Risk-Assignment Consistency Constraints

• Risk-assignment consistency constraints are conditions ensuring that risk evaluations remain coherent when transported across time, stages, groups, or assignment structures, preventing paradoxes in dynamic risk assessments.
• They require structural properties like recursive compatibility, multi-portfolio time consistency, and additive invariance, which are vital in stochastic programming, routing, and set-valued risk measures.
• These constraints also underpin fairness in statistical learning and stable assignments, integrating techniques such as orthogonality conditions and convex-order solidity to maintain consistent decision-making.

Searching arXiv for recent and foundational papers relevant to risk-assignment consistency constraints across dynamic risk, stochastic programming, routing, fairness, and constrained optimization. Risk-assignment consistency constraints are conditions that make a risk evaluation, risk budget, or risk-based decision rule remain coherent when it is transported across time, stages, horizons, groups, feasible sets, or assignment structures. Across the literature, the expression does not denote a single universal axiom; it denotes a family of constraints that prevent re-optimization paradoxes, nonrecursive dynamic assessments, unstable rankings after appending common uncertainty, group-dependent error allocation, or feasibility distortions under exogenous constraints. In dynamic risk theory this family includes multi-portfolio time consistency, recursivity, restriction, and h-longevity; in multistage stochastic programming it includes decomposability together with conditions that preserve conditional optimality; in statistical learning it includes approximate constancy of conditional risk and second-best surrogate consistency; and in constrained risk sharing and rating it includes convex-order solidity, quasi-convexity, pooling-effect consistency, and diversification consistency (Feinstein et al., 2012, Shapiro et al., 2018, Cominetti et al., 2013, Donini et al., 2018, Nunno et al., 2023, Guo et al., 16 Jun 2025).

1. General structural patterns

A recurrent pattern is recursive compatibility: a present evaluation should agree with a future evaluation pulled backward through the same risk machinery. In multistage stochastic programming, this appears in the decomposable representation

$\mathcal Q = p_2 \circ p_3 \circ \cdots \circ p_T,$

or, more explicitly,

$$\mathcal Q(Z_1+\cdots+Z_T) = Z_1 + p_2\big(Z_2 + p_3(\cdots + p_T(Z_T))\big),$$

with each $p_t$ a coherent conditional risk mapping. In stochastic control, the same idea appears through a Bellman recursion on an augmented state carrying a risk threshold $r_k$, together with feasibility constraints of the form

$d(x_k,u)+\rho_k(r'(x_{k+1}))\le r_k.$

These formulations encode stagewise risk assignment rather than a single static ex ante bound (Shapiro et al., 2018, Chow et al., 2015).

A second pattern is invariance under common continuation or horizon extension. In routing, additive consistency requires

$$\rho(X)\le \rho(Y)\ \Rightarrow\ \rho(X+Z)\le \rho(Y+Z)\quad \text{for all } Z\perp (X,Y),$$

so a common independent downstream segment cannot reverse a preference ordering. In fully-dynamic risk measurement, the restriction property

$$P_{st}(X)=P_{su}(X),\qquad X\in L^p(\mathcal F_t),\ u>t,$$

requires that once a position is already $\mathcal F_t$-measurable, a later horizon should not alter its risk value. These are different manifestations of the same structural demand: a consistency constraint should neutralize artifacts introduced by a common continuation that carries no new position-specific content (Cominetti et al., 2013, Nunno et al., 2023).

A third pattern is ordering preservation under constraints. In fairness-constrained empirical risk minimization, the conditional risk on positive instances is required to be approximately constant across sensitive groups: $\big|L^{+,a}(f)-L^{+,b}(f)\big|\le \epsilon.$ In set-valued dynamic risk measures, ordering must be strong enough to survive backward propagation through sets of eligible portfolios. In Choquet rating criteria, quasi-convexity,

$r(\lambda X+(1-\lambda)Y)\le \max\{r(X),r(Y)\},$

plays the analogous role for categorical risk scores. This suggests a common interpretation: risk-assignment consistency constraints regulate how comparisons are propagated when the objects being compared are no longer single real numbers but groups, sets, portfolios, or discretized ratings (Donini et al., 2018, Feinstein et al., 2012, Guo et al., 16 Jun 2025).

