An optimised certainty equivalent makes the cash adjustment and the preference model visible instead of hiding risk attitude inside one score
The optimised certainty equivalent evaluates an uncertain outcome by choosing a current cash amount and applying a utility or loss function to what remains, yielding a flexible family of convex risk measures when signs and assumptions are handled carefully.
Classical certainty equivalents ask what sure amount is as desirable as an uncertain outcome. Ben-Tal and Teboulle’s optimised certainty equivalent adds an explicit optimisation over a cash shift. This construction can generate familiar and less familiar convex risk measures and connect preferences, penalties and computation.
The formula is not self-justifying. Results depend on whether the random quantity represents gains or losses, the utility or loss function, probability model, units, sample, constraints and stakeholder authority. Sign conventions must be declared before any interpretation.
Chapter contents
- 1. Define gains, losses and the sign convention
- 2. Contrast ordinary and optimised certainty equivalents
- 3. Construct the OCE from a cash shift and expected utility
- 4. Generate a convex risk measure with a loss function
- 5. Read the dual representation and probability penalty
- 6. Worked example: compare two projects under one elicited loss function
- 7. Estimate the OCE from finite samples
- 8. Solve the cash optimisation reliably
- 9. Elicit utility or loss functions without disguising value choices
- 10. Handle ambiguity in probabilities and preferences
- 11. Embed OCE criteria in optimisation and compare alternatives
- 12. Govern use, review failures and preserve contestability
Choose a reading route
- Route 1: Set the convention — outcomes, losses, cash and utility
- Route 2: Build the measure — optimisation, convexity and duality
- Route 3: Work an example — discrete outcomes and sensitivity
- Route 4: Elicit preferences — data, ambiguity and governance
- Route 5: Use and compare — optimisation, limits and review
How to use this guide
Start with the chapters that match your immediate decision, then return to the worked example and audit chapters. Each chapter separates a model from the evidence needed to use it responsibly. A claim is not strengthened merely by being technical; it is strengthened when assumptions, observations, alternatives and limits can be inspected.
| Layer | Question | Minimum record |
|---|---|---|
| Concept | What exactly is being represented? | Definitions and boundary |
| Method | How will the reasoning be carried out? | Steps, inputs and assumptions |
| Transfer | Would it work in a changed setting? | A deliberately altered case |
| Audit | What could prove it wrong? | Evidence, limits and revision trigger |
Chapter 1. Define gains, losses and the sign convention
The idea
Risk-measure formulas change meaning when a positive number is treated as gain in one source and loss in another. The OCE should be stated in one convention and translated carefully before comparison.
A disciplined method
Name the random variable, units and direction of preference; write the formula; test a constant outcome and a cash shift; document how reported risk maps back to the decision.
A useful working record makes the sequence visible: first state the question and unit of analysis; next record the observations and their provenance; then apply the stated rule; finally compare the result with plausible alternatives. For this chapter, the decisive choice is whether larger values are better or worse and how cash invariance appears. Keeping that choice explicit prevents a later reader from mistaking a convenient convention for a fact about the world.
Worked example
A project net payoff uses gains-positive notation, while an operational loss model uses losses-positive notation; the team writes an explicit transformation before comparing scores.
Work the example twice. On the first pass, follow the stated assumptions. On the second, reverse one assumption, remove one piece of evidence and ask whether the conclusion, ranking or boundary changes. This small stress test matters because polished outputs can hide sensitivity. The objective is not to force certainty; it is to locate the evidence on which the conclusion actually turns.
Changed-context transfer check
Reverse the sign of every outcome and verify the translated result rather than reusing the same interpretation.
A transfer succeeds only when the structure of the problem survives the change. Match roles, dependencies and measurement conditions rather than surface vocabulary. Record what was preserved, what changed, and why the method still applies—or why it does not. If the new context changes the data-generating process or the people bearing the burden, re-specify the model instead of importing the old answer.
Limits, evidence and revision
Limit: Published conventions vary, and a sign error can invert rankings while leaving calculations apparently smooth.
