Let $A$ be a stochastic matrix. Thus $A$ has nonnegative entries, and the sum of the elements in each row is 1. This implies that the vector $\begin{bmatrix} 1 & 1 & \cdots & 1 \end{bmatrix}^T$ is an eigenvector corresponding to the eigenvalue 1. Is it true that there exists a vector $b$ such that
$$(A - I)x \geq b$$
has no solutions in $x$? If so, is there a simple proof?
Motivation: I've been trying to construct an answer to another question using linear programming duality (as the OP implies he is interested in). If my reasoning is correct, this is the only step I need to complete the argument. I feel like this should be an easy question to answer, but I've been working on it for a while with no success.