Bi-Compositional Division Rules

Christoph Schlegel

Papers from arXiv.org

Abstract: 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 $\{r^\theta\}_{\theta\in[-\infty,+\infty]}$, 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 $u_\theta=\log\varphi_\theta$, where \[ \varphi_\theta(x):=\frac{e^{\theta x}-1}{\theta}\quad(\theta\ne0), \qquad \varphi_0(x):=x, \] 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.

Date: 2026-09
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