Evidence to retain: variable definition, units, formula, constant test, cash-shift test and translation
Common misconception: Risk measures use one universal sign convention.
A gold-standard conclusion is therefore conditional. It should name the observation period, excluded cases, uncertainty and the event that would trigger review. Negative findings and unresolved contradictions belong in the record. They protect future readers from repeating a hidden assumption and make correction possible without reconstructing the entire analysis from memory.
Practical evidence check
Can another reader recover the input, reproduce the intermediate decision and identify at least one observation that would change the result? If not, the chapter has produced an explanation but not yet an auditable finding.
Chapter 2. Contrast ordinary and optimised certainty equivalents
The idea
An ordinary certainty equivalent in expected-utility theory inverts expected utility. An optimised certainty equivalent instead chooses a cash amount that balances present value with expected utility of the residual outcome. The two are related but not interchangeable.
A disciplined method
Write both definitions, apply them to a constant and a simple lottery, identify normalisation assumptions and state which decision interpretation is intended.
A useful working record makes the sequence visible: first state the question and unit of analysis; next record the observations and their provenance; then apply the stated rule; finally compare the result with plausible alternatives. For this chapter, the decisive choice is what the optimisation over the cash shift changes. Keeping that choice explicit prevents a later reader from mistaking a convenient convention for a fact about the world.
Worked example
For a two-outcome payoff, the ordinary certainty equivalent solves an inverse-utility equation; the OCE searches over a cash shift and can imply a different numerical criterion.
Work the example twice. On the first pass, follow the stated assumptions. On the second, reverse one assumption, remove one piece of evidence and ask whether the conclusion, ranking or boundary changes. This small stress test matters because polished outputs can hide sensitivity. The objective is not to force certainty; it is to locate the evidence on which the conclusion actually turns.
Changed-context transfer check
Use linear utility and confirm that optimisation does not manufacture risk aversion absent from the function.
A transfer succeeds only when the structure of the problem survives the change. Match roles, dependencies and measurement conditions rather than surface vocabulary. Record what was preserved, what changed, and why the method still applies—or why it does not. If the new context changes the data-generating process or the people bearing the burden, re-specify the model instead of importing the old answer.
Limits, evidence and revision
Limit: Different utility normalisations or domains can make comparisons misleading.
Evidence to retain: utility function, inverse or optimiser, normalisation, results and interpretation
Common misconception: Adding the word optimised merely improves numerical computation of the same certainty equivalent.
A gold-standard conclusion is therefore conditional. It should name the observation period, excluded cases, uncertainty and the event that would trigger review. Negative findings and unresolved contradictions belong in the record. They protect future readers from repeating a hidden assumption and make correction possible without reconstructing the entire analysis from memory.
Practical evidence check
Can another reader recover the input, reproduce the intermediate decision and identify at least one observation that would change the result? If not, the chapter has produced an explanation but not yet an auditable finding.
Chapter 3. Construct the OCE from a cash shift and expected utility
The idea
In a common gains convention, the OCE takes the best value of a cash amount plus expected utility of the remaining uncertain payoff after that amount is removed. The optimisation exposes a one-dimensional allocation between sure cash and residual risk.
A disciplined method
Choose a concave nondecreasing utility with appropriate normalisation, define the probability model, optimise the cash variable and check existence, units and boundary behaviour.
A useful working record makes the sequence visible: first state the question and unit of analysis; next record the observations and their provenance; then apply the stated rule; finally compare the result with plausible alternatives. For this chapter, the decisive choice is which cash shift maximises the criterion and why. Keeping that choice explicit prevents a later reader from mistaking a convenient convention for a fact about the world.
Worked example
A discrete three-outcome payoff is evaluated over candidate cash shifts; the optimum balances immediate cash against utility of the shifted residual outcomes.
Work the example twice. On the first pass, follow the stated assumptions. On the second, reverse one assumption, remove one piece of evidence and ask whether the conclusion, ranking or boundary changes. This small stress test matters because polished outputs can hide sensitivity. The objective is not to force certainty; it is to locate the evidence on which the conclusion actually turns.
