Optimal Marketing Strategies over Social Networks Jason Hartline (Northwestern), Vahab Mirrokni (Microsoft Research) Mukund Sundararajan (Stanford)

Network Affects Value JOHN VAHAB JASON zune $20 A person’s value for an item depends on others who own the item

Network Affects Value JOHN VAHAB JASON zune $30 A person’s value for an item depends on others who own the item

Examples  Early phone system Value proportional to #subscribers Monthly fee doubles every year for first four years  CompuServe Initially, small sign up fee

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?

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

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

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 +

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

Value with Network Effects  Set B of buyers  If set S of buyers has adopted, v iS drawn from distribution F iS.

Directed Graph Setting v i (S) = w ii + ∑ j in S w ji w ii w ji

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?

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)

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

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

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

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

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)

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

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

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)

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)

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)

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)

Recap We propose model where adoption depends on price, study revenue maximization Identify Influence and Exploit Strategies Show they are reasonable Discuss optimization techniques

Further Work Pricing model: set prices once and for all (no traveling salesman) No price discrimination Dynamics ?

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