Consider two $n \times n$ stochastic matrices $A$ and $B$. If for any two probability vectors $x$, $y$ in $R^n$, we have $xA=yA$ implies $xB=yB$, what can we say about the relationship of $A$ and $B$?
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Any vector in $R^n$ whose coordinates sum to $0$ is a scalar multiple of the difference $x-y$ between probability vectors $x$ and $y$. So, your condition on $A$ and $B$ can be restated as $$\mathbf1^\perp \cap\text{ker}\,A\subseteq\text{ker}\,B,$$ where $\mathbf1:=[1,\dots,1]^\top$, $\mathbf1^\perp$ is the orthogonal complement of $\{\mathbf1\}$, and $\text{ker}\,A$ and $\text{ker}\,B$ are the null spaces of $A$ and $B$, respectively.
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$\begingroup$ In the first sentence, do you want to restrict to vectors whose coordinates sum to 0? $\endgroup$ Commented yesterday
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$\begingroup$ @PaulLarson : Thank you for your comment. This is now fixed. $\endgroup$ Commented yesterday