Changed-context transfer check
Widen the loss tail and observe how the optimal shift and value respond.
A transfer succeeds only when the structure of the problem survives the change. Match roles, dependencies and measurement conditions rather than surface vocabulary. Record what was preserved, what changed, and why the method still applies—or why it does not. If the new context changes the data-generating process or the people bearing the burden, re-specify the model instead of importing the old answer.
Limits, evidence and revision
Limit: An optimiser may not exist or may be non-unique without suitable assumptions; utility outside its intended domain can be invalid.
Evidence to retain: scenario outcomes, probabilities, utility, objective curve, optimiser and boundary checks
Common misconception: The optimal cash shift must equal the reported OCE value.
A gold-standard conclusion is therefore conditional. It should name the observation period, excluded cases, uncertainty and the event that would trigger review. Negative findings and unresolved contradictions belong in the record. They protect future readers from repeating a hidden assumption and make correction possible without reconstructing the entire analysis from memory.
Practical evidence check
Can another reader recover the input, reproduce the intermediate decision and identify at least one observation that would change the result? If not, the chapter has produced an explanation but not yet an auditable finding.
Chapter 4. Generate a convex risk measure with a loss function
The idea
The negative OCE or an equivalent loss-based formulation can define a convex risk measure. Convexity reflects diversification within the model, while monotonicity and cash invariance depend on the admissible loss or utility function and convention.
A disciplined method
Translate to a loss function explicitly, verify convexity and monotonicity conditions, test cash shifts and mixtures, and avoid claiming coherence unless positive homogeneity also holds.
A useful working record makes the sequence visible: first state the question and unit of analysis; next record the observations and their provenance; then apply the stated rule; finally compare the result with plausible alternatives. For this chapter, the decisive choice is which axioms the chosen function actually satisfies. Keeping that choice explicit prevents a later reader from mistaking a convenient convention for a fact about the world.
Worked example
An exponential-type loss yields a convex risk criterion with a smooth penalty for adverse outcomes but need not behave like a coherent spectral measure.
Work the example twice. On the first pass, follow the stated assumptions. On the second, reverse one assumption, remove one piece of evidence and ask whether the conclusion, ranking or boundary changes. This small stress test matters because polished outputs can hide sensitivity. The objective is not to force certainty; it is to locate the evidence on which the conclusion actually turns.
Changed-context transfer check
Scale every loss by two and test whether the score doubles; if not, do not label it positively homogeneous.
A transfer succeeds only when the structure of the problem survives the change. Match roles, dependencies and measurement conditions rather than surface vocabulary. Record what was preserved, what changed, and why the method still applies—or why it does not. If the new context changes the data-generating process or the people bearing the burden, re-specify the model instead of importing the old answer.
Limits, evidence and revision
Limit: Convexity is a mathematical property under the assumed model and does not prove social acceptability or physical diversification.
Evidence to retain: function shape, axiom tests, mixture test, scaling test and counterexample
Common misconception: Every convex risk measure is coherent.
A gold-standard conclusion is therefore conditional. It should name the observation period, excluded cases, uncertainty and the event that would trigger review. Negative findings and unresolved contradictions belong in the record. They protect future readers from repeating a hidden assumption and make correction possible without reconstructing the entire analysis from memory.
Practical evidence check
Can another reader recover the input, reproduce the intermediate decision and identify at least one observation that would change the result? If not, the chapter has produced an explanation but not yet an auditable finding.
Chapter 5. Read the dual representation and probability penalty
The idea
Convex OCE risk measures can be represented through alternative probability weights penalised according to a convex conjugate or divergence-like term. The dual view shows which distributional distortions drive the risk value.
A disciplined method
Derive or cite the conjugate, define admissible densities, compute the penalty and inspect the worst or influential reweightings rather than treating dual variables as literal forecasts.
A useful working record makes the sequence visible: first state the question and unit of analysis; next record the observations and their provenance; then apply the stated rule; finally compare the result with plausible alternatives. For this chapter, the decisive choice is which scenarios receive additional weight and at what penalty. Keeping that choice explicit prevents a later reader from mistaking a convenient convention for a fact about the world.
