THE GOLD STANDARD OF JUDGEMENT · SPECIALIST STUDY
Compare whole outcome distributions without inventing one utility score
Stochastic dominance asks whether one uncertain prospect is preferred by every decision maker in a stated utility class. First-order dominance represents people who prefer more; second-order dominance adds risk aversion. When curves cross, the result may correctly remain partial. Statistical tests and optimization can extend the method, but sampling uncertainty and normative assumptions must stay visible.
This guide belongs to the Judgement subject guide, the Gold Standard of Education, and the How X Works learning library. Choose a reading route, work the examples, then use the transfer checks to test whether the method survives a changed context.
Choose a reading route
Build the foundations · chapters 1–2
Apply the core method · chapters 3–5
Test evidence and alternatives · chapters 6–7
Handle limits and disagreement · chapters 8–9
Transfer and govern · chapters 10
The whole problem at a glance
| Order | Distribution condition | Utility class |
|---|---|---|
| First order | F_A(x) ≤ F_B(x) for all x, with strict inequality somewhere | All increasing utility functions |
| Second order | The integral of F_A up to x is no greater than B for all x, with suitable endpoint conditions | All increasing, concave utility functions |
| Restricted or higher order | Additional bounds or repeated integrals | A narrower or more shape-constrained class |
| No dominance | Required inequalities fail or evidence is inconclusive | Preference remains partial without more assumptions |
For costs or losses, signs and directions must be transformed carefully. Always state whether larger or smaller outcomes are better.
CHAPTER 1 OF 10 · Build the foundations
1. Define outcomes, direction and the utility class
Core idea. Stochastic dominance orders distributions relative to a class of admissible utility functions. The analyst must define what the outcome means, whether more is better and which preference restrictions are justified. The important discipline is to name the object being assessed, the boundary around it and the kind of conclusion the evidence can support. Without those three choices, technical vocabulary creates confidence without creating a checkable claim.
Method. Specify support, units, population, horizon and utility class before examining curves. Transform losses consistently and avoid mixing individual welfare with institutional objectives without explanation. Write the procedure before looking at the preferred result, retain intermediate evidence and record who can challenge each assumption. A method becomes reusable when another reader can repeat the steps, locate disagreements and understand why a different answer emerged.
Worked example. Two investments return the same units over one year. The question is whether every increasing-utility investor, or every increasing risk-averse investor, would prefer one distribution. The example is deliberately small enough to inspect. Its purpose is not to imitate every real case but to expose the decision rule, the evidence it consumes and the point at which judgement enters.
Changed-context transfer. Change the outcome from household income to waiting time, where smaller is better. The CDF inequalities must be reoriented rather than copied. Keep the underlying principle while changing the actors, incentives or data. If the conclusion survives only because the original labels were familiar, the learner has memorised a case rather than acquired a transferable method.
Limits. People may care about multiple attributes, reference points or catastrophic thresholds not captured by one scalar outcome. A responsible conclusion therefore states what remains unknown, what evidence could reverse it and when the analysis must be repeated. Precision about limits is part of the result, not a disclaimer added after it.
Misconception to avoid. Stochastic dominance discovers preferences without assumptions. It reduces assumptions relative to one utility function but still depends on an entire utility class. Test the tempting shortcut against a counterexample and ask which stakeholder bears the cost when it fails. The correction should alter the workflow, not merely add cautious wording.
Practical evidence check
- Write the preference class in words and mathematical restrictions, then ask whether every stakeholder accepts that normative boundary.
- Return to chapter contents
CHAPTER 2 OF 10 · Build the foundations
2. Read cumulative distributions as threshold evidence
Core idea. A cumulative distribution function gives the probability of an outcome at or below each threshold. Comparing CDFs checks many thresholds at once rather than focusing on mean, variance or selected quantiles. The important discipline is to name the object being assessed, the boundary around it and the kind of conclusion the evidence can support. Without those three choices, technical vocabulary creates confidence without creating a checkable claim.
Method. Plot empirical CDFs on a common support, inspect endpoints and discontinuities, and calculate differences with a clear sign convention. Pair graphs with a table at decision-relevant thresholds. Write the procedure before looking at the preferred result, retain intermediate evidence and record who can challenge each assumption. A method becomes reusable when another reader can repeat the steps, locate disagreements and understand why a different answer emerged.
Worked example. At every threshold, prospect A has no more probability in the lower tail than B. A therefore offers at least as much chance of exceeding every threshold. The example is deliberately small enough to inspect. Its purpose is not to imitate every real case but to expose the decision rule, the evidence it consumes and the point at which judgement enters.
Changed-context transfer. If outcomes are costs, use survival or transformed benefit distributions so lower cost is interpreted correctly. Keep the underlying principle while changing the actors, incentives or data. If the conclusion survives only because the original labels were familiar, the learner has memorised a case rather than acquired a transferable method.
