Let $P,Q$ be two distributions on a finite set $X$. Consider the following metric*
​$$ d(P,Q) = \frac12\max_{\emptyset\neq A\subseteq X} \|P​(\cdot\mid A)-Q(\cdot\mid A)\|_1. $$
Obviously, the total variation metric $\frac12\|P-Q\|_1$ is majorized by $d(P,Q)$.

Question: has anyone encountered $d(P,Q)$ in the literature? Does it have a name?

*It's not immediately obvious that $d$ satisfies the triangle inequality, but I think this can be shown.