2. Set-valued dynamic risk measures and transaction-cost markets

In markets with transaction costs, capital requirements are naturally set-valued because compensation can be made in a basket of currencies or assets rather than a single numeraire. The basic object is a conditional set-valued risk measure

$$\mathcal Q(Z_1+\cdots+Z_T) = Z_1 + p_2\big(Z_2 + p_3(\cdots + p_T(Z_T))\big),$$0

where $$\mathcal Q(Z_1+\cdots+Z_T) = Z_1 + p_2\big(Z_2 + p_3(\cdots + p_T(Z_T))\big),$$1 is the closed linear subspace of eligible portfolios. The map must satisfy $$\mathcal Q(Z_1+\cdots+Z_T) = Z_1 + p_2\big(Z_2 + p_3(\cdots + p_T(Z_T))\big),$$2-translativity,

$$\mathcal Q(Z_1+\cdots+Z_T) = Z_1 + p_2\big(Z_2 + p_3(\cdots + p_T(Z_T))\big),$$3

monotonicity,

$$\mathcal Q(Z_1+\cdots+Z_T) = Z_1 + p_2\big(Z_2 + p_3(\cdots + p_T(Z_T))\big),$$4

and finiteness at zero. Its primal representation is

$$\mathcal Q(Z_1+\cdots+Z_T) = Z_1 + p_2\big(Z_2 + p_3(\cdots + p_T(Z_T))\big),$$5

where $$\mathcal Q(Z_1+\cdots+Z_T) = Z_1 + p_2\big(Z_2 + p_3(\cdots + p_T(Z_T))\big),$$6 is the associated conditional acceptance set, and the correspondence between $$\mathcal Q(Z_1+\cdots+Z_T) = Z_1 + p_2\big(Z_2 + p_3(\cdots + p_T(Z_T))\big),$$7 and $$\mathcal Q(Z_1+\cdots+Z_T) = Z_1 + p_2\big(Z_2 + p_3(\cdots + p_T(Z_T))\big),$$8 is one-to-one (Feinstein et al., 2012).

The central consistency question is whether the scalar equivalence between time consistency and recursion survives in the set-valued setting. The natural extension of scalar time consistency is

$$\mathcal Q(Z_1+\cdots+Z_T) = Z_1 + p_2\big(Z_2 + p_3(\cdots + p_T(Z_T))\big),$$9

The paper shows that this is too weak. A risk measure can be time consistent in this sense and still fail to satisfy a Bellman-type recursion. The reason is that set inclusion among risk compensating portfolios does not control how many incomparable eligible portfolios propagate backward. The usual scalar implication “time consistency $p_t$0 recursion” therefore fails in the set-valued framework (Feinstein et al., 2012).

To restore recursion, the paper introduces multi-portfolio time consistency: $p_t$1 For a normalized dynamic risk measure, this property is equivalent to the recursive form

$p_t$2

and, under $p_t$3, also equivalent to an additive decomposition of the acceptance sets. A common misconception in this area is that the scalar notion of time consistency extends without loss to the vector-valued or set-valued case; the paper shows that the correct recursive notion is strictly stronger. In the scalar embedding, by contrast, the two notions coincide because risk sets reduce to upper intervals of the form

$p_t$4

The set-valued theory therefore replaces claim-by-claim consistency with consistency for collections of claims and collections of portfolios (Feinstein et al., 2012).

3. Multistage programs and risk-constrained control

In risk-averse multistage stochastic programming, decomposability provides the stagewise risk assignment: $p_t$5 A policy is time consistent when the tail policy remains optimal for every conditional subproblem. The crucial result is negative: decomposability of $p_t$6 is not by itself enough to guarantee time consistency of all optimal solutions. The implication from decomposability to time consistency holds under additional assumptions, specifically uniqueness of the optimal solution or strict monotonicity of the nested risk mappings. The latter is expressed as

$p_t$7

and for a coherent risk measure $p_t$8 it is characterized by

$p_t$9

The paper’s counterexample with $r_k$0 shows that a decomposable but non-strictly-monotone risk measure can admit one optimal policy that is time consistent and another with the same objective value that is not. Nestedness alone is therefore not a universal risk-assignment consistency guarantee (Shapiro et al., 2018).

