I'm having trouble completing the above question, as my working knowledge of "limiting probabilities" is not very good. For the 1-step transition matrix, I have $$P= \begin{pmatrix} 0.0 & 0.0 & 0.6 & 0.4\\ 0.5 & 0.5 & 0.0 & 0.0\\ 0.4 & 0.6 & 0.0 & 0.0\\ 0.0 & 0.0 & 0.7 & 0.3\end{pmatrix}$$
To me, a "successful bid" is represented by going from $sf$ to $fs$, $ss$ to $ss$, $fs$ to $ss$ or $ff$ to $fs$. Is that a correct interpretation?
I've been looking through my professor's slides and trying very hard to understand how to apply $$\mathbf{\pi}^{(n)}=\mathbf{\pi}^{(0)}P^{n}$$ to my current problem. I think I understand that if I know the initial distribution, I know the distribution at time $n$, but I'm not sure what the initial distribution is for this problem...
Any help with how I can continue with this problem would be a great help.
Thanks in advance for your patience, all.