 Constant nonzero drift is impossible on a finite Markov chain

For a real function $f$ on a finite state space, a state minimizing $f$ satisfies $Pf\ge f$, so strictly negative drift cannot hold at every state. Equivalently, telescoping the expected value would give ${\mathbb{E}}_{x}f(X_n)=f(x)-n\delta$, contradicting boundedness of $f$. Constant negative drift is useful before a target is reached; it must be restricted to the states outside that target.