

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?

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.

Problem Definition

-hypergraph pricing: each buyer is interested in at most of the items.

Graph pricing: each buyer is interested in at most of the items.

Special case: -hypergraph pricing

Items are aligned on a line and each buyer is interested in buying a path (consecutive items).

Special case: highway pricing

Driver 1 Driver 2 Driver 3

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 problemPTAS [Grandoni-Rothvoss-11] NP-hard[Elbassioni-Raman-Ray-09]

Previous Work

All the previous work assumes that the profit margin is positive for every item.

Example

1

3

2

10 10

30

Optimal Positive Pricing Strategy

1

3

2

10 10

3030

10

0

Profit is 40.

Even better strategy

1

3

2

10 10

3015

-5

15

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

Loss leaders

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.

Discount model

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

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

Coupon Model

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

Profitability gap

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

Open Problem [Baclan-Blum 06]

[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.”

Make a guess:

Main 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 constant approximation under the coupon model. [Popat-W-11]

Our results:

Comparison

Positive profit prices

Loss leaders

Item pricing -approxmation-hard

3-hyper graph vertex pricing

8.1-approxmiatoinAPX-hard, 2-UGhard -hard

Graph vertex pricing

4-approximation2-UGhard Super-constant UG-

hardness

Highway pricing PTASNP-hard Super-constant UG-

hardness

Proof

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:

Item pricing: a special Max-CSP

A instance of item pricing with items indexed by

A pricing function is a function defined on

Dictator Test for item pricing

Completeness ◦ There exists some function such that for every ,

the pricing function has a good profit .

Soundness ◦ For non-dictator function, it has profit .

(c,s)-dictator Test.

[Khot-Kindler-Mossel-O’Donnell-07]:assuming the Unique Games Conjecture, it is NP-hard to get better

than -approximation.

Dictator Test for 3-hypergraph pricing

Generate and randomly. Generate such that each with probability

and random from with probability . Randomly generated a and add a equation

Hastad’s (1-Dictator Test for

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 .

Analysis of Hastad’s Test(informal proof)

Generate and randomly

Add a equation

Equivalent Test for non-dictator (1)

Generate and randomly

Add a equation

Equivalent Test for non-dictator (2)

Passing probability is 1/q.

Generate and randomly. Generate such that each with probability

and random with probability . For every Add a buyer interested in )with

budget .

The Dictator Test for 3-hypergraph pricing

For , we know that with probability we have that and Then for

The profit is then at least

Completeness

Completeness c = q log q.

Generate randomly. Add a buyer interested in with budget for

every

Soundness Analysis:Equivalent test for non-dictator (1)

Generate randomly. Add a buyer interested in with budget for

every .

Then for any , suppose , then the profit is at most

Equivalent test for non-dictator (2)

Soundness is q.

Real valued price function.

NP-hardness reduction

Discount model

Things not covered

Dictator Test for graph pricing and highway problem

Generate randomly and such that with probability and random in with probability

For every add a equation

Khot-Kindler-Mossel-O’Donnell’s Dictator Test for

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

Informal Proof KKMO (1)

Informal Proof of KKMO(2)

Generate randomly and such that with probability and random in with probability

For every add a buyer interested in with budget

A Candidate Test for graph pricing

We can not prove the soundness claim for this test.

Generate randomly and such that with probability and random in with probability

For every add a buyer interested in with budget

Dictator Test for graph pricing

Unbalanced price function

Real value price function

Thing not covered

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.

Highway problem

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

Proof of Lemma1.

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.

Proof of Lemma 2.

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

Conclusion

Getting better upper and lower bound for hypergraph pricing problem

Can we have a -dictator test for CSP of the form for

Open Problem