Slide 15.1 Stochastic Choice MathematicalMarketing 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.

Slide 15.2 Stochastic Choice MathematicalMarketing Key Chapter Terminology  Consumer Panel Data  Stationarity  Purchase Incidence Data  Brand Switching Data

Slide 15.3 Stochastic Choice MathematicalMarketing 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

Slide 15.4 Stochastic Choice MathematicalMarketing Definition of Stationarity For a parameter  we have  t =  t´ for all t, t´ = 1, 2, …, T.

Slide 15.5 Stochastic Choice MathematicalMarketing Purchase Incidence Data rNumber of Households 0f0f0 1f1f1 2f2f2 ··· TfTfT Totaln

Slide 15.6 Stochastic Choice MathematicalMarketing Brand Switching Data Purchase Occasion Two ABC Purchase Occasion One A B C

Slide 15.7 Stochastic Choice MathematicalMarketing Three Kinds of Probabilities What are the differences between the following types of probabilities?  Joint Probability  Marginal Probability  Conditional Probability AB A12 B34

Slide 15.8 Stochastic Choice MathematicalMarketing Three Kinds of Probabilities What are the differences between the following types of probabilities?  Joint Probability – Pr(A 1 and B 2 ) = 2/10  Marginal Probability – Pr(A 1 ) = (1+2)/10  Conditional Probability – Pr(A 2 | A 1 ) = 1/3 AB A12 B34

Slide 15.9 Stochastic Choice MathematicalMarketing Notation for the Three Kinds of Probabilities  Joint Probability  Marginal Probability  Conditional Probability

Slide Stochastic Choice MathematicalMarketing Bayes Theorem Pr(A, B) = Pr(B | A) · Pr(A) = Pr(A | B) · Pr(B)

Slide Stochastic Choice MathematicalMarketing Bayesian Terminology Normalizing Constant Prior Probability Conditional Probability or Likelihood Posterior Probability

Slide Stochastic Choice MathematicalMarketing Combinations (Order Does Not Matter) The number of combinations of T things taken r at a time is given by this expression What is T!? Alternative notation -

Slide Stochastic Choice MathematicalMarketing 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

Slide Stochastic Choice MathematicalMarketing The Zero-Order Property How many ways are there of “r out T” happening? What is the probability of any one of them happening?

Slide Stochastic Choice MathematicalMarketing Zero-Order Homogeneous Bernoulli Model Joint Probabilities Occasion Two AB Occasion One Ap2p2 p (1 - p) B(1 - p) p(1 - p) 2

Slide Stochastic Choice MathematicalMarketing Zero-Order Homogeneous Bernoulli Model Probabilities Conditional on Occasion 1 Occasion Two AB Occasion One Ap(1 - p) Bp(1 – p)

Slide Stochastic Choice MathematicalMarketing 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) = c 1 p  -1 (1 - p)  -1

Slide Stochastic Choice MathematicalMarketing Likelihood or Conditional Probability of r Purchases out of T Occasions Pr(r, T | p) = c 2 p r (1 - p) T- r with c 2 =

Slide Stochastic Choice MathematicalMarketing Invoking Bayes Theorem

Slide Stochastic Choice MathematicalMarketing Posterior Probabilities The posterior probabilities look like a beta distribution that depends on r and T:  * =  + r and  * =  + T - r. The Posterior  The Likelihood  The Prior

Slide Stochastic Choice MathematicalMarketing Touching Data We assert without proof that So for example, for r = 1 and T = 3:

Slide Stochastic Choice MathematicalMarketing 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 As the objective function

Slide Stochastic Choice MathematicalMarketing Markov Models  Single Period Memory (vs. Bernoulli model with zero memory)  Stationarity  Characterized by a Transition Matrix and an Initial State Vector

Slide Stochastic Choice MathematicalMarketing Single Period Memory in Markov Chains Define y t as the brand chosen on occasion t. With Markov Chains we have Pr(y t = j | y t-1, y t-2, ···, y 0 ) = Pr(y t = j | y t-1 ).

Slide Stochastic Choice MathematicalMarketing Stationarity in Markov Chains Pr(y t = j | y t-1 ) = Pr(y t = j | y t-1 ) for all t, t

Slide Stochastic Choice MathematicalMarketing Transition Matrix Occasion t + 1 AB Occasion t A.7.3 B.5 The elements of the transition matrix are the Pr(k | j) such that

Slide Stochastic Choice MathematicalMarketing 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,

Slide Stochastic Choice MathematicalMarketing The Law of Total Probability and Discrete Variables An additive law that loops through all the ways that an event (like A) could happen

Slide Stochastic Choice MathematicalMarketing Law Applied to Market Shares of brand 1 at time (1) Pr(Buy k given a previous purchase of 2) Pr(Previous Purchase of 2 at time 0)

Slide Stochastic Choice MathematicalMarketing Summation Notation and Matrix Notation for Law of Total Probability

Slide Stochastic Choice MathematicalMarketing Two Markov Models before We Seek Variety Zero –order homogeneous Bernoulli Superior-Inferior Brand Model

Slide Stochastic Choice MathematicalMarketing Variety Seeking Model 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?

Slide Stochastic Choice MathematicalMarketing 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

Slide Stochastic Choice MathematicalMarketing Purchase Incidence Data rNumber of Households 0f0f0 1f1f1 2f2f2 ··· TfTfT Totaln

Slide Stochastic Choice MathematicalMarketing Straw Man Model – Binomial We collect the panel data for T weeks and assume one purchase opportunity per week How would you test this model? The r+1 st term from expanding (q + p) T where q = 1 - p

Slide Stochastic Choice MathematicalMarketing Straw Man Model - Poisson We let T   p  0 But hold Tp =

Slide Stochastic Choice MathematicalMarketing Poisson Prediction Equation

Slide Stochastic Choice MathematicalMarketing 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

Slide Stochastic Choice MathematicalMarketing NBD Prediction Equation Where the  function (not gamma distribution) is defined as  (q) acts like a factorial function for non-integers, i. e. if q is an integer then  (q) = (q + 1)!