Four competing companies (A, B, C, and D) are competing in a finite market of 5000 clients. Each month, the clients can either renew their contract with the company they were with or switch companies, according to the following ratios:
• Company A retains 35% of its clients; 25% go to company B, 15% go to company C, and 25% go to company D.
• Company B retains 50% of its clients; 10% go to company A, 20% go to company C, and 20% go to company D.
• Company C retains 40% of its clients; 40% go to company A, 5% go to company B, and 15% go to company D.
• Company D retains 40% of its clients; 35% go to company A, 15% go to company B, and 10% go to company C.
How many clients should each company expect to end up with?
So first what I did was put the numbers into a nice matrix as follows (which each column is a company. first column = company A and last column = company D:
$$ \begin{bmatrix} 0.35 & 0.1 & 0.4 & 0.35 \\ 0.25 & 0.5 & 0.05 & 0.15\\ 0.15 & 0.2 & 0.4 & 0.1 \\ 0.25 & 0.2 & 0.15 & 0.4 \\ \end{bmatrix} \quad $$
Then I reduced it, but I just got a trinagluar matrix with 1s and 0s which doesn't help. Here is the answer:
But I have no clue how they got that. Any ideas? I tried finding the eigen value but that didn't do anything either.
here's my closest I got with my steady-state eigenvector:
Hopefully that helps.