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Hardness of pricing loss leaders

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Hardness of pricing loss leaders

Yi Wu

IBM Almaden Research

Joint work with PreyasPopat

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$

- 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.

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

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

Driver 2

Driver 3

Driver 1

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.

15

30

15

1

2

10

10

Loss leader

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.
- Example of loss leader
- Printer and ink
- E-book reader and E-book
- Movie ticket and popcorn and drink

- 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
[Balcan-Blum-Chan-Hajiaghayi-07]

The seller assign a profit margin to each item and have profit with the buyer interested in set

[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.”

- 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]

- 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:

- A instance of item pricing with items indexed by
- A pricing function is a function defined on

- 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.

- Generate and randomly.
- Generate such that each with probability and random from with probability .
- Randomly generated a and add a equation

- 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 .

- Generate and randomly
- Add a equation

- Generate and randomly
- Add a equation

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 .

- For , we know that with probability we have that and Then for
The profit is then at least

Completeness c = q log q.

- Generate randomly.
- Add a buyer interested in with budget for every

- Generate randomly.
- Add a buyer interested in with budget for every .
Then for any , suppose , then the profit is at most

Soundness is q.

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

- Generate randomly and such that with probability and random in with probability
- For every add a equation

- 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

- 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.

- Generate randomly and such that with probability and random in with probability
- For every add a buyer interested in with budget

- 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.

- 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

- 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.

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

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