Download Marketing Decision Making

Chapter 15: Stochastic Choice
The sequence of coverage is:
 Key Terminology
 The Brand Switching Matrix
 Zero-Order Bernoulli Model
 Population Heterogeneity
 Markov Chains
 Learning Models (Not Covered)
 Purchase Incidence
 Negative Binomial Model
This chapter follows the development in Lilien, Gary L. and Philip Kotler
(1983) Marketing Decision Making. New York: Harper and Row.
Mathematical
Marketing
Slide 15.1
Stochastic Choice
Key Chapter Terminology
 Consumer Panel Data
 Stationarity
 Purchase Incidence Data
 Brand Switching Data
Mathematical
Marketing
Slide 15.2
Stochastic Choice
A Typical Record in Consumer Panel Data





Household ID
Date and Time
Household Member Buying
Household Member Using
Product Purchased





Package Size
Price
Promotion Information
Local Media Feature
Where Bought
Collected via diaries, scanners in the home, or all local stores have scanners
Equivalent data for online behavior are called clickstream data
Mathematical
Marketing
Slide 15.3
Stochastic Choice
Definition of Stationarity
For a parameter  we have
t = t´ for all t, t´ = 1, 2, …, T.
Mathematical
Marketing
Slide 15.4
Stochastic Choice
Purchase Incidence Data
Mathematical
Marketing
r
Number of Households
0
f0
1
f1
2
f2
···
···
T
fT
Total
n
Slide 15.5
Stochastic Choice
Brand Switching Data
Purchase Occasion Two
Purchase
Occasion
One
Mathematical
Marketing
A
B
C
A
10
5
10
25
B
8
12
5
25
C
10
10
30
50
Slide 15.6
Stochastic Choice
Three Kinds of Probabilities
What are the differences between the following types
of probabilities?

Joint Probability

Marginal Probability

Conditional Probability
Mathematical
Marketing
A
B
A
1
2
B
3
4
Slide 15.7
Stochastic Choice
Three Kinds of Probabilities
What are the differences between the following types
of probabilities?

Joint Probability – Pr(A1 and B2) = 2/10

Marginal Probability – Pr(A1) = (1+2)/10

Conditional Probability – Pr(A2 | A1) = 1/3
Mathematical
Marketing
A
B
A
1
2
B
3
4
Slide 15.8
Stochastic Choice
Notation for the Three Kinds of Probabilities


Joint Probability
Marginal Probability
Pr(j, k)
m   Pr(j, k)
1
j
k

Conditional Probability
Mathematical
Marketing
Pr(j, k)
p jk  Pr(k | j) 
Pr(j)
Slide 15.9
Stochastic Choice
Bayes Theorem
Pr(A, B)
Pr(B | A) 
Pr(A)
Pr(A | B) 
Pr(A, B)
Pr(B)
Pr(A, B) =
Pr(B | A) · Pr(A) = Pr(A | B) · Pr(B)
Pr(B | A)  Pr(A)
Pr(A | B) 
Pr(B)
Mathematical
Marketing
Slide 15.10
Stochastic Choice
Bayesian Terminology
Conditional Probability
or Likelihood
Prior Probability
Posterior Probability
Pr(B | A)  Pr(A)
Pr(A | B) 
Pr(B)
Normalizing Constant
Mathematical
Marketing
Slide 15.11
Stochastic Choice
Combinations (Order Does Not Matter)
T
 
r
The number of combinations of T things taken r at a time
is given by this expression
T!
r! (T  r)!
What is T!?
Mathematical
Marketing
Alternative notation - C Tr
Slide 15.12
Stochastic Choice
The Zero-Order Property
Pr(A, B, A, A, B, ···) = p · (1 - p) · p · p (1 - p) · ···
So overall, r purchases of A out of T occasions would be
T r
T r
 p (1  p)
r
Mathematical
Marketing
Slide 15.13
Stochastic Choice
The Zero-Order Property
T r
T r
 p (1  p)
r
How many ways are there of
“r out T” happening?
Mathematical
Marketing
What is the probability of
any one of them happening?
Slide 15.14
Stochastic Choice
Zero-Order Homogeneous Bernoulli Model
Joint Probabilities
Occasion Two
Occasion
One
Mathematical
Marketing
A
B
A
p2
p (1 - p)
B
(1 - p) p
(1 - p)2
Slide 15.15
Stochastic Choice
Zero-Order Homogeneous Bernoulli Model
Probabilities Conditional on Occasion 1
Occasion Two
Occasion
One
Mathematical
Marketing
A
B
A
p
(1 - p)
B
p
(1 – p)
Slide 15.16
Stochastic Choice
Population Heterogeneity
 p itself is a random variable that differs from household to household
 We assume p is distributed according to the Beta distribution, which acts as
a mixing distribution
 We call this the prior distribution of p
Pr(p) = c1p-1(1 - p)-1
Mathematical
Marketing
Slide 15.17
Stochastic Choice
Likelihood or Conditional Probability of
r Purchases out of T Occasions
Pr(r, T | p) = c2 pr (1 - p)T- r
T
with c2 =  
r
Mathematical
Marketing
Slide 15.18
Stochastic Choice
Invoking Bayes Theorem
Pr( r, T | p) Pr( p)
Pr( p | r, T) 
Pr( r, T)
Pr(p | r, T)  c3  Pr(r, T | p)  Pr(p)
Mathematical
Marketing
Slide 15.19
Stochastic Choice
Posterior Probabilities
The Posterior  The Likelihood  The Prior
Pr (p | r, T)  c 4  p r (1  p) T r  p α 1 (1  p)β 1
 c 4  p α  r 1 (1  p)β T r 1
The posterior probabilities look like a beta distribution that depends on r and T:
* =  + r and * =  + T - r.
Mathematical
Marketing
Slide 15.20
Stochastic Choice
Touching Data
We assert without proof that
αr
p̂  E(p | r, T) 
α β T
So for example, for r = 1 and T = 3:
α 1
p̂  E(p | 1, 3) 
α β 3
Mathematical
Marketing
Slide 15.21
Stochastic Choice
Testing the Model
So for each of the triples AAA, AAB, ABA, ABB, BAA, BAB, BBA, BBB, we can use
Minimum Pearson Chi Square and use
8
(p j  p̂ j ) 2
j
p̂ j
χ̂ 62  
As the objective function
Mathematical
Marketing
Slide 15.22
Stochastic Choice
Markov Models
 Single Period Memory
(vs. Bernoulli model with zero memory)
 Stationarity
 Characterized by a
• Transition Matrix and an
• Initial State Vector
Mathematical
Marketing
Slide 15.23
Stochastic Choice
Single Period Memory in Markov Chains
Define yt as the brand chosen on occasion t. With
Markov Chains we have
Pr(yt = j | yt-1, yt-2, ···, y0) = Pr(yt = j | yt-1).
Mathematical
Marketing
Slide 15.24
Stochastic Choice
Stationarity in Markov Chains
Pr(yt = j | yt-1 ) = Pr(yt = j | yt-1) for all t, t
Mathematical
Marketing
Slide 15.25
Stochastic Choice
Transition Matrix
Occasion t + 1
Occasion t
A
B
A
.7
B
.3
.5
.5
The elements of the transition matrix are the Pr(k | j) such that
 Pr(k | j)  1
k
Mathematical
Marketing
Slide 15.26
Stochastic Choice
Initial State Vector
m(0) is a J by 1 vector of shares at “time 0”. A typical element
provides the share for brand j,
(0)
j
{m }
m (0)  [m1(0)
Mathematical
Marketing
(0)
m (0)

