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

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 vi drawn from distribution Fi

Revenue(p) = p(1 - F(p))

pi* is optimal price, Ri is optimal revenue

ContributionsPropose 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 DefinitionGiven:

a monopolist seller and set V of potential buyersdigital goods (zero manufacturing cost)value of buyer for good vi = 2V 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, viS drawn from distribution FiS.

Directed Graph Setting

vi(S) = wii + ∑j in S wji

wii

wji

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

viS drawn from distribution FiS

Player specific revenue function Ri(S)

Ri(S) is monotone

∑i Ri(B/i) is an upper bound on revenueOptimal price no longer optimal (myopic optimal price)

Optimizing Symmetric Casevi(S) drawn from distr. Fk(k=|S|)

Define: p*(#bought, #remain), E*(.,.)

E(k, t) = (1 - Fk(p))[p + E*(k+1, t-1)] + Fk(p)[E*(k,t-1)]

optimal price is myopic

Initial discounts or freebies are reasonable

Hardness of General Case?

vi(S) = wii + ∑j in S Wji

Even when weights are known,Maximizing Revenue =Maximizing feedback arc set

Approximation-ratio of 1/2Random ordering achieves approx ratio of 1/2

wii

wij

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 Ri(S) is submodular

Ri(S) - Ri(S/j) >= Ri(T) - Ri(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 ½Offer 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 IEDefine Revenue(I)

Lemma: If Ri 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 StrategiesShow they are reasonableDiscuss optimization techniques

Further Work

Pricing model: set prices once and for all (no traveling salesman)

No price discrimination

Dynamics ?

Thanks