Pricing Strategies for Social Networks - under Additive Influence Model
Transcript of Pricing Strategies for Social Networks - under Additive Influence Model
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Pricing Strategies for Social Networksunder Additive Influence Model
Paris Siminelakisjoint work with Dimitris Fotakis
School of Electrical and Computer Engineering at the National TechnicalUniversity of Athens
ERICE 2011
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Social Structure Today
Figure: World Map and Friendship network of Facebook. Visualizing Facebook Friends, Paul Butler
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Network Dynamics
How can we utilize the knowledge of the Social Graph?
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Influence Maximization
Domingos and Richardson[Mining the Network Value of
Customers ’01].
How can we select a limited number of people to adoptour “product”, so as to maximize the spread of it?
Kempe,Kleinberg,Tardos[Maximizing the spread of Influence ’03]
formalized this question in algorithmic terms under theInfluence Maximization problem.
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Threshold Phenomena
Figure: Scene from the Riots in Athens in 2009
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Threshold Phenomena
Figure: Scene from one of Obama’s Speaches
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What about Price?
Figure: The New York Stock Exchange
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Pricing Strategies
Hartline, Mirrokni, Sundararajan[Optimal Marketing Strategies over Social Networks, WWW’08]
Singler Seller promoting Single Product
Influence Model with only Positive Externalities
Myopic Buyers
Marketing Strategy
A marketing strategy identifies which buyer to visit next and theprice to offer the product. The buyer can accept or reject the offer.
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Uniform Additive Model
Figure: Node v accepts an offer of value (1− r)·(w1 + w2 + w3) withprobability r .
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Example
Figure: Social Network with Buyers(Nodes) and Influence(Edges)between them.
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Example
Figure: The first node visited is offered the product for free(p = 1). Heaccepts the offer paying 0.
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Example
Figure: The second node is offered the price 2/3(p = 1/3). He rejectsthe offer. The third node is offered the product for free.
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Example
Figure: In general if the total influence∑
j∈A wji for a node is Mi,A. Heaccepts an offer of (1− r)·Mi,A with probability r .
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Example
Figure: We have a permutation π of buyers and vector of prices xoffered, which correspond to probabilities of accepting p.
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Expected Revenue
Let Ai be the set of buyers that have accepted the offer wheni is considered. We offer him the price xi s.t. he accepts withprobability pi :
xi (pi ,Ai ) = (1− pi )∑j∈Ai
wji
The expected revenue from node i is:
E[Ri ] = pi · (1− pi )E[Mi ,A] = pi · (1− pi )∑
j :π(j)<π(i)
pjwji
Consider a marketing strategy (π,p). The total expectedrevenue is:
R(π,p) =∑i
pi (1− pi )(∑
j :π(j)<π(i)
pjwji )
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Expected Revenue
We can rewrite the expected revenue in terms of edges:
R(π,p) =∑
(i ,j)∈E :π(i)<π(j)
wij · pipj(1− pj)
Optimize this function in the discrete space of permutationsand the continuous space of probabilities p ∈ [0.5, 1]n.
We generally aim to obtain a large fraction of all edges aspossible. A trivial upper bound would be:
Rupper =∑i>j
wij ·1
4(undirected)
Rupper =∑i>j
max{wij ,wji} ·1
4(directed)
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Revenue Maximization
Revenue Maximization
Given a weighted graph G (V ,E ) and an influence model theproblem of Revenue Maximization consists of designing amarketing strategy that maximizes the expected revenue.
Hartline et.al showed for a very simple case (linear, determenistic) theproblem is NP-Hard using a reduction from Maximum Acyclic SubgraphProblem.
They provided an optimal algorithm for a fully symmetric case based onDynamic Programming(complete graph, identical weights).
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Influence and Exploit
Motivated by hardness results Hartline et.al considered a class ofapproximation strategies called Influence and Exploit:
Influence Step: Select a set I ⊂ V of buyers and offer theproduct for free.
Exploit Step: Visit buyers in some order and offer theoptimal myopic price based on the distribution Fi ,S .
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Approximation with IE
The strategy they proposed constructed the influence set bysampling randomly vertices with probability q.
For the Uniform Additive Model this strategy yields a 23
approximation algorithm.
For general monotone and sub-modular valuation functions1/3(deterministic) and 2/5(randomized) approximationalgorithm for OPT-IE using results from Local Search.
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Questions and Answers
Question:Consider models where the seller doesn’t visit buyers ina sequence but posts prices.
