Coherent Risk Metrics

Naming note. “Coherent risk metric (CRM)” and “coherent risk measure” denote the same
object — the class characterized by Artzner et al.’s four axioms. majumdar2017how
writes metric; dixit2023risk and the original finance literature
write measure. This wiki’s canonical page is coherent_risk_measures;
this page is kept as the metric-spelling entry point and should be read as a synonym, not a
distinct concept (see Open Questions).

Definition

A risk metric is a map $\rho :\mathcal{Z}\to \mathbb{R}$ from a random cost $Z$ (over outcomes
$\omega$ in a probability space) to a real number — the perceived risk of incurring that cost. A risk
metric is coherent when it satisfies four axioms: monotonicity, translation invariance, positive
homogeneity, and subadditivity (Artzner et al. 1999; dixit2023risk Def. 1).
Coherent metrics interpolate between risk-neutral (expected cost) and worst-case assessments while
remaining convex and rational; non-coherent metrics (e.g. mean–variance, or Value-at-Risk) can prefer a
controller that is dominated outcome-by-outcome, which is unsafe (majumdar2017how §3.2).
The axioms are domain-agnostic — both sources develop them for terrestrial/automotive/aerial robots, not
for any space regime — so they apply unchanged to a free-flying space manipulator; only the cost $Z$
(e.g. collision distance, pointing error) is application-specific.

Key Equations

Symbols per notation.md. Risk-theory symbols $\rho ,Z,\mathcal{P},{\mathbb{E}}_p,c$
are reproduced source-faithfully and are not in the registry; $\alpha$ is the CVaR confidence level
from notation.md. The positive-homogeneity scalar $\beta \ge 0$ is the source’s symbol
(majumdar2017how A3; dixit2023risk writes
$\alpha$, which would collide with the CVaR level, so $\beta$ is used here) and is a local scalar here,
not the registry’s $\beta$ (the isotropic damping floor in the singularity layer). (No load-bearing
matrix/region glyph — $\mathbf{\Gamma }$, $\mathbf{\Lambda }$, $\Omega$ as the singularity-free region —
is reused here; $\Omega$ below is the outcome space, per the sources.)

The four coherence axioms, for costs $Z,Z^{\prime }\in \mathcal{Z}$, $c\in \mathbb{R}$, $\beta \ge 0$
(dixit2023risk Def. 1; majumdar2017how A1–A4):

$$\begin{array}{rlrl}& \text{Monotonicity:}& & Z\le Z^{\prime }\Rightarrow \rho (Z)\le \rho (Z^{\prime })\\ & \text{Translation invariance:}& & \rho (Z+c)=\rho (Z)+c\\ & \text{Positive homogeneity:}& & \rho (\beta Z)=\beta \rho (Z)\\ & \text{Subadditivity:}& & \rho (Z+Z^{\prime })\le \rho (Z)+\rho (Z^{\prime })\end{array}$$

Positive homogeneity and subadditivity together imply convexity of $\rho$
(majumdar2017how). Every coherent risk metric admits the dual
(risk-envelope) representation as a worst-case expectation over a compact convex set $\mathcal{P}$ of
probability measures (Artzner et al. 1999; majumdar2017how Eq. for CRMs;
dixit2023risk Def. 2):

$$\rho (Z)=\underset{p\in \mathcal{P}}{\max }{\mathbb{E}}_p[Z].$$

This makes coherent risk equivalent to distributional robustness: the risk envelope $\mathcal{P}$ is
the adversary’s set of admissible distributions.

Source Support

• majumdar2017how — primary: axiomatizes risk metrics for robotics,
  states Axioms A1–A4 as the coherent class (plus A5–A6 for the narrower distortion class), gives the
  risk-envelope representation, and shows by counterexample that mean–variance and VaR violate the
  axioms and behave unsafely.
• dixit2023risk — support: gives the same four axioms as Def. 1
  (“coherent risk measures”, crediting Artzner et al. 1999) and the dual representation as Def. 2,
  then builds a risk-averse MPC for obstacle avoidance on general coherent measures (CVaR, EVaR, total
  variation, g-entropic).

Related Topics

• coherent_risk_measures — the canonical page for this exact concept; this
  page is the synonymous metric-spelling entry. Content should not diverge.
• conditional_value_at_risk — the most-used coherent metric and the
  building block from which all distortion (spectral) risk metrics are composed.
• chance_constraints — equivalent to a constraint on VaR (${\mathrm{VaR}}_{\alpha }(Z)\le 0\iff \mathbb{P}[Z>0]\le \alpha$); VaR is not coherent (it fails subadditivity), motivating
  the move to coherent metrics.
• motion_planning_under_uncertainty — the planning setting where
  a coherent metric replaces expected cost to make the plan risk-aware.
• risk_aware_mpc — how dixit2023risk embeds a general
  coherent metric in a tractable (convex mixed-integer) receding-horizon scheme.

Open Questions

• Both sources develop coherence over a scalar cost $Z$ on a static outcome space. Multi-stage,
  free-flying inspection raises the time-consistency issue (nested vs. static risk over the horizon)
  that majumdar2017how §5 flags but does not resolve here — see
  risk_aware_mpc.
• For low-level control (vs. high-level planning), majumdar2017how
  Discussion 1 questions whether subadditivity (A4) is even meaningful when sub-costs (e.g. base
  attitude error vs. EE pointing error on our coupled system) cannot be diversified independently.
• Schema: this page and coherent_risk_measures are the same concept under
  two spellings — should one be a pure redirect rather than a parallel topic? (Flagged for human review;
  not resolved by an agent.)