Let $S$ be a hyperbolic surface, which is not the punctured torus or $4$-holed sphere. I am interested in finding a ``geometrically optimal'' pants decomposition on $S$.
Here is a candidate definition. Given a pants decomposition $P$, order the curves of $P$ from longest to shortest (in the hyperbolic metric). Then, pants decompositions $P$ and $P'$ can be compared by comparing the lengths of their curves lexicographically. That is: if the longest curve $c_1$ of $P$ is shorter than the longest curve $c'_1$ of $P'$, then $P$ is better. Or, if $\ell(c_1) = \ell(c'_1)$ and $\ell(c_2) < \ell(c'_2)$, where $c_2, c'_2$ are the second-longest curves of $P$ and $P'$, then $P$ is better. And so on, lexicographically.
With this definition, the induced ordering on pants decompositions becomes a well-ordering. More precisely: given a fixed pants decomposition $P$, there are finitely many curves shorter than the longest curve of $P$, hence finitely many better pants decompositions. In particular, there exists an ``optimal'' decomposition, whose longest curve is no longer than the Bers constant.
It is clear that optimal decompositions are not necessarily unique (otherwise, the optimal decomposition would never change as we move in Teichmuller space). But if $P$ and $P'$ are both optimal on a given surface, what can be said about how far apart they are? For example: are their shortest curves necessarily disjoint?