• Written – A paper copy of your answers to questions 1 -> 6, will be submitted as a formal report. This report must be turned in by class time on Apr. 20, 2017; no late work will be accepted. Questions should be answered sequentially, with your work in arriving at the answers either typed or hand-written legibly.

Market Share and Markov Chains

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A Markov chain is a simple concept which can explain and model complicated real time processes. A simple Markov chain is based upon the premise that the next state in the process only depends upon the previous state and that the probabilities to change from state to state remain fixed (constant) at all time.

In this project let’s assume that you are an analyst, in a market research company, who has been hired to examine which cola company another soda company should merge with in the future. The soda company wishes to join that cola company which will have the largest market share in the future.

Coke, Pepsi, and RC Cola are the only cola companies in operation and the ones which are being considered for the merger. Currently (at time zero), Pepsi owns 45% of market share, Coke owns 33% of market share and RC Cola owns 22% of market share.

As you start your analysis you find that individual customers change their preferences or choices for cola drinks over each month and find that:

1. The probability of a customer staying with the brand Pepsi over a month is 60%.
2. The probability of a Pepsi customer switching to Coke over a month is 15%.
3. The probability of a Pepsi customer switching to RC Cola over a month is 25%
4. The probability of a customer staying with the brand Coke over a month is 87%
5. The probability of a Coke customer switching to Pepsi over a month is 8%
6. The probability of a Coke customer switching to RC Cola over a month is 5%
7. The probability of a customer staying with RC Cola over a month is 50%
8. The probability of a RC Cola customer switching to Coke over a month is 30%
9. The probability of a RC Cola customer switching to Pepsi over a month is 20%.

The following diagram depicts the changing preference transitions of cola customers during a one month transition reflecting the information above:

The notation

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