I consider the case when company II can't place its restaurants at the same location. I haven't an idea how to compose a $3\times3$ matrix. I have some thoughts but I'm doubtful. Please correct me if I'm wrong.
I want to fill in the $a_{11}$ element of the matrix which shows that company I set up a restaurant in $L$ and company II in $L$ and $C$. But the value in $a_{11}$ depends on the walker's position.
Case 1: $L$ is the closest location to the walker, therefore the probability to visit I's restaurant is $p$ and that of II's is $(1-p)$. Therefore $a_{11} = (p, 1-p)$.
Case 2: $C$ is the closest location to the walker. I cannot be visited and II always gets the walker; $a_{11} = (0,1)$.
Case 3: $R$ is the closest location to the walker, therefore the closest restaurant to the walker is II's restaurant located in $C$. $a_{11} = (0,1)$.
Can I just write $(p, 1-p)$ as $a_{11}$?
How do I do the $(L, LR)$ case (I in $L$, II in $L$ and $R$)? There are 2 values: $(p, 1-p)$ if the walker in the left or in the right and $(p, 2*(1-p))$ if he is in centre. Which one to pick?
Do I need to compose three $3\times3$ matrices related to the position of walker?