# Algebra 1073

*label*Algebra

*timer*Asked: Apr 11th, 2017

**Question description**

- Written β 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.
- Also please answer the questions as well

Market Share and Markov Chains

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 letβs 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:

- The probability of a customer staying with the brand Pepsi over a month is 60%.
- The probability of a Pepsi customer switching to Coke over a month is 15%.
- The probability of a Pepsi customer switching to RC Cola over a month is 25%
- The probability of a customer staying with the brand Coke over a month is 87%
- The probability of a Coke customer switching to Pepsi over a month is 8%
- The probability of a Coke customer switching to RC Cola over a month is 5%
- The probability of a customer staying with RC Cola over a month is 50%
- The probability of a RC Cola customer switching to Coke over a month is 30%
- 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 ππ ππππ π0 is read as the market share of Pepsi at the current time (t = zero), ππ ππππ π1 is read as the market share of Pepsi at the end of one month, and so on such that ππ ππππ ππ is the market share of Pepsi at the end of the nth month after time zero.

If you want to compute the market share of Pepsi after one month it would be calculated as:

ππ ππππ π1 = ππ ππππ π0(. 60) + ππ πΆπππ0(. 08) + ππ π πΆ0(.20)

If you want to compute the market share of Coke after one month it would be calculated as:

ππ πΆπππ1 = ππ ππππ π0(. 15) + ππ πΆπππ0(. 87) + ππ π πΆ0(.30)

If you want to compute the market share of RC Cola after one month it would be calculated as:

ππ π πΆ1 = ππ ππππ π0(. 25) + ππ πΆπππ0(. 05) + ππ π πΆ0(.50)

This notation and calculations make it possible for us to use matrix notation and operations to describe how the distribution of market share change over the months from one state to the next. Let the current (time zero) distribution of market share be represented by the row matrix:

π£0 = [ππ ππππ π0 ππ πΆπππ0 ππ π
πΆ0] = [. 45 . 33 . 22] .

The notation π£0 , π£1 , π£2 , ... , π£π is a monthly sequence of distributions of market share.

The information and the preference transition diagram given before, lead to a βTransition Matrixβ which represents how over each month the distribution of market share is changing from one state of distribution of market share to the next. Since the probabilities to change brands of cola are assumed to not change as time progresses we find that the βTransition Matrixβ is a matrix in which the elements are constants. The transition matrix in this case is square (3x3) and it is represented by P as it is a matrix containing probabilities:

π11 π = [π21 π31

π12 π13 π22 π23] π32 π33

The first column of entries is determined by the probabilities associated with Pepsi customers. The second column of entries is determined by the probabilities associated with Coke customers. The third column of entries is determined by the probabilities associated with RC Cola customers.

For example, π11 , is the probability that a Pepsi customer stays with Pepsi for the month, π22 is the probability that a Coke customer stays with Coke for the month, and π33 is the probability that a RC Cola customer stays with RC Cola for the month.

We can now model the process of the changing distributions of market share by using matrix multiplication:

π£1 = π£0π

π£2 = π£1π = (π£0π)π = π£0π2
π£3 = π£2π = (π£0π2)π = π£0π3
β

π£π = π£0ππ

So that we can now see that the monthly distribution of market share is dependent upon the starting (time zero) distribution of market share and the transition matrix raised to the nth power.

Question 1. What is the transition matrix in this case? This is, I would like you to write out the
matrix P showing me all nine of the constants which make up the matrix P *.
*

Question 2. Sum across the rows of the transition matrix. What do you notice? Explain why you think that you obtained this result.

Question 3. What is the distribution of market share after six month from the current (time zero) distribution?

Question 4. Which cola company is the biggest loser of market share and which cola company is the biggest gainer of market share at the end of 12 months?

In a Markov Chain, such as in this project, you will find that as you continue to raise the transition matrix P to powers you will find that the entries in ππ stop changing and become constants. To see this calculate for yourself π3, π6, π12,When this occurs you have arrived at what is called the βLong Termβ , βSteady Stateβ or βEnd Stateβ transition matrix.

Question 5. To four decimals of accuracy what are the entries in the βSteady Stateβ transition matrix and what do you notice about these entries? Roughly how many months did it take to arrive at this βSteady Stateβ transition matrix?

Question 6. As the analyst, which cola company should the soda company merge with?