Worked example
A tail-heavy alternative distribution raises the score but pays a divergence penalty; the optimum reveals the balance set by the loss function.
Work the example twice. On the first pass, follow the stated assumptions. On the second, reverse one assumption, remove one piece of evidence and ask whether the conclusion, ranking or boundary changes. This small stress test matters because polished outputs can hide sensitivity. The objective is not to force certainty; it is to locate the evidence on which the conclusion actually turns.
Changed-context transfer check
Add an extreme rare outcome and inspect whether its reweighting dominates.
A transfer succeeds only when the structure of the problem survives the change. Match roles, dependencies and measurement conditions rather than surface vocabulary. Record what was preserved, what changed, and why the method still applies—or why it does not. If the new context changes the data-generating process or the people bearing the burden, re-specify the model instead of importing the old answer.
Limits, evidence and revision
Limit: Dual representations require technical conditions, and penalty choices inherit the preference model.
Evidence to retain: conjugate, alternative weights, penalties, normalisation, optimiser and sensitivity
Common misconception: The dual distribution is the true probability distribution.
A gold-standard conclusion is therefore conditional. It should name the observation period, excluded cases, uncertainty and the event that would trigger review. Negative findings and unresolved contradictions belong in the record. They protect future readers from repeating a hidden assumption and make correction possible without reconstructing the entire analysis from memory.
Practical evidence check
Can another reader recover the input, reproduce the intermediate decision and identify at least one observation that would change the result? If not, the chapter has produced an explanation but not yet an auditable finding.
Chapter 6. Worked example: compare two projects under one elicited loss function
The idea
A discrete example makes cash optimisation, preference sensitivity and model limits visible without suggesting a universal risk appetite.
A disciplined method
Define two payoff distributions with equal mean, choose and justify one normalised loss function, solve the one-dimensional optimisation, compare results and repeat with alternatives.
A useful working record makes the sequence visible: first state the question and unit of analysis; next record the observations and their provenance; then apply the stated rule; finally compare the result with plausible alternatives. For this chapter, the decisive choice is which outcomes and curvature change the ranking. Keeping that choice explicit prevents a later reader from mistaking a convenient convention for a fact about the world.
Worked example
Project A has stable moderate payoffs; Project B has higher upside and a severe downside. Under one risk-averse function A ranks higher despite equal mean; weaker curvature narrows the gap.
Work the example twice. On the first pass, follow the stated assumptions. On the second, reverse one assumption, remove one piece of evidence and ask whether the conclusion, ranking or boundary changes. This small stress test matters because polished outputs can hide sensitivity. The objective is not to force certainty; it is to locate the evidence on which the conclusion actually turns.
Changed-context transfer check
Add a shared shock to both projects and reassess rankings and diversification claims.
A transfer succeeds only when the structure of the problem survives the change. Match roles, dependencies and measurement conditions rather than surface vocabulary. Record what was preserved, what changed, and why the method still applies—or why it does not. If the new context changes the data-generating process or the people bearing the burden, re-specify the model instead of importing the old answer.
Limits, evidence and revision
Limit: The fictional preferences are illustrative and should not govern real finance, safety or public policy.
Evidence to retain: outcomes, probabilities, function, optimiser, OCE value, alternative functions and ranking trace
Common misconception: One OCE ranking reveals the objectively best project.
A gold-standard conclusion is therefore conditional. It should name the observation period, excluded cases, uncertainty and the event that would trigger review. Negative findings and unresolved contradictions belong in the record. They protect future readers from repeating a hidden assumption and make correction possible without reconstructing the entire analysis from memory.
| Step | Calculation | Judgement check |
|---|---|---|
| Model | outcomes and probabilities | credible scenarios? |
| Preference | loss or utility function | whose values? |
| Optimise | cash shift and objective | global and stable? |
| Compare | values and sensitivity | ranking robust? |
Practical evidence check
Can another reader recover the input, reproduce the intermediate decision and identify at least one observation that would change the result? If not, the chapter has produced an explanation but not yet an auditable finding.
