Expectation of hitting time of a markov chain
In the question above we can find a system that allows us to calculate the expected time to hit a state $i$ , from every other state
$$ \psi(j) = \begin{cases} 0\text{,} & j = i\\ 1 + \sum\limits_{k \neq j} p_{j k}\psi(k)\text{,} & j \neq i \end{cases} $$
That it has a solution seems clear (the expected values respect all those equations)
Why is this solution unique?