In finite-horizon stochastic optimal control, the same issue appears as inconsistency of a static ex ante risk budget under re-optimization. The state is augmented by a risk-to-go threshold $r_k$1, and the value function is defined by

$r_k$2

The Bellman operator uses feasible pairs $r_k$3 satisfying

$r_k$4

A key refinement shows that the inequality can be replaced by equality without loss of optimality, which yields an exact stagewise risk assignment. The time-consistent reformulation imposes

$r_k$5

and the analytical risk-to-go update is

$r_k$6

The associated consistency relation,

$r_k$7

is the explicit stagewise constraint that prevents the re-optimization paradox illustrated by Haviv’s counterexample. In this formulation, time consistency is obtained not by weakening the risk constraint, but by endogenizing the evolution of the remaining risk budget (Chow et al., 2015).

4. Additivity, horizon matching, and dynamic risk evaluation

For route choice with random travel times, additive consistency is defined by

$r_k$8

A key lemma shows that for a risk measure this is equivalent to exact additivity on independent risks: $r_k$9 If arc travel times $d(x_k,u)+\rho_k(r'(x_{k+1}))\le r_k.$0 are independent and a path $d(x_k,u)+\rho_k(r'(x_{k+1}))\le r_k.$1 has travel time $d(x_k,u)+\rho_k(r'(x_{k+1}))\le r_k.$2, then

$d(x_k,u)+\rho_k(r'(x_{k+1}))\le r_k.$3

Risk-averse routing is thereby reduced to a deterministic shortest path problem with arc lengths $d(x_k,u)+\rho_k(r'(x_{k+1}))\le r_k.$4. The paper emphasizes that mean-standard deviation, Value-at-Risk, and Average Value-at-Risk fail this property, whereas the entropic risk measure

$d(x_k,u)+\rho_k(r'(x_{k+1}))\le r_k.$5

is additive on independent random variables. Within expected utility, dual theory, and rank-dependent expected utility, the only preferences satisfying additive consistency are those induced by the entropic risk measures, with expectation as the distortion-only special case (Cominetti et al., 2013).

Fully-dynamic risk measures introduce a second axis of consistency: horizon matching. A fully-dynamic risk measure is a two-parameter family

$d(x_k,u)+\rho_k(r'(x_{k+1}))\le r_k.$6

and the paper identifies horizon risk as the error arising when a measure designed for a later horizon $d(x_k,u)+\rho_k(r'(x_{k+1}))\le r_k.$7 is used to evaluate $d(x_k,u)+\rho_k(r'(x_{k+1}))\le r_k.$8 with $d(x_k,u)+\rho_k(r'(x_{k+1}))\le r_k.$9. Strong time-consistency is

$$\rho(X)\le \rho(Y)\ \Rightarrow\ \rho(X+Z)\le \rho(Y+Z)\quad \text{for all } Z\perp (X,Y),$$0

order time-consistency is

$$\rho(X)\le \rho(Y)\ \Rightarrow\ \rho(X+Z)\le \rho(Y+Z)\quad \text{for all } Z\perp (X,Y),$$1

and weak time-consistency is

$$\rho(X)\le \rho(Y)\ \Rightarrow\ \rho(X+Z)\le \rho(Y+Z)\quad \text{for all } Z\perp (X,Y),$$2

Weak and order time-consistency are equivalent, and for normalized fully-dynamic risk measures strong time-consistency is equivalent to restriction plus order time-consistency. The restriction property,

$$\rho(X)\le \rho(Y)\ \Rightarrow\ \rho(X+Z)\le \rho(Y+Z)\quad \text{for all } Z\perp (X,Y),$$3

eliminates horizon risk; its failure makes horizon dependence observable (Nunno et al., 2023).

The quantitative measure of horizon risk is the h-longevity indicator defined through

$$\rho(X)\le \rho(Y)\ \Rightarrow\ \rho(X+Z)\le \rho(Y+Z)\quad \text{for all } Z\perp (X,Y),$$4

Restriction is equivalent to $$\rho(X)\le \rho(Y)\ \Rightarrow\ \rho(X+Z)\le \rho(Y+Z)\quad \text{for all } Z\perp (X,Y),$$5. In BSDE-generated risk measures

$$\rho(X)\le \rho(Y)\ \Rightarrow\ \rho(X+Z)\le \rho(Y+Z)\quad \text{for all } Z\perp (X,Y),$$6

the paper proves

$$\rho(X)\le \rho(Y)\ \Rightarrow\ \rho(X+Z)\le \rho(Y+Z)\quad \text{for all } Z\perp (X,Y),$$7

