Bi-Compositional Division Rules

We characterise the division rules for claims problems that satisfy equal treatment of equals, bilateral consistency, composition down, and composition up. The rules are precisely the members of a one-parameter log-exponential family $\{r^{\theta }{\}}_{\theta \in [-\mathrm{\infty },+\mathrm{\infty }]}$, with constrained equal awards (CEA) and constrained equal losses (CEL) as its endpoints. For finite $\theta$, $r^{\theta }$ is the equal-sacrifice rule in awards for $u_{\theta }=\log {\phi }_{\theta }$, where $${\phi }_{\theta }(x):=\frac{e^{\theta x}-1}{\theta }(\theta \mathrm{\ne }0),{\phi }_0(x):=x,$$ and simultaneously the equal-sacrifice rule in losses for the dual utility $u_{-\theta }$. The proportional rule is the midpoint, $\theta =0$. Continuity of the rules are not assumed but a consequence of the other axioms.