Math help expected share question

Question

Consumers in a certain state choose between three long distance telephone services. GTT, NCJ, and Dash. Aggressive marketing by all three companies results in a continual shift of customers among the three services. Each year, GTT loses 5% of its customers to NCJ and 20% to Dash, NCJ loses 25% of its customers to GTT and 25% to Dash, and Dash loses 30% of its customers to GTT and 30% to NCJ. Assuming that these percentages remain valid over a long period of time, what is each company's expected market share in the long run?

 

GTT's expected market share is ____ %

(round to the nearest tenth as needed)

Answer 1

This time I will say more about how to set up the Markov matrix.

Let's let the first column be what happens to GTT customers.

75% stay with GTT (g)

5% go to NCJ (n)

20% got to Dash (d)

The second column is what happens to NCJ customers

25% go to g

50% stay with n

25% go to d

The third column is what happens to Dash customers

30% got to g

30% go to n

40% stay with d

 

This gives us 

Matrix A           vector u

0.75 0.25 0.30   g
0.05 0.50 0.30   n
0.20 0.25 0.40   d

 

Since all the columns add up to 1, this is a Markov matrix and there exists an eigenvector corresponding to the eigenvalue of 1. This means that A - I = 0 where I is the identity matrix.

 

A - I              

-0.25  0.25  0.30 | 0
 0.05 -0.50  0.30 | 0
 0.20  0.25 -0.60 | 0 

 

This reduces to 

-0.25  0.25 0.30 | 0
 0.00 -0.45 0.36 | 0
 0.00  0.00 0.00 | 0

 

Letting d be a free variable we get

-.45n + 0.36d = 0 => n =0.8d

-0.25g + 0.25*0.8d + 0.3d = 0 => g = 2d

 

This gives the eigenvector (2, 0.8, 1) or normalized (10/19, 4/19, 5/19).

 

After a long time 10/19 will be with GTT, 4/19 will be with NCJ, and 5/19 will be with Dash.

 

Answer 2

I set this up in an Excel spreadsheet.  I started with each one having an identical 33.33 customers.  (But anything would work if the numbers added up to 100.)  After about ten iterations, the answers converged.  Do you know how to set this up on a spreadsheet?