  Every nontrivial action of the braid group $B_n$ on $\mathbb{R}$ by
orientation-preserving homeomorphisms yields, up to conjugation by a
homeomorphism of $\mathbb{R}$, a representation $\rho : B_n \rightarrow
\mathrm{H\widetilde{ome}o}_+(S^1)$ and therefore determines a translation
number for every element of $B_n$. In this manuscript we offer a simple
characterisation of which actions of $B_n$ on $\mathbb{R}$ produce translation
numbers that agree with those arising from the standard Nielsen-Thurston action
on $\mathbb{R}$. Our approach is to prove an analogous statement concerning
left orderings of $B_n$ via a technique that uses the space of left orderings
of $B_n$, the isolated points in this space, and the natural conjugacy action
of $B_n$. We use this result to extend recent rigidity results of Mann and
Wolff concerning mapping class group actions on $S^1$ to the case of low-genus
surfaces with marked points.