Chapter 7. Estimate the OCE from finite samples
The idea
In practice the expectation is often replaced by a sample average. Estimation error affects both the objective and the optimal cash shift, especially with heavy tails or sparse extreme losses.
A disciplined method
Preserve independent evaluation data, calculate empirical OCE, use suitable uncertainty analysis, inspect tail coverage and repeat across resamples and plausible distribution models.
A useful working record makes the sequence visible: first state the question and unit of analysis; next record the observations and their provenance; then apply the stated rule; finally compare the result with plausible alternatives. For this chapter, the decisive choice is how sampling variation changes the score and optimiser. Keeping that choice explicit prevents a later reader from mistaking a convenient convention for a fact about the world.
Worked example
A sample with no severe event produces a reassuring empirical value; stress augmentation and resampling reveal that the ranking is unstable.
Work the example twice. On the first pass, follow the stated assumptions. On the second, reverse one assumption, remove one piece of evidence and ask whether the conclusion, ranking or boundary changes. This small stress test matters because polished outputs can hide sensitivity. The objective is not to force certainty; it is to locate the evidence on which the conclusion actually turns.
Changed-context transfer check
Double the sample with observations from a shifted environment and test for distribution change before pooling.
A transfer succeeds only when the structure of the problem survives the change. Match roles, dependencies and measurement conditions rather than surface vocabulary. Record what was preserved, what changed, and why the method still applies—or why it does not. If the new context changes the data-generating process or the people bearing the burden, re-specify the model instead of importing the old answer.
Limits, evidence and revision
Limit: Ordinary bootstrap methods can fail for heavy tails or dependent data; domain-specific validation is needed.
Evidence to retain: sample definition, dependence, tail diagnostics, estimates, uncertainty and stress results
Common misconception: Optimising a sample average removes estimation uncertainty.
A gold-standard conclusion is therefore conditional. It should name the observation period, excluded cases, uncertainty and the event that would trigger review. Negative findings and unresolved contradictions belong in the record. They protect future readers from repeating a hidden assumption and make correction possible without reconstructing the entire analysis from memory.
Practical evidence check
Can another reader recover the input, reproduce the intermediate decision and identify at least one observation that would change the result? If not, the chapter has produced an explanation but not yet an auditable finding.
Chapter 8. Solve the cash optimisation reliably
The idea
The core optimisation is often one-dimensional and convex or concave under appropriate formulation, but numerical errors, poor domains and nonsmooth loss functions can still mislead.
A disciplined method
Derive first-order or subgradient conditions, bracket plausible cash shifts, compare analytic and numerical solutions, test endpoints and report tolerance and solver status.
A useful working record makes the sequence visible: first state the question and unit of analysis; next record the observations and their provenance; then apply the stated rule; finally compare the result with plausible alternatives. For this chapter, the decisive choice is whether the returned point is globally optimal and numerically stable. Keeping that choice explicit prevents a later reader from mistaking a convenient convention for a fact about the world.
Worked example
A piecewise-linear loss produces an interval of optimisers; the implementation reports the set or a documented tie rule rather than fake precision.
Work the example twice. On the first pass, follow the stated assumptions. On the second, reverse one assumption, remove one piece of evidence and ask whether the conclusion, ranking or boundary changes. This small stress test matters because polished outputs can hide sensitivity. The objective is not to force certainty; it is to locate the evidence on which the conclusion actually turns.
Changed-context transfer check
Rescale currency units and verify the formulation and tolerances behave consistently.
A transfer succeeds only when the structure of the problem survives the change. Match roles, dependencies and measurement conditions rather than surface vocabulary. Record what was preserved, what changed, and why the method still applies—or why it does not. If the new context changes the data-generating process or the people bearing the burden, re-specify the model instead of importing the old answer.
Limits, evidence and revision
Limit: Solver convergence does not validate the loss function, probability model or decision context.
Evidence to retain: objective plot, conditions, bracket, solver logs, tolerances and independent check
Common misconception: A solver success flag proves the OCE calculation is correct.