m
2
J ]
Slide 15.27
Stochastic Choice
The Law of Total Probability and Discrete Variables
An additive law that loops through all the ways that an event (like A)
could happen
Pr(A)   Pr(A | B j )Pr(B j )
j
Mathematical
Marketing
Slide 15.28
Stochastic Choice
Law Applied to Market Shares of brand 1 at time (1)
m(k1)  Pr(k | 1)  m1(0)  Pr(k | 2)  m(20)   Pr(k | J)  m(J0)
Pr(Buy k given a previous purchase of 2)
Pr(Previous Purchase of 2 at time 0)
Mathematical
Marketing
Slide 15.29
Stochastic Choice
Summation Notation and Matrix Notation
for Law of Total Probability
J
(0)
m(1)

p
m
 jk j
k
j
[m (1)]  [m (0)]P
[m(2)]  [m(1)]P  [m(1)]P  [m(0)]PP
[m (3)]  [m (2)]P  [m (1)]PP  [m (0)]PPP
    
Mathematical
Marketing
Slide 15.30
Stochastic Choice
Two Markov Models before We Seek Variety
Zero –order homogeneous Bernoulli
p 1  p 
p 1  p 


Mathematical
Marketing
Superior-Inferior Brand Model
0 
1
p 1  p 


Slide 15.31
Stochastic Choice
Variety Seeking Model

p  vp

 

(1  p)  v(1  p)

1. What values go in the cells of the above transition matrix that are marked with the “-”?
2. What does the model predict for Pr(AAA)?
3. How could we estimate the model from the 8 triples, AAA, AAB, ABA, …, BBB?
Mathematical
Marketing
Slide 15.32
Stochastic Choice
Purchase Incidence Data
The goal is to predict or explain the number of households who will purchase our brand r times
Or the number of Web surfers who will visit our site r times or purchase at our site r times
Or in general the number of population members who will exhibit a discrete behavior r times
Mathematical
Marketing
Slide 15.33
Stochastic Choice
Purchase Incidence Data
Mathematical
Marketing
r
Number of Households
0
f0
1
f1
2
f2
···
···
T
fT
Total
n
Slide 15.34
Stochastic Choice
Straw Man Model – Binomial
We collect the panel data for T weeks and assume one purchase opportunity per week
The r+1st term from expanding
(q + p)T where q = 1 - p
T
T r r


f̂ r  T   (1  p) p
r
How would you test this model?
Mathematical
Marketing
Slide 15.35
Stochastic Choice
Straw Man Model - Poisson
Mathematical
Marketing
We let
But hold
T 
p0
Tp = 
Slide 15.36
Stochastic Choice
Poisson Prediction Equation
f̂ r  n  e
Mathematical
Marketing
λ
λ
r!
r
Slide 15.37
Stochastic Choice
Negative Binomial Distribution
Named after the terms in the expansion of (q - p)-r
Can arise from
 A binomial where the number of tosses is itself random
 A Poisson where  changes over time due to contagion
 A Poisson where  varies across households with the gamma distribution
Mathematical
Marketing
Slide 15.38
Stochastic Choice
NBD Prediction Equation
 k   m  Γ(k  r)
f̂ r  n  
 

 k  m   k  m  Γ(k) r!
k
m
Where the  function (not gamma distribution) is defined as

Γ(q)   x q 1e  x dx
0
(q) acts like a factorial function for non-integers,
i. e. if q is an integer then (q) = (q + 1)!
Mathematical
Marketing
Slide 15.39
Stochastic Choice