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Optimal Marketing Strategies over Social Networks Jason Hartline (Northwestern), Vahab Mirrokni (Microsoft Research) Mukund Sundararajan (Stanford)
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Network Affects Value JOHN VAHAB JASON zune $20 A person’s value for an item depends on others who own the item
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Network Affects Value JOHN VAHAB JASON zune $30 A person’s value for an item depends on others who own the item
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Examples Early phone system Value proportional to #subscribers Monthly fee doubles every year for first four years CompuServe Initially, small sign up fee
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Standard Influence Models (See [Kempe+03], its citations) Probability of adoption depends on who else has item No dependence on price Maximize adoption: Which k players would you give item away to?
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Standard Optimal Pricing Set B of buyers No network effect or externalities Value v i drawn from distribution F i Revenue(p) = p(1 - F(p)) p i * is optimal price, R i is optimal revenue
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Contributions Propose model where adoption is based on price and network effects Study Revenue maximization Identify a family of strategies called influence and exploit strategies that are easy to implement and optimize over
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Problem Definition Given: a monopolist seller and set V of potential buyers digital goods (zero manufacturing cost) value of buyer for good v i = 2 V R +
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Problem Definition (cont.) Assumptions: buyer’s decision to buy an item depends on other buyers who own the item and the price seller does not know the buyer’s value function but instead has a distributional information about them
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Value with Network Effects Set B of buyers If set S of buyers has adopted, v iS drawn from distribution F iS.
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Directed Graph Setting v i (S) = w ii + ∑ j in S w ji w ii w ji
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Marketing Strategy Seller visits buyers in a sequence and offers each buyer a price Order and price can depend on history of sales Seller earns the price as revenue when buyer accepts Goal: maximize expected revenue Marketing Strategy: sequence of offer to buyers and the prices that we offer Question: algorithmic techniques?
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Upper Bound on Revenue v iS drawn from distribution F iS Player specific revenue function R i (S) R i (S) is monotone ∑ i R i (B/i) is an upper bound on revenue Optimal price no longer optimal (myopic optimal price)
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Optimizing Symmetric Case v i (S) drawn from distr. F k (k=|S|) Define: p*(#bought, #remain), E*(.,.) E(k, t) = (1 - F k (p))[p + E*(k+1, t-1)] + F k (p)[E*(k,t-1)] optimal price is myopic Initial discounts or freebies are reasonable
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Hardness of General Case? v i (S) = w ii + ∑ j in S W ji Even when weights are known, Maximizing Revenue = Maximizing feedback arc set Approximation-ratio of 1/2 Random ordering achieves approx ratio of 1/2 w ii w ij
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Influence and Exploit(IE) Give buyers in set I item for free. Recall freebies by symmetric strategy Visit remaining buyers in random sequence, offer each(adaptively) myopic optimal price Motivated by max feedback arc set heuristic and optimal pricing
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Diminishing Returns We assume R i (S) is submodular R i (S) - R i (S/j) >= R i (T) - R i (T/j), if S is a subset of T Studies indicate this is reasonable assumption
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Easy 0.25-Approximation Building I: Pick each buyer with probability ½ Offer remaining myopic optimal price Sub-modularity implies: Pick each element in set S with prob. p, then: E[f(S)] >= p f(S)
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Monotone Hazard Rate Monotone Hazard Rate: f(t)/(1-F(t)) is increasing in t Buyers accepts offer with non-trivial probability Can be used to improve the bounds to 2/3 Satisfied by exponential, uniform and Gaussian distributions Nice closure properties
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Optimizing over IE Define Revenue(I) Lemma: If R i s are submodular, so is revenue as a function of influence set. But, it is not monotone Use Feige, Mirrokni, Vondrak, to get a 0.4 approximation
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Local Search Add to S/Delete from S, if F(S) improves S = {5} F(S) = 5 Maximizing non-monotone sub-modular functions (Feige et. al., 08)
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Local Search S = {3,5} F(S) = 10 Add to S/Delete from S, if F(S) improves Maximizing non-monotone sub-modular functions (Feige et. al., 08)
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Local Search S = {2, 3, 5} F(S) = 11 Add to S/Delete from S, if F(S) improves Maximizing non-monotone sub-modular functions (Feige et. al., 08)
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Local Search S = {2, 5} F(S) = 12 Add to S/Delete from S, if F(S) improves Maximizing non-monotone sub-modular functions (Feige et. al., 08)
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Recap We propose model where adoption depends on price, study revenue maximization Identify Influence and Exploit Strategies Show they are reasonable Discuss optimization techniques
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Further Work Pricing model: set prices once and for all (no traveling salesman) No price discrimination Dynamics ?
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Thanks
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