Consider $A$ as a matrix, that when normalized represents an finite-state time-homogeneous Markov chain $M$ with entries $0\leq p_{i,j}\leq 1$, where each row sums up to $1$. We can also assume that $M$ is irreducible and aperiodic, hence it has an unique steady-state distribution $\pi$ ($\pi=\pi P$).
Now consider a parameter $0<\delta\leq 1$ is added to the diagonal elements of the matrix $A$ Markov chain. So $a'_{i,j}=a_{i,j}+\delta$, from where we get $A'$. All rows of $A'$ are then normalized to sum up to one. From it we get new transition matrix $P'$. We again compute the steady-state distribution of this newly constructed chain $M'$ and call it $\pi'$.
If we observe what is happening with the elements of $\pi'$ if $\delta$ is increased, we can see that each value in the distribution monitonically increases or decreases.
Now my question. How could I prove that this is true for all $M$ with above properties (or does anyone have an counter example?)?
It also seems that $\pi'$ converges to some $\pi_C$ if I set $\delta$ to some arbitrarily large number (not compute the limit). Is there a direct formula to compute such $\pi_C$?
For example, lets take this matrix $A$ for out initial matrix:
$$ A = \begin{bmatrix} 0.056&0.084&0.242&0.255\\ 0.071&0.056&0.249&0.210\\ 0.086&0.095&0.056&0.080\\ 0.115&0.102&0.101&0.056 \end{bmatrix} $$
Matrix $P$ for our Markov chain $M$, that we get by normalizing rows of $A$ (note that values are rounded to three decimal places):
$$ P = \begin{bmatrix} 0.087&0.132&0.380&0.401\\ 0.121&0.095&0.426&0.358\\ 0.271&0.300&0.175&0.254\\ 0.309&0.272&0.270&0.149 \end{bmatrix} $$
Steady state distribution for chain $M$ is $$\pi= \begin{bmatrix} 0.211&0.213&0.298&0.278 \end{bmatrix}$$
$P'$ for $\delta=0.5$ (diagonal elements of $A$ are increased by $0.5$ and $A'$ is normalized by rows)
$$ P' = \begin{bmatrix} 0.489&0.074&0.213&0.224\\ 0.065&0.512&0.229&0.193\\ 0.105&0.116&0.681&0.098\\ 0.132&0.117&0.116&0.636 \end{bmatrix} $$
Corresponding steady-state distribution for $M'$ (with $\delta=0.5$) is: $$\pi'=\begin{bmatrix}0.172 & 0.180 & 0.351 & 0.296\end{bmatrix}$$ And also I computed $\pi_C$ (with $\delta=100$) which is: $$\pi_C= \begin{bmatrix} 0.139&0.153&0.395&0.312 \end{bmatrix}$$
Below is a plot of steady-state distribution with increasing $\delta$. We can observe that each value in $\pi$ either monotonically increases or monotonically decreases.
And a little background... I am constructing a new multi-criteria decision making method, which is in some sense simmilar to PageRank algorithm. We are trying to evaluate given alternatives, based on user preference. I discovered that increasing $\delta$ in some sense increases the separation between the alternatives.
-edit comment- Sorry for the not so clear explanation and the error that I made. Hope we can now understand eachother.