yi wu ibm almaden research joint work with preyas popat
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Hardness of pricing loss leaders
Yi WuIBM Almaden Research
Joint work with Preyas Popat
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Introduction
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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?
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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
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-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
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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
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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.
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Example
1
3
2
10 10
30
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Optimal Positive Pricing Strategy
1
3
2
10 10
3030
10
0
Profit is 40.
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Even better strategy
1
3
2
10 10
3015
-5
15
Profit is 50.
Loss leader
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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
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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?
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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
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[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
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What kind of approximation is achievable for the item pricing problems with prices below cost allowed?
Open Problem [Baclan-Blum 06]
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[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:
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Main Results
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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:
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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
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Proof
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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
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A instance of item pricing with items indexed by
A pricing function is a function defined on
Dictator Test for item pricing
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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.
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Dictator Test for 3-hypergraph pricing
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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
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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)
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Generate and randomly
Add a equation
Equivalent Test for non-dictator (1)
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Generate and randomly
Add a equation
Equivalent Test for non-dictator (2)
Passing probability is 1/q.
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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
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For , we know that with probability we have that and Then for
The profit is then at least
Completeness
Completeness c = q log q.
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Generate randomly. Add a buyer interested in with budget for
every
Soundness Analysis:Equivalent test for non-dictator (1)
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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.
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Real valued price function.
NP-hardness reduction
Discount model
Things not covered
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Dictator Test for graph pricing and highway problem
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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
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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)
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Informal Proof of KKMO(2)
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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.
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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
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Unbalanced price function
Real value price function
Thing not covered
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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
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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.
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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.
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Pricing loss leaders is hard even for the those tractable cases under the positive profit prices model.
Conclusion
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Getting better upper and lower bound for hypergraph pricing problem
Can we have a -dictator test for CSP of the form for
Open Problem