A gold-standard conclusion is therefore conditional. It should name the observation period, excluded cases, uncertainty and the event that would trigger review. Negative findings and unresolved contradictions belong in the record. They protect future readers from repeating a hidden assumption and make correction possible without reconstructing the entire analysis from memory.
Practical evidence check
Can another reader recover the input, reproduce the intermediate decision and identify at least one observation that would change the result? If not, the chapter has produced an explanation but not yet an auditable finding.
Chapter 9. Elicit utility or loss functions without disguising value choices
The idea
The function determines how outcomes are traded. It may be inferred from choices, calibrated to policy thresholds or selected for mathematical convenience, but those routes answer different questions.
A disciplined method
Identify the decision owner and affected groups, use interpretable reference gambles or trade-offs, test consistency, allow ranges or multiple functions and record authority and dissent.
A useful working record makes the sequence visible: first state the question and unit of analysis; next record the observations and their provenance; then apply the stated rule; finally compare the result with plausible alternatives. For this chapter, the decisive choice is whose preferences and constraints the function represents. Keeping that choice explicit prevents a later reader from mistaking a convenient convention for a fact about the world.
Worked example
A risk committee compares choices over small and large stakes; inconsistent curvature leads to a set of plausible functions rather than one fitted curve.
Work the example twice. On the first pass, follow the stated assumptions. On the second, reverse one assumption, remove one piece of evidence and ask whether the conclusion, ranking or boundary changes. This small stress test matters because polished outputs can hide sensitivity. The objective is not to force certainty; it is to locate the evidence on which the conclusion actually turns.
Changed-context transfer check
Ask affected users rather than only executives and test whether the acceptable function set changes.
A transfer succeeds only when the structure of the problem survives the change. Match roles, dependencies and measurement conditions rather than surface vocabulary. Record what was preserved, what changed, and why the method still applies—or why it does not. If the new context changes the data-generating process or the people bearing the burden, re-specify the model instead of importing the old answer.
Limits, evidence and revision
Limit: Observed choices can reflect framing, liquidity, misunderstanding or strategic behaviour rather than stable utility.
Evidence to retain: elicitation protocol, participants, stakes, responses, fitted set, inconsistencies and approval
Common misconception: A technically convenient loss function is value-neutral.
A gold-standard conclusion is therefore conditional. It should name the observation period, excluded cases, uncertainty and the event that would trigger review. Negative findings and unresolved contradictions belong in the record. They protect future readers from repeating a hidden assumption and make correction possible without reconstructing the entire analysis from memory.
Practical evidence check
Can another reader recover the input, reproduce the intermediate decision and identify at least one observation that would change the result? If not, the chapter has produced an explanation but not yet an auditable finding.
Chapter 10. Handle ambiguity in probabilities and preferences
The idea
An OCE calculated under one distribution and one utility function can be fragile. Robust variants consider sets of distributions or functions, but additional robustness choices create additional assumptions.
A disciplined method
Define credible ambiguity sets, separate probability and preference uncertainty, compute best and worst cases, inspect drivers and compare with transparent scenario sensitivity.
A useful working record makes the sequence visible: first state the question and unit of analysis; next record the observations and their provenance; then apply the stated rule; finally compare the result with plausible alternatives. For this chapter, the decisive choice is which uncertainty changes the decision rather than only the decimal score. Keeping that choice explicit prevents a later reader from mistaking a convenient convention for a fact about the world.
Worked example
A Wasserstein neighbourhood protects against nearby distribution shifts while a set of utility functions captures elicitation uncertainty; their effects are reported separately.
Work the example twice. On the first pass, follow the stated assumptions. On the second, reverse one assumption, remove one piece of evidence and ask whether the conclusion, ranking or boundary changes. This small stress test matters because polished outputs can hide sensitivity. The objective is not to force certainty; it is to locate the evidence on which the conclusion actually turns.
Changed-context transfer check
Increase the ambiguity radius and locate the point where the decision flips.