Arthur et.al[WINE’09] only recommendations about products cascadethrough the network from an initial seed of early adopters.
Akhlaghpour et.al[WINE’10] considered two variants of a posted pricesetting and provided approximation algorithms.
Anari et.al[WINE’10] posted price with historical externalities and study
equilibria.
Bimpikis et.al[WINE’10] similar setting with divisible good andpositive externalities.
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Questions and Answers
Q: Are there intelligent ways to construct the sequence rather thanrandom orderings?
Q: Are there strategies that yield higher revenue?
Q: Are there better approximation algorithms for the OptimalInfluence and Exploit strategy for special cases of the generalmodel?
Our Work
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Why does Influence and Exploit work?
Consider the simple 1/2 approximation algorithm for Max-Cut whereevery edge is cut with probability 1/2.
Influence and Exploit is in fact the same algorithm with the differencethat 1/4 of the edges in the Exploit Set are also cut. Therefore, oneadjusts the sampling probability q to strike balance between the two cuts.
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Why does Influence and Exploit work?
Consider the simple 1/2 approximation algorithm for Max-Cut whereevery edge is cut with probability 1/2.
Influence and Exploit is in fact the same algorithm with the differencethat 1/4 of the edges in the Exploit Set are also cut. Therefore, oneadjusts the sampling probability q to strike balance between the two cuts.
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Influence and Exploit
Limitations:
Effectively Disregard Network Structure
‘‘Naive” Upper Bounds
Questions:
What is the true cost of restricting ourselves to two pricingclasses?
How can we utilize network structure to design more efficientIE strategies?
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Marketing Strategies
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Marketing Strategies
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Marketing Strategies
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Marketing Strategies
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Marketing Strategies
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Sequence
Directed Graphs:
Fixed Prices ⇒ Maximum Acyclic Subgraph Problem.
Random Ordering ⇒ 0.5-approximation for MAS.
Guruswami et.al.(FOCS’08) ⇒ UGC-Hard to do better ↑.Undirected Graphs:
Hardness?
Fixed Prices ⇒ Find the optimal ordering.
Focus on Pricing
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Marketing Strategies
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Marketing Strategies
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Pricing
Hardness? Approximation?
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Influence and Exploit
Hardness of finding an Optimal IE strategy?
Limits of IE in approximating the optimal revenue?
Approximation Ratio?
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Influence and Exploit
Hardness of finding an Optimal IE strategy?
Limits of IE in approximating the optimal revenue?
Approximation Ratio?
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Influence and Exploit
Hardness of finding an Optimal IE strategy?
Limits of IE in approximating the optimal revenue?
Approximation Ratio?
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Influence and Exploit
Hardness of finding an Optimal IE strategy?
Limits of IE in approximating the optimal revenue?
Approximation Ratio?
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NP-Hardness of Optimal IE
Reduction from monotone One-in-Three SAT[Schaefer, STOC’78]
m-clauses with at most three literals(3-SAT).
All variables are without negation(monotone).
Exactly one true literal in every clause(One-in-Three).
Satisfying Assignment ⇔ Revenue of IE ≥ m · p(1− p) · (2 + p)
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NP-Hardness for Undirected Case
Extend Hardness proof when allowing any price.
One direction is trivial(One-in-Three SAT ⇒ Rev ≥ f (m, p))
The other isn’t ⇒ Strong Coupling between DecisionVariables
Solution: Decouple the clauses ⇒ extra decision variables
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Optimal Influence and Exploit
Even if we could allocate buyers optimally into two pricing classeshow close to the optimal revenue?(NP-Hard, Naive Upper Bounds)
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Optimal Influence and Exploit
Even if we could allocate buyers optimally into two pricing classeshow close to the optimal revenue?(NP-Hard, Naive Upper Bounds)
Local Analysis
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Randomized Rounding
Assume an optimal marketing strategy (π,p) with revenue:∑i>j
wijpipj(1− pj)
Assign vertices in Influence set by randomized rounding of theprobabilities. For a vertex i with probability p we have:
I (p) = A(p)(p − 1/2), E (p) = 1− I (p)
The expected revenue w.r.t. the rounding procedure from anedge wij(w .l .o.g . pi = x > pj = y) is:
E[R(wij)] =1
4[I (x)E (y) + I (y)E (x) +
1
2E (x)E (y)]
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Lower Bounds for OPT-IE
By linearity of expectation this analysis is valid for every edge.
