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
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