I am presented with $P =\begin{bmatrix} 0.5 & \alpha & \beta \\ \alpha & \beta & 0.5 \\ \beta & 0.5 & \alpha \end{bmatrix}$
where $\alpha+\beta=0.5$ and $\alpha,\beta \in (0,0.5)$.
Performing $[\pi_1,\pi_2,\pi_3]\times P=$ $\begin{bmatrix} 0.5\cdot\pi_1+\alpha\cdot\pi_2+\beta\cdot\pi_3 \\ \alpha\cdot\pi_1+\beta\cdot\pi_2+0.5\cdot\pi_3 \\ \beta\cdot\pi_1+0.5\cdot\pi_2+\alpha\cdot\pi_3 \end{bmatrix}^T$
Working through I find:
$\pi=(\frac{\alpha+2\beta(1-\beta)}{1-\beta-2\alpha^2}\cdot\pi_3,\frac{2\beta\alpha+0.5}{1-\beta-2\alpha^2}\cdot\pi_3,\pi_3)$
Experimenting with values of $\alpha$ and $\beta$ leads me to believe this always turns out to be $\pi = (1/3,1/3,1/3)$ under the convention $\sum \pi_i=1$. This and the fact the transition matrix appears to be irreducible and positive recurrent leads me to believe it is unique.
However the question goes on to ask how/whether I would change my answer if $\alpha=0$. I don't see how this would change things but the question implies it would.
Any advice or tips on the correctness of my solution so far and where to go from here?
Thanks!