Obtain lower bound by comparing term-wise the two revenuesand looking for the worst case.
minx ,y
αsym(A) =
[I (x)E (y) + I (y)E (x) + 1
2E (x)E (y)]
4[xy(1− y)]
s.t x > y , x , y ∈ [0.5, 1]
Theorem
The above rounding procedurea approximates the optimal revenuewithin a factor of 0.8875(undirected) and 0.553(directed)
aFor appropriately selected piecewise linear functions Asym(p) and Adir (p).
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Strength of IE
Cost of restricting ourselves to two prices is small.
Obtain improved approximation strategies by approximatingOPT-IE.
Randomized Rounding ⇒ Geometry
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SDP Relaxation
Generalize Feige and Goemans[STOC’95] algorithm forMAX-DICUT.
OPT-IE can be encoded as a quadratic program withxi ∈ {−1,+1}.Construct an appropriate geometry via solving theSDP-relaxation.
Final Decision is made by Hyperplane Rounding.
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Quadratic Program
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Relaxation
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An edge wij
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Rotation
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Hyperplane Rounding
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Analysis
Local analysis where the approximation ratio is derived from:
α(fλ) = minθk1,θk2,θ12
2
π
3g(θ12, θk1, θk2) + 5f (θk1) − 3f (θk2)
5 + 3 cos θk1 − 5 cos θk2 − 3 cos θ12s.t cos θ12 + cos θk1 + cos θk2 ≥ −1
− cos θ12 − cos θk1 + cos θk2 ≥ −1
− cos θ12 + cos θk1 − cos θk2 ≥ −1
cos θ12 − cos θk1 − cos θk2 ≥ −1
0 ≤ θk1, θk2, θ12 ≤ π
Theorem
SDP-IE strategy is a 0.8999(undirected) and 0.8947(directed) approximationalgorithm for OPT-IE.
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Parametrized Analysis for varying pricing probability
Figure: Approximation Ratio α(p)for Undirected case
Figure: Approximation Ratio α(p)for Directed case
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Pricing
What if we used multiplepricing classes?
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Generalizing Influence and Exploit
Motivated by the previous observations, we naturally generalize Influence andExploit from Max-Cut to Multi-Way-Cut. Consider m “buckets” where nodesare sampled with probability qi . We associated every bucket with a price pi andconsider random ordering inside every bucket.
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Generalizing Influence and Exploit
Motivated by the previous observations, we naturally generalize Influence andExploit from Max-Cut to Multi-Way-Cut. Consider m “buckets” where nodesare sampled with probability qi . We associated every bucket with a price pi andconsider random ordering inside every bucket.
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Quadratic Programming
Using the sequence lemma for the undirected case we can cast theselection of the appropriate sampling probabilities qi as aQuadratic Program.
Lemma
There are sampling probabilities q such that the generalized IE strategy yieldsan 0.7059 approx-ratio for the symmetric case.
R(q) =m∑i=1
pi (1 − pi )qi [1
2piqi +
i−1∑j=1
piqi ]
max R(q1, . . . , qm) = qTQq
s.tm∑i=1
qi = 1, qi ≥ 0, i = 1, . . . ,m
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Directed Case
Lemma
Any α approximation algorithm based on Uniform Sampling for the symmetriccase results in α/2 approximation algorithm for the directed case.
.Proof: In Influence and Exploit where only uniform sampling is performed, thesequence is essentially random. That means that even if there is only onenon-zero edge wij or wji , we will have 1/2 probability of hitting it.
Corollary
The generalized IE strategy yields at least 0.353 of the optimal revenue in thedirected case.
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Eigen-Distributions
In the same framework we can fit sampling eigen- distributions separately forevery node. These distributions reveal their structural role.
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Summary
Optimal Influnce and Exploit Strategies
I NP-Hard even for undirected case.
I OPT-IE approximates the optimal revenue within a factor of0.8875(undirected) and 0.553(directed).
I SDP-Approximation algorithms for OPT-IE with approx-ratio0.8999(undirected) and 0.8947(directed).
.Revenue Maximization
I NP-Hard even for undirected case.
I Using the approximation algorithms for OPT-IE obtain a0.7970(undirected) and 0.501(directed) approximation strategy(previously 0.667 and 0.333).
I Generalize IE strategies for more than two classes and obtain 0.7059 and0.353 approximation.
I UGC-Hard to approximate directed case more than 0.75(with IE) and0.844(any strategy).
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Future Work
Explore possibility of using SDP for partitioning vertices ink-classes.
Consider posted price mechanism that are network aware.
Mixed models of information and influence propagation.
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Questions?
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Figure: San Vito Lo Capo beach near Erice