Time Consistency

Definition

Time consistency is a property of a risk assessment over a sequential (multi-period) decision
problem. Intuitively: if one situation is judged less risky than another in every state of the world
at time-step $k+1$, it must also be judged less risky at time-step $k$ (majumdar2017how).
A single static risk metric applied to the total cost $\rho ({\sum }_k{Z}_k)$ generally fails this
property; obtaining it requires a sequence of risk metrics $\{{\rho }_{k,N}{\}}_{k=0}^N$ — a dynamic
risk metric that re-assesses risk from each time-step. Violating it admits “irrational” behaviour,
e.g. deeming a plan infeasible at $k=0$ even though it satisfies the risk threshold in every successor
state under any realization of the uncertainty. The notion is regime-agnostic (it lives in the
risk/decision layer, not the dynamics), so it applies unchanged to our free-flying manipulator.

Key Equations

Symbols per notation.md.

Let $Z_k$ be the random stage cost and ${\rho }_{k,N}$ the risk assessed at step $k$ of the cost stream
$(Z_k,\dots ,Z_N)$. The dynamic metric $\{{\rho }_{k,N}{\}}_{k=0}^N$ is time-consistent if, for all
$0\le l\le k\le N$ and all stage-cost sequences $\{Z_i\},\{Z_i^{\prime }\}$,

$$(Z_i=Z_i^{\prime }\forall i=l,\dots ,k-1)\wedge {\rho }_{k,N}(Z_k,\dots ,Z_N)\le {\rho }_{k,N}(Z_k^{\prime },\dots ,Z_N^{\prime })⟹{\rho }_{l,N}(Z_l,\dots ,Z_N)\le {\rho }_{l,N}(Z_l^{\prime },\dots ,Z_N^{\prime }).$$

Any such metric (under mild conditions) is a nested composition of one-step risk metrics
${\rho }_k(\cdot )$ — and compounding them this way constructs a time-consistent metric:

$${\rho }_{k,N}=Z_k+{\rho }_k(Z_{k+1}+{\rho }_{k+1}(Z_{k+2}+\cdots +{\rho }_{N-1}(Z_N)\left)\right).$$

Notation note: here $\rho$ is a generic risk functional and $Z_i$ a random stage cost — neither is in
notation.md (that registry is mechanics-focused); $\alpha$ is the CVaR confidence
level already listed there. $\Omega$ here is the source’s probability sample space, not the
singularity-free region $\Omega$ in notation.md — a glyph collision, flagged below.

Source Support

• majumdar2017how — §5 (“Sequential Decision Making and Time
  Consistency”): defines time consistency and the local property, gives the CVaR scenario-tree
  counter-example where a static metric is inconsistent, and states the nested-composition theorem
  (after Ruszczyński 2010) characterizing time-consistent dynamic risk metrics. A robotics paper;
  regime-agnostic, no space-dynamics assumption.

Related Topics

• conditional_value_at_risk — CVaR is the one-step metric whose static,
  total-cost use furnishes the paper’s counter-example; nesting CVaR per step restores consistency.
• coherent_risk_measures — the axioms (A1–A4) for a single-period metric;
  time consistency is the additional multi-period requirement layered on top.
• coherent_risk_metrics — near-synonym page for the coherent-metric class;
  the one-step ${\rho }_k$ in the composition are drawn from it (specifically its distortion subclass).
• motion_planning_under_uncertainty — the sequential-planning
  setting (an MDP over the horizon) in which a time-inconsistent objective produces irrational plans.
• risk_aware_mpc — receding-horizon control re-solves the OCP each step, so a
  time-consistent (nested) risk objective is exactly what keeps successive horizons from contradicting.

Open Questions

• The source flags a tension between time consistency and interpretability: $\rho ({\sum }_k{Z}_k)$ is
  easy to read but inconsistent, while the nested composition is consistent but hard to interpret — is
  this trade-off avoidable for a risk-aware inspection plan?
• For our free-flying trajectory tracking, what are the natural per-step stage costs $Z_k$ (pointing
  error, collision proximity, singularity margin ${\sigma }_G$) to nest, and does nesting CVaR over them
  remain tractable in a receding-horizon solve?
• The paper proves the nested form is the only time-consistent class under mild conditions — do those
  conditions hold for the chance-constrained / continuous-state setting we actually plan in?