If $$\rho(X)\le \rho(Y)\ \Rightarrow\ \rho(X+Z)\le \rho(Y+Z)\quad \text{for all } Z\perp (X,Y),$$8, h-longevity holds, and the horizon-risk indicator becomes

$$\rho(X)\le \rho(Y)\ \Rightarrow\ \rho(X+Z)\le \rho(Y+Z)\quad \text{for all } Z\perp (X,Y),$$9

An analogous equivalence between vanishing-at-zero drivers, normalization, and restriction is established for BSVIE-based risk measures as well. The conceptual implication is sharp: recursive consistency over evaluation times and correct matching of the risk horizon are distinct constraints, and only the fully-dynamic framework makes that separation explicit (Nunno et al., 2023).

5. Statistical learning, fairness, and constrained surrogate risk

In fairness-constrained classification, risk-assignment consistency is imposed by requiring the positive-class conditional risk to be approximately constant across sensitive groups: $$P_{st}(X)=P_{su}(X),\qquad X\in L^p(\mathcal F_t),\ u>t,$$0 This defines $$P_{st}(X)=P_{su}(X),\qquad X\in L^p(\mathcal F_t),\ u>t,$$1-fairness and generalizes Equal Opportunity. With the hard loss and $$P_{st}(X)=P_{su}(X),\qquad X\in L^p(\mathcal F_t),\ u>t,$$2, the condition becomes equality of true positive rates across groups; with the linear loss and $$P_{st}(X)=P_{su}(X),\qquad X\in L^p(\mathcal F_t),\ u>t,$$3, it becomes

$$P_{st}(X)=P_{su}(X),\qquad X\in L^p(\mathcal F_t),\ u>t,$$4

The corresponding constrained learning problem is

$$P_{st}(X)=P_{su}(X),\qquad X\in L^p(\mathcal F_t),\ u>t,$$5

and its empirical counterpart is the Fair ERM problem. Under a uniform learnability bound, if

$$P_{st}(X)=P_{su}(X),\qquad X\in L^p(\mathcal F_t),\ u>t,$$6

then with probability at least $$P_{st}(X)=P_{su}(X),\qquad X\in L^p(\mathcal F_t),\ u>t,$$7,

$$P_{st}(X)=P_{su}(X),\qquad X\in L^p(\mathcal F_t),\ u>t,$$8

and

$$P_{st}(X)=P_{su}(X),\qquad X\in L^p(\mathcal F_t),\ u>t,$$9

The fairness constraint therefore has both risk consistency and fairness consistency guarantees. In RKHS form, it becomes an orthogonality constraint,

$\mathcal F_t$0

and for $\mathcal F_t$1 it reduces to $\mathcal F_t$2, with a corresponding projected kernel

$\mathcal F_t$3

For linear models, the same condition becomes a data preprocessing step (Donini et al., 2018).

A different constrained-consistency problem arises when the classifier class is restricted by fairness, monotonicity, or interpretability and the Bayes rule need not be feasible. In that setting, the paper shows that when the constraint restricts only the prediction set,

$\mathcal F_t$4

hinge losses are the only surrogate losses that preserve consistency in second-best scenarios. The key proportionality condition is

$\mathcal F_t$5

and among common surrogate losses only the hinge loss satisfies it. Once functional-form restrictions are added, hinge loss is no longer sufficient by itself; consistency requires a classification-preserving reduction satisfying a sublevel-set condition and inclusion of the step classifier

$\mathcal F_t$6

This paper directly refutes the common expectation that a classification-calibrated surrogate remains valid under arbitrary feasible-set restrictions (Kitagawa et al., 2021).

Consistency under localization addresses yet another constrained regime. Localized SVMs combine local empirical SVMs over a regionalization

$\mathcal F_t$7

with measurable weights $\mathcal F_t$8 summing to one. Under a uniform overlap bound, kernel-family control by $\mathcal F_t$9, moment conditions compatible with the loss growth type $\big|L^{+,a}(f)-L^{+,b}(f)\big|\le \epsilon.$0, and regularization/sample-size balances such as

$\big|L^{+,a}(f)-L^{+,b}(f)\big|\le \epsilon.$1

the empirical localized predictor converges to the theoretical localized predictor in $\big|L^{+,a}(f)-L^{+,b}(f)\big|\le \epsilon.$2, and the risk converges to the Bayes risk: $\big|L^{+,a}(f)-L^{+,b}(f)\big|\le \epsilon.$3 The result shows that localization, overlapping regions, and varying kernels do not by themselves destroy risk consistency, provided the regionalization and regularization constraints are scaled appropriately (Köhler, 2023).

