joseph

Report
Optimal Marketing Strategies
over Social Networks
Jason Hartline (Northwestern),
Vahab Mirrokni (Microsoft Research)
Mukund Sundararajan (Stanford)
Network Affects Value
JOHN
JASON
$20
zune
VAHAB
A person’s value for
an item depends on
others who own the
item
Network Affects Value
JOHN
JASON
$30
zune
VAHAB
zune
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
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 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
wii
wji
vi(S) = wii + ∑j in S 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 revenue
Optimal price no longer optimal
(myopic optimal price)
Optimizing Symmetric Case
vi(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
wii
Even when weights are known,
Maximizing Revenue =
Maximizing feedback arc set
wij
Approximation-ratio of 1/2
Random ordering achieves approx ratio of 1/2
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 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 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
Add to S/Delete from
S, if F(S) improves
S = {3,5}
F(S) = 10
Maximizing non-monotone sub-modular functions
(Feige et. al., 08)
Local Search
Add to S/Delete from
S, if F(S) improves
S = {2, 3, 5}
F(S) = 11
Maximizing non-monotone sub-modular functions
(Feige et. al., 08)
Local Search
Add to S/Delete from
S, if F(S) improves
S = {2, 5}
F(S) = 12
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 ?
Thanks

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