The group of quasi-isometries of the real line cannot act effectively on the line

We prove that the group $\mathrm{QI}^{+}(\mathbb{R})$ of orientation-preserving quasi-isometries of the real line is a left-orderable, non-simple group, which cannot act effectively on the real line $\mathbb{R}.$
Minor changes; final version, to appear in Algebraic and Geometric Topology