Algebra 1073

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timer Asked: Apr 11th, 2017

Question description

  • 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.
  • 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 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:

page1image18424

The notation π‘šπ‘ π‘ƒπ‘’π‘π‘ π‘–0 is read as the market share of Pepsi at the current time (t = zero), π‘šπ‘ π‘ƒπ‘’π‘π‘ π‘–1 is read as the market share of Pepsi at the end of one month, and so on such that π‘šπ‘ π‘ƒπ‘’π‘π‘ π‘–π‘› 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:

π‘šπ‘ π‘ƒπ‘’π‘π‘ π‘–1 = π‘šπ‘ π‘ƒπ‘’π‘π‘ π‘–0(. 60) + π‘šπ‘ πΆπ‘œπ‘˜π‘’0(. 08) + π‘šπ‘ π‘…πΆ0(.20)

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

π‘šπ‘ πΆπ‘œπ‘˜π‘’1 = π‘šπ‘ π‘ƒπ‘’π‘π‘ π‘–0(. 15) + π‘šπ‘ πΆπ‘œπ‘˜π‘’0(. 87) + π‘šπ‘ π‘…πΆ0(.30)

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

π‘šπ‘ π‘…πΆ1 = π‘šπ‘ π‘ƒπ‘’π‘π‘ π‘–0(. 25) + π‘šπ‘ πΆπ‘œπ‘˜π‘’0(. 05) + π‘šπ‘ π‘…πΆ0(.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:

𝑣0 = [π‘šπ‘ π‘ƒπ‘’π‘π‘ π‘–0 π‘šπ‘ πΆπ‘œπ‘˜π‘’0 π‘šπ‘ π‘…πΆ0] = [. 45 . 33 . 22] .
The notation 𝑣0 , 𝑣1 , 𝑣2 , ... , 𝑣𝑛 is a monthly sequence of distributions of market share.

The information and the preference transition diagram given before, lead to a β€œTransition Matrix” 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 β€œTransition Matrix” 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:

page2image16144

𝑝11 𝑃 = [𝑝21 𝑝31

𝑝12 𝑝13 𝑝22 𝑝23] 𝑝32 𝑝33

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, 𝑝11 , is the probability that a Pepsi customer stays with Pepsi for the month, 𝑝22 is the probability that a Coke customer stays with Coke for the month, and 𝑝33 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:

𝑣1 = 𝑣0𝑃
𝑣2 = 𝑣1𝑃 = (𝑣0𝑃)𝑃 = 𝑣0𝑃2 𝑣3 = 𝑣2𝑃 = (𝑣0𝑃2)𝑃 = 𝑣0𝑃3 ⁞
𝑣𝑛 = 𝑣0𝑃𝑛

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 𝑃𝑛 stop changing and become constants. To see this calculate for yourself 𝑃3, 𝑃6, 𝑃12,When this occurs you have arrived at what is called the β€œLong Term” , β€œSteady State” or β€œEnd State” transition matrix.

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

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

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