In a study of the domestic market share ofthe three major au

In a study of the domestic market share ofthe three major automobile manufactuers A, B, and C in a certain country, it was found that of the customers who bought a car manufactured by A, 75% would again buy a car manufactured by A, 15% would buy a car manufactured by B, and 10% would buy a car manufactured by C. Of the customers who bought a car manufactured by B, 90% would again buy a car manufactured by B, and 5% would buy a car manufactured by A and C. Finally, of the customers who bought a car manufactured by C, 85% would again buy a car manufactured by C, 5% would buy a car manufactured by A, and 10% would buy a car manufactured by B. Assuming that these sentiments reflect the buying habits of customers in the future model years, determine the market share that will be held by each manufacturer in the long run.

Solution

The above is an example of 3X3 markov chain, we need to create first 3X3 transition matrix for this problem. The columns denote the conditional probability of buying a car manufactured by three companies and rows represents the respective manufacturer of the car

Hence the matrix T will be equal to

We can define T^(n) as an n step transition matrix, now we can find T^(2)

Now let us assume the current share in the market for different manufacturers as s0 = [0.56 0.24 0.20], then the companies share after two years

s2 = s0 * T2

=> [0.3612, 0.37885, 0.25995]

Hence percentage share will be A = 36.12%, B = 37.885% and C = 25.995%