6. Assignment, feasibility, and constrained operational systems

In stable matching with assignment constraints, the central consistency question is whether designer-imposed constraints are compatible with stability. The original worker- and firm-based constraints are reduced to a pair-based problem: $\big|L^{+,a}(f)-L^{+,b}(f)\big|\le \epsilon.$4 The structural tool is the iterated deletion of unattractive alternatives (IDUA), which computes a normal form $\big|L^{+,a}(f)-L^{+,b}(f)\big|\le \epsilon.$5 preserving stable matchings: $\big|L^{+,a}(f)-L^{+,b}(f)\big|\le \epsilon.$6 A crucial consistency lemma states that if a vertex $\big|L^{+,a}(f)-L^{+,b}(f)\big|\le \epsilon.$7 has zero out-degree in one direction, then a stable matching in the current digraph avoids $\big|L^{+,a}(f)-L^{+,b}(f)\big|\le \epsilon.$8 if and only if it is a stable matching after deleting $\big|L^{+,a}(f)-L^{+,b}(f)\big|\le \epsilon.$9. The paper’s point is that forbidden pairs cannot be deleted arbitrarily; they can be deleted only when they are currently safe to delete. The main algorithm solves the reduced problem in time

$r(\lambda X+(1-\lambda)Y)\le \max\{r(X),r(Y)\},$0

with $r(\lambda X+(1-\lambda)Y)\le \max\{r(X),r(Y)\},$1 time between consecutive outputs, where $r(\lambda X+(1-\lambda)Y)\le \max\{r(X),r(Y)\},$2 is the number of feasible stable matchings output. Here feasibility consistency is expressed as compatibility between pair-level assignment constraints and the blocking-pair structure of stable outcomes (Gutin et al., 2022).

In risk-aware multi-robot assignment and planning, consistency is enforced structurally through the propagation of uncertainty from perception to planning to assignment. Bayesian SegNet with dropout produces a pixelwise uncertainty

$r(\lambda X+(1-\lambda)Y)\le \max\{r(X),r(Y)\},$3

and the risk-aware traversal cost is

$r(\lambda X+(1-\lambda)Y)\le \max\{r(X),r(Y)\},$4

Candidate paths are generated by A*, and assignment is performed by maximizing CVaR of the total efficiency

$r(\lambda X+(1-\lambda)Y)\le \max\{r(X),r(Y)\},$5

under a partition matroid constraint $r(\lambda X+(1-\lambda)Y)\le \max\{r(X),r(Y)\},$6: $r(\lambda X+(1-\lambda)Y)\le \max\{r(X),r(Y)\},$7 The objective is

$r(\lambda X+(1-\lambda)Y)\le \max\{r(X),r(Y)\},$8

and the algorithm removes a selected vehicle from further consideration after each greedy choice,

$r(\lambda X+(1-\lambda)Y)\le \max\{r(X),r(Y)\},$9

The consistency mechanism is not a separate symbolic constraint linking perception and assignment; it is the fact that uncertainty modifies the cost map, which modifies path costs and path efficiencies, and those efficiencies are exactly the random inputs seen by the CVaR assignment problem (Sharma et al., 2020).

Heavy-traffic redundancy systems with assignment constraints offer a queueing-theoretic analogue. Jobs of type $$\mathcal Q(Z_1+\cdots+Z_T) = Z_1 + p_2\big(Z_2 + p_3(\cdots + p_T(Z_T))\big),$$00 can only be assigned to servers in $$\mathcal Q(Z_1+\cdots+Z_T) = Z_1 + p_2\big(Z_2 + p_3(\cdots + p_T(Z_T))\big),$$01, and stability is characterized by

$$\mathcal Q(Z_1+\cdots+Z_T) = Z_1 + p_2\big(Z_2 + p_3(\cdots + p_T(Z_T))\big),$$02

