I am stuck with the following problem in research.
Let $A_{1}$, $A_{2}$ and $B$ be stochastic matrices. Let $B = f(A_{1},A_{2})$. Let $\pi =[\pi_{1},\pi_{2},\pi_{3}]$ be a vector such that $\sum_{i} \pi = 1$.
Let $R_{1}$ denote the region $c^{T}_{1}\pi<c^{T}_{2}\pi$, where $c_{1}$ and $c_{2}$ are constant column vectors.
Find $A_{1}$ such that $\forall$ $\pi \in R_{1}$
$f_{1}(A_{1},B,\pi) \in R_{1}$
$f$ and $f_{1}$ are known functions.
Some direction will be useful. Thanks.