Problem definition

Tina Belcher, brand-new manager of the new Bob´s burger parlour downtown San Francisco, wants to analyse the impact on the market and the loyalty of the customers of Bob’s Burger and Pesto´s, an Italian restaurant nearby. It is assumed that one customer dines once a week at Bob’s or at Pesto´s, but not at both. Tina has collected information from 100 consumers over a 10-week period. When checking the data, she discovered that of all the customers who consumed at Bob’s in a given week, 90 % returned to the restaurant following week, while 10 % switched to Pesto’s. Of all the consumers who consumed at Pesto’s in a given week, 80 % returned to the same place the following week, but 20 % switched to Bob´s. Assuming that the transition probabilities are the same for every customer and that they not change over time, we can get insights modelling customer fidelity with Markov chains.

Model customer fidelity as a Markov process and answer the following questions:

a) If at week 0, a customer dines at Bob’s, what is the probability of the same customer dining at Bob´s in week 1?

b) What is the probability of a customer going back to Bob´s two weeks running?

c) What is the probability of a customer changing to Pesto´s after dining at Bob´s and then going back to Bob´s

d) Consider the customer of section a) what is the probability that this customer comes back to Bob´s in two weeks time?

e) What is the expected market share of customers between the two restaurants?