The decisive structural condition is the local stability or Complete Resource Pooling condition,

$$\mathcal Q(Z_1+\cdots+Z_T) = Z_1 + p_2\big(Z_2 + p_3(\cdots + p_T(Z_T))\big),$$03

Under this condition, as $$\mathcal Q(Z_1+\cdots+Z_T) = Z_1 + p_2\big(Z_2 + p_3(\cdots + p_T(Z_T))\big),$$04,

$$\mathcal Q(Z_1+\cdots+Z_T) = Z_1 + p_2\big(Z_2 + p_3(\cdots + p_T(Z_T))\big),$$05

Thus strict assignment constraints do not prevent full resource pooling in the heavy-traffic limit so long as no local bottleneck appears before the global one. The paper’s message is not that assignment constraints are irrelevant, but that under the explicit CRP inequalities they become asymptotically insensitive at the scale of the heavy-traffic collapse (Cardinaels et al., 2020).

7. Feasible-set stability, distributional robustness, and rating consistency

In constrained risk sharing, the decisive property is componentwise convex-order solidity. For a feasible set $$\mathcal Q(Z_1+\cdots+Z_T) = Z_1 + p_2\big(Z_2 + p_3(\cdots + p_T(Z_T))\big),$$06, this means

$$\mathcal Q(Z_1+\cdots+Z_T) = Z_1 + p_2\big(Z_2 + p_3(\cdots + p_T(Z_T))\big),$$07

Under this condition, every feasible allocation admits a feasible comonotonic improvement: $$\mathcal Q(Z_1+\cdots+Z_T) = Z_1 + p_2\big(Z_2 + p_3(\cdots + p_T(Z_T))\big),$$08 and for convex-order-consistent risk measures,

$$\mathcal Q(Z_1+\cdots+Z_T) = Z_1 + p_2\big(Z_2 + p_3(\cdots + p_T(Z_T))\big),$$09

This restores the classical comonotonic reduction under constraints. The paper also identifies important failures: Value-at-Risk caps and idiosyncratic deductibles are excluded, and VaR caps can lead to optimal allocations that are non-comonotonic in the aggregate loss. The structural lesson is that a feasible set must be stable under componentwise risk reduction if it is to preserve the economic logic of constrained Pareto optima (Blier-Wong et al., 27 Apr 2026).

Distributionally robust risk constraints address consistency from a sampling perspective. In the risk-constrained program

$$\mathcal Q(Z_1+\cdots+Z_T) = Z_1 + p_2\big(Z_2 + p_3(\cdots + p_T(Z_T))\big),$$10

the Wasserstein-robust counterpart replaces the unknown law $$\mathcal Q(Z_1+\cdots+Z_T) = Z_1 + p_2\big(Z_2 + p_3(\cdots + p_T(Z_T))\big),$$11 by

$$\mathcal Q(Z_1+\cdots+Z_T) = Z_1 + p_2\big(Z_2 + p_3(\cdots + p_T(Z_T))\big),$$12

Under continuity, convexity in $$\mathcal Q(Z_1+\cdots+Z_T) = Z_1 + p_2\big(Z_2 + p_3(\cdots + p_T(Z_T))\big),$$13, boundedness and uniform Lipschitz continuity in $$\mathcal Q(Z_1+\cdots+Z_T) = Z_1 + p_2\big(Z_2 + p_3(\cdots + p_T(Z_T))\big),$$14, a finite-sample radius guarantee with $$\mathcal Q(Z_1+\cdots+Z_T) = Z_1 + p_2\big(Z_2 + p_3(\cdots + p_T(Z_T))\big),$$15 and $$\mathcal Q(Z_1+\cdots+Z_T) = Z_1 + p_2\big(Z_2 + p_3(\cdots + p_T(Z_T))\big),$$16, nonempty interior of the original feasible set, and compactness of optimizer sets, the optimal values and optimizers of the distributionally robust CVaR-constrained and chance-constrained programs converge almost surely to those of the original stochastic problems. In particular, eventually

$$\mathcal Q(Z_1+\cdots+Z_T) = Z_1 + p_2\big(Z_2 + p_3(\cdots + p_T(Z_T))\big),$$17

and any accumulation point of robust optimizers is an optimizer of the original problem. The robust constraint is therefore asymptotically conservative but statistically consistent (Cherukuri et al., 2020).