A transfer succeeds only when the structure of the problem survives the change. Match roles, dependencies and measurement conditions rather than surface vocabulary. Record what was preserved, what changed, and why the method still applies—or why it does not. If the new context changes the data-generating process or the people bearing the burden, re-specify the model instead of importing the old answer.
Limits, evidence and revision
Limit: Oversized sets can make results unusably conservative; data-driven radii and preference sets still need justification.
Evidence to retain: ambiguity construction, calibration, robust values, worst cases, flip points and rationale
Common misconception: Robust OCE eliminates model risk.
A gold-standard conclusion is therefore conditional. It should name the observation period, excluded cases, uncertainty and the event that would trigger review. Negative findings and unresolved contradictions belong in the record. They protect future readers from repeating a hidden assumption and make correction possible without reconstructing the entire analysis from memory.
Practical evidence check
Can another reader recover the input, reproduce the intermediate decision and identify at least one observation that would change the result? If not, the chapter has produced an explanation but not yet an auditable finding.
Chapter 11. Embed OCE criteria in optimisation and compare alternatives
The idea
OCE objectives can be embedded in portfolio, allocation or control problems. Comparison with expected utility, expected shortfall, spectral measures and distributionally robust objectives reveals which behaviour comes from the criterion.
A disciplined method
State the decision variables and constraints, solve matched alternatives on the same scenarios, evaluate out of sample and explain practical differences in selected actions.
A useful working record makes the sequence visible: first state the question and unit of analysis; next record the observations and their provenance; then apply the stated rule; finally compare the result with plausible alternatives. For this chapter, the decisive choice is which modelling choice drives the action rather than merely the score. Keeping that choice explicit prevents a later reader from mistaking a convenient convention for a fact about the world.
Worked example
An allocation chosen by OCE is compared with mean-variance and expected-shortfall allocations under common costs and holdout scenarios.
Work the example twice. On the first pass, follow the stated assumptions. On the second, reverse one assumption, remove one piece of evidence and ask whether the conclusion, ranking or boundary changes. This small stress test matters because polished outputs can hide sensitivity. The objective is not to force certainty; it is to locate the evidence on which the conclusion actually turns.
Changed-context transfer check
Add a hard safety constraint and check whether all risk criteria respect it.
A transfer succeeds only when the structure of the problem survives the change. Match roles, dependencies and measurement conditions rather than surface vocabulary. Record what was preserved, what changed, and why the method still applies—or why it does not. If the new context changes the data-generating process or the people bearing the burden, re-specify the model instead of importing the old answer.
Limits, evidence and revision
Limit: Optimisation may exploit model errors; a preferred in-sample objective can perform poorly after shift.
Evidence to retain: formulations, common data, solutions, active constraints, holdout outcomes and explanation
Common misconception: A flexible risk objective can replace hard feasibility and safety constraints.
A gold-standard conclusion is therefore conditional. It should name the observation period, excluded cases, uncertainty and the event that would trigger review. Negative findings and unresolved contradictions belong in the record. They protect future readers from repeating a hidden assumption and make correction possible without reconstructing the entire analysis from memory.
Practical evidence check
Can another reader recover the input, reproduce the intermediate decision and identify at least one observation that would change the result? If not, the chapter has produced an explanation but not yet an auditable finding.
Chapter 12. Govern use, review failures and preserve contestability
The idea
A risk score can influence people and resources. Governance should expose convention, function, data, uncertainty, authority, thresholds and routes to challenge or override the result.
A disciplined method
Maintain a model card and decision record, predefine review triggers, monitor realised outcomes, audit disparate effects, require human authority for material uses and provide appeal or correction.
A useful working record makes the sequence visible: first state the question and unit of analysis; next record the observations and their provenance; then apply the stated rule; finally compare the result with plausible alternatives. For this chapter, the decisive choice is who can stop or revise the model when preferences or conditions change. Keeping that choice explicit prevents a later reader from mistaking a convenient convention for a fact about the world.
Worked example
A service-allocation model is paused when tail outcomes fall outside validation range and affected users challenge the elicited trade-offs.
