# Hardness of pricing loss leaders - PowerPoint PPT Presentation

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Hardness of pricing loss leaders. Yi Wu IBM Almaden Research Joint work with Preyas Popat. Introduction. Example: supermarket pricing. Buy coffee and alcohol if under 15\$. Buy cereal and milk if under 10\$. How to price items to maximize profit?.

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## Hardness of pricing loss leaders

Yi Wu

Joint work with PreyasPopat

### 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\$

### 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.

### 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.

### 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

### 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.

30

1

2

10

10

3

30

30

0

1

2

10

10

3

10

Profit is 40.

### Even better strategy

15

30

15

1

2

10

10

3

-5

Profit is 50.

• Definition: Aloss leaderis a product sold at a low price (at cost or below cost) to stimulate other profitable sales.

• Printer and ink

• Movie ticket and popcorn and drink

### 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?

### 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

### 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).

### Open Problem [Baclan-Blum 06]

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

### 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.”

### 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]

### 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:

### Dictator Test for item pricing

• A instance of item pricing with items indexed by

• A pricing function is a function defined on

### (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.

### 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

### 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 .

### Equivalent Test for non-dictator (1)

• Generate and randomly

### Equivalent Test for non-dictator (2)

• Generate and randomly

Passing probability is 1/q.

### The Dictator Test for 3-hypergraph pricing

• Generate and randomly.

• Generate such that each with probability and random with probability .

### Completeness

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

The profit is then at least

Completeness c = q log q.

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

• Generate randomly.

### Equivalent test for non-dictator (2)

• Generate randomly.

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

Soundness is q.

### Things not covered

• Real valued price function.

• NP-hardness reduction

• Discount model

### 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

### 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

### A Candidate Test for graph pricing

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

We can not prove the soundness claim for this test.

### Dictator Test for graph pricing

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

### Thing not covered

• Unbalanced price function

• Real value price function

### 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.

### 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

### 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.

### Conclusion

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

### Open Problem

• Getting better upper and lower bound for hypergraph pricing problem

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