Linear Algebra Diagonalization & Probability

Question

There are 3 telecommunication service providers in a certain country, namely telco A,

B, and C. Every year,

• about 4% of the subscribers of telco A will change to telco B, and 4% of subscribers

of telco A will change to telco C;

• about 4% of the subscribers of telco B will change to telco A, and 2% of subscribers

of telco B will change to telco C; and

• about 4% of the subscribers of telco C will change to telco A, and 1% of subscribers

of telco C will change to telco B.

Assume that everyone in the country is subscribed to exactly 1 telco, and that there are

no change in the population size every year.

(a) Suppose that exactly 1/3 of the population are subscribed to each telco

this year. What is the proportion of the population that are subscribed to each

telco after 5 years?

(b) In the long run, which telco will have the most subscribers? Explain

how you derive your answer.

Answer 1

Let us denote the number of people who are subscribed to A at some date (say, 1st January) in a given year Y be 'nA(Y)'. Similarly, nB(Y) and nC(Y) are the number of people subscribed to B and C respectively on 01/01/Y.

The first piece of information we are given is that every person is subscribed to exactly 1 company, and there is no change in population year after year. So,

nA(Y)+ nB(Y) + nC(Y) = nA(Y+1)+ nB(Y+1) + nC(Y+1) = constant.

Now, let us consider the people who change their telecom providers during the year. Let us denote the proportion of people who change from A → B to be pAB = 0.04 from the given information. Similarly, we may define pBA = 0.04 , pAC = 0.04 , pCA = 0.04 , pBC = 0.02 and pCB = 0.01.

Now, let us think about how to calculate the number of people subscribed to A in year Y+1. Clearly, we need to take the number of people subscribed to A at the start of year Y, subtract the number of people who changed from A to B or C during the year, and add the people who changed from B or C to A. Algebraically, we write: nA(Y+1) = nA(Y) (1-pAB -pAC) + nB(Y) pBA + nC(Y) pCA. Similarly, we may also write:

nB(Y+1) = nB(Y) (1 - pBA - pBC) + nA(Y) pAB + nC(Y) pCB, and

nC(Y+1) = nC(Y) (1 - pCA - pCB) + nA(Y) pAC + nB(Y) pBC.

We can see the algebraic structure of matrix multiplication. Abbreviating nA(Y) as nA, nA(Y+1) as nA1,

nA1 = mAA nA + mBA nB + mCA nC

nB1 = mAB nA + mBB nB + mCB nC

nC1 = mAC nA + nBC nB + nCC nC

where the diagonal element mAA is given as (1-pAB -pAC), whereas the off-diagonal elements like mAB are just equal to pAB.

We can check that :

nA1 + nB1 + nC1 = (mAA + mAB + mAC) nA + (mBA + mBB + mBC) nB + (mCA + mCB + mCC) nC

= nA + nB + nC

holds for arbitrary nA , nB and nC, because each of the individual brackets sum to 1 (e.g. mAA + mAB + mAC = 1 - pAB - pAC + pAB + pAC = 1, etc.)

We may denote the set of equations we have above as a matrix transformation on a vector representing the numbers of subscribers in the different companies.

N(Y+1) = M N(Y), where N(Y) is the column vector (nA(Y), nB(Y) , nC(Y))^T (T is transpose), and the matrix M is denoted by :

0.92 0.04 0.04

0.04 0.94 0.01

0.04 0.02 0.95

We can see that to compute the number of subscribers after multiple years, we merely need to carry out this matrix multiplication operation multiple times :

N(Y + y) = M^y N(Y).

If you have followed this far, you should be able to finish the exercise. You should find that starting from nA(Y0) = nB(Y0) = nC(Y0) , nA is a constant. nB + nC is also a constant year after year, but nC is increasing and nB is decreasing (so that C will end up with the largest share of the market).

Hope this helps.