Loading in 5 sec....

Hardness of pricing loss leadersPowerPoint Presentation

Hardness of pricing loss leaders

- 93 Views
- Uploaded on

Download Presentation
## PowerPoint Slideshow about ' Hardness of pricing loss leaders' - alaina

**An Image/Link below is provided (as is) to download presentation**

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript

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.

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

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.

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

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
- Add a equation

Equivalent Test for non-dictator (2)

- Generate and randomly
- Add a equation

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 .
- For every Add a buyer interested in )with budget .

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.
- Add a buyer interested in with budget for every

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.

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
- For every add a buyer interested in with budget

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
- For every add a buyer interested in with budget

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

Download Presentation

Connecting to Server..