Choquet rating criteria formulate consistency for categorical risk scores. A rating criterion $$\mathcal Q(Z_1+\cdots+Z_T) = Z_1 + p_2\big(Z_2 + p_3(\cdots + p_T(Z_T))\big),$$18 is represented by a risk measure $$\mathcal Q(Z_1+\cdots+Z_T) = Z_1 + p_2\big(Z_2 + p_3(\cdots + p_T(Z_T))\big),$$19 through

$$\mathcal Q(Z_1+\cdots+Z_T) = Z_1 + p_2\big(Z_2 + p_3(\cdots + p_T(Z_T))\big),$$20

and in the Choquet case

$$\mathcal Q(Z_1+\cdots+Z_T) = Z_1 + p_2\big(Z_2 + p_3(\cdots + p_T(Z_T))\big),$$21

The paper studies three notions: risk aversion consistency $$\mathcal Q(Z_1+\cdots+Z_T) = Z_1 + p_2\big(Z_2 + p_3(\cdots + p_T(Z_T))\big),$$22, pooling effect consistency $$\mathcal Q(Z_1+\cdots+Z_T) = Z_1 + p_2\big(Z_2 + p_3(\cdots + p_T(Z_T))\big),$$23, and diversification consistency $$\mathcal Q(Z_1+\cdots+Z_T) = Z_1 + p_2\big(Z_2 + p_3(\cdots + p_T(Z_T))\big),$$24. Under law invariance, the hierarchy is

$$\mathcal Q(Z_1+\cdots+Z_T) = Z_1 + p_2\big(Z_2 + p_3(\cdots + p_T(Z_T))\big),$$25

and under scenario-based law invariance,

$$\mathcal Q(Z_1+\cdots+Z_T) = Z_1 + p_2\big(Z_2 + p_3(\cdots + p_T(Z_T))\big),$$26

For a distortion risk measure

$$\mathcal Q(Z_1+\cdots+Z_T) = Z_1 + p_2\big(Z_2 + p_3(\cdots + p_T(Z_T))\big),$$27

the characterizations are

$$\mathcal Q(Z_1+\cdots+Z_T) = Z_1 + p_2\big(Z_2 + p_3(\cdots + p_T(Z_T))\big),$$28

and

$$\mathcal Q(Z_1+\cdots+Z_T) = Z_1 + p_2\big(Z_2 + p_3(\cdots + p_T(Z_T))\big),$$29

For $$\mathcal Q(Z_1+\cdots+Z_T) = Z_1 + p_2\big(Z_2 + p_3(\cdots + p_T(Z_T))\big),$$30-distortion risk measures

$$\mathcal Q(Z_1+\cdots+Z_T) = Z_1 + p_2\big(Z_2 + p_3(\cdots + p_T(Z_T))\big),$$31

quasi-convexity is equivalent to coherence and to $$\mathcal Q(Z_1+\cdots+Z_T) = Z_1 + p_2\big(Z_2 + p_3(\cdots + p_T(Z_T))\big),$$32 being componentwise concave and submodular. Because the Choquet representation is unique, these properties transfer exactly between the rating criterion and the representing risk measure. The case studies on collateralized loan obligations and catastrophe bonds show that these consistency notions are operational: criteria such as Average ES and Average MAXVAR reward pooling and diversification, whereas Average VaR, Max VaR, and Average PD can behave non-monotonically or perversely (Guo et al., 16 Jun 2025).

Taken together, these strands support a unified interpretation. Risk-assignment consistency constraints do not merely ask whether a risk functional is well defined; they ask whether its induced decisions remain coherent after recursion, concatenation, conditioning, discretization, sampling, or feasibility restriction. The strongest negative results are correspondingly structural: standard time consistency is too weak in the set-valued case, decomposability is too weak without strict monotonicity, many popular route-choice risk measures are not additively consistent, hinge consistency fails under unrestricted functional-form constraints, and VaR-type feasible sets need not be stable under risk reduction. The positive results are equally structural: multi-portfolio time consistency restores recursion, exact risk-to-go updates restore time-consistent control, restriction removes horizon risk, orthogonality constraints control group-conditioned error allocation, convex-order solidity restores comonotonic improvement, Wasserstein ambiguity sets yield asymptotically correct robust constraints, and concavity or submodularity of Choquet distortions restores prudent rating orderings.