Work the example twice. On the first pass, follow the stated assumptions. On the second, reverse one assumption, remove one piece of evidence and ask whether the conclusion, ranking or boundary changes. This small stress test matters because polished outputs can hide sensitivity. The objective is not to force certainty; it is to locate the evidence on which the conclusion actually turns.
Changed-context transfer check
Transfer the method to a new population and require new elicitation and validation instead of assuming portability.
A transfer succeeds only when the structure of the problem survives the change. Match roles, dependencies and measurement conditions rather than surface vocabulary. Record what was preserved, what changed, and why the method still applies—or why it does not. If the new context changes the data-generating process or the people bearing the burden, re-specify the model instead of importing the old answer.
Limits, evidence and revision
Limit: Governance cannot make an unsuitable quantification fair; some harms should not be traded through one scalar objective.
Evidence to retain: model record, authority, monitoring, incidents, appeals, overrides and retirement decision
Common misconception: Documenting the formula makes every use legitimate.
A gold-standard conclusion is therefore conditional. It should name the observation period, excluded cases, uncertainty and the event that would trigger review. Negative findings and unresolved contradictions belong in the record. They protect future readers from repeating a hidden assumption and make correction possible without reconstructing the entire analysis from memory.
Practical evidence check
Can another reader recover the input, reproduce the intermediate decision and identify at least one observation that would change the result? If not, the chapter has produced an explanation but not yet an auditable finding.
Field workbook
Create two three-outcome payoff distributions with the same mean. Declare a sign convention, choose a normalised utility or loss function, plot the cash-shift objective, find the optimum, compare with ordinary certainty equivalent and expected shortfall, then repeat across two plausible functions and one distributional stress.
- Write the decision question in one sentence and name the decision owner.
- List observations separately from interpretations, with dates and provenance.
- Record at least two plausible alternatives before choosing a preferred account.
- Run the changed-context test and note which assumptions fail.
- State the limit, confidence and next review trigger in plain language.
| Layer | Evidence | Failure signal |
|---|---|---|
| Convention | gain or loss definition and cash test | sign reversal |
| Preference | elicited function or set | hidden value choice |
| Computation | objective, optimiser and tolerance | solver-only assurance |
| Decision | sensitivity and governance | fragile scalar ranking |
Continue through the verified ecosystem
- Expected utility and certainty equivalents — classical foundations and nearby terminology
- Conditional value at risk — expected shortfall and coherent tail limits
- Spectral risk measures — quantile weights and tail preferences
- Distributionally robust optimisation — ambiguity sets and out-of-sample protection
- SMAA — missing preferences and robust ranking regions
Frequently asked questions
What is an optimised certainty equivalent?
A criterion that chooses a cash shift and combines it with expected utility of the remaining uncertain outcome.
Is it the same as an ordinary certainty equivalent?
No. Ordinary certainty equivalents invert expected utility; OCE adds an optimisation over a cash amount and has different representation properties.
Why does the sign convention matter?
Sources may model gains or losses, so a missing sign translation can reverse interpretation and rankings.
Is every OCE risk measure coherent?
No. Appropriate constructions are convex, while positive homogeneity and full coherence require additional structure.
Can the utility or loss function be estimated from choices?
It can be elicited or calibrated, but observed choices may be noisy, framed or inconsistent, so ranges and sensitivity are often safer.
Does a robust OCE remove model risk?
No. It makes selected ambiguity explicit but still depends on the ambiguity set, data, function, constraints and decision context.
Authoritative sources and further reading
- Ben-Tal and Teboulle: The Optimized Certainty Equivalent — definition, properties and convex-risk representation
- Tel Aviv University research record: Optimized Certainty Equivalent — bibliographic and publication record
- Artzner et al.: Coherent Measures of Risk — coherence axioms and comparison
- Bartl, Drapeau and Tangpi: Robust Optimized Certainty Equivalents — model uncertainty and computational representation
- Wu and Xu: Preference-Robust Modified OCE — uncertain utility and preference robustness
This is an educational account of a risk and preference framework, not financial advice. It does not validate any probability model, utility function, ambiguity set, constraint, stakeholder mandate or use for non-compensable harms.