Limits. Plots can hide tiny violations, ties or measurement error. Numerical evaluation and inference remain necessary. A responsible conclusion therefore states what remains unknown, what evidence could reverse it and when the analysis must be repeated. Precision about limits is part of the result, not a disclaimer added after it.
Misconception to avoid. A higher mean implies a lower CDF everywhere. Distributions can share or reverse means while their CDFs cross. Test the tempting shortcut against a counterexample and ask which stakeholder bears the cost when it fails. The correction should alter the workflow, not merely add cautious wording.
Practical evidence check
- Choose five thresholds and manually calculate both cumulative probabilities before trusting a software plot.
- Return to chapter contents
CHAPTER 3 OF 10 · Apply the core method
3. Apply first-order stochastic dominance
Core idea. For outcomes where more is preferred, A first-order dominates B when its CDF is everywhere no greater and strictly lower somewhere. Every increasing utility function then weakly prefers A. The important discipline is to name the object being assessed, the boundary around it and the kind of conclusion the evidence can support. Without those three choices, technical vocabulary creates confidence without creating a checkable claim.
Method. Compare the full support, handle equality carefully, and verify endpoint probability. For discrete data, evaluate jump points; for continuous data, use justified grids plus analytical or numerical safeguards. Write the procedure before looking at the preferred result, retain intermediate evidence and record who can challenge each assumption. A method becomes reusable when another reader can repeat the steps, locate disagreements and understand why a different answer emerged.
Worked example. A moves probability mass from outcome 0 to outcome 5 without reducing any higher outcome. Its CDF falls at low thresholds and never rises above B. The example is deliberately small enough to inspect. Its purpose is not to imitate every real case but to expose the decision rule, the evidence it consumes and the point at which judgement enters.
Changed-context transfer. Add a small chance of a very poor outcome to A. One CDF crossing can remove first-order dominance even if A retains a higher mean. Keep the underlying principle while changing the actors, incentives or data. If the conclusion survives only because the original labels were familiar, the learner has memorised a case rather than acquired a transferable method.
Limits. Finite grids may miss narrow crossings, and estimated CDFs contain sampling error. A responsible conclusion therefore states what remains unknown, what evidence could reverse it and when the analysis must be repeated. Precision about limits is part of the result, not a disclaimer added after it.
Misconception to avoid. First-order dominance means every person prefers A. It covers increasing expected-utility preferences over the chosen scalar outcome, not every behavioural or multi-attribute preference. Test the tempting shortcut against a counterexample and ask which stakeholder bears the cost when it fails. The correction should alter the workflow, not merely add cautious wording.
Practical evidence check
- Compute all discrete jump points and identify exactly where the inequality is strict.
- Return to chapter contents
CHAPTER 4 OF 10 · Apply the core method
4. Apply second-order dominance for risk aversion
Core idea. Second-order dominance integrates CDF differences and represents all increasing concave utility functions. It can order prospects whose CDFs cross when one offers a sufficiently favourable distribution to risk-averse decision makers. The important discipline is to name the object being assessed, the boundary around it and the kind of conclusion the evidence can support. Without those three choices, technical vocabulary creates confidence without creating a checkable claim.
Method. Compute cumulative integrals across the support with consistent endpoints and means. Verify orientation from the chosen definition and test the inequalities over all relevant thresholds. Write the procedure before looking at the preferred result, retain intermediate evidence and record who can challenge each assumption. A method becomes reusable when another reader can repeat the steps, locate disagreements and understand why a different answer emerged.
Worked example. A and B share a mean, but A concentrates probability near the centre while B puts more in both tails. A may second-order dominate B for increasing concave utility. The example is deliberately small enough to inspect. Its purpose is not to imitate every real case but to expose the decision rule, the evidence it consumes and the point at which judgement enters.
Changed-context transfer. If B has a much higher mean, reduced spread alone may not give A dominance. Both location and dispersion enter the integrated condition. Keep the underlying principle while changing the actors, incentives or data. If the conclusion survives only because the original labels were familiar, the learner has memorised a case rather than acquired a transferable method.
Limits. Second-order dominance assumes global concavity and expected utility. Loss aversion, ambiguity or changing risk attitudes may not fit. A responsible conclusion therefore states what remains unknown, what evidence could reverse it and when the analysis must be repeated. Precision about limits is part of the result, not a disclaimer added after it.
Misconception to avoid. Lower variance proves second-order dominance. Variance is only one summary and can miss distribution shape and mean differences. Test the tempting shortcut against a counterexample and ask which stakeholder bears the cost when it fails. The correction should alter the workflow, not merely add cautious wording.
Practical evidence check
- Numerically integrate the empirical CDF difference and verify the result against expected utility for several concave functions.
- Return to chapter contents
CHAPTER 5 OF 10 · Apply the core method
5. Treat no dominance as a valid partial preference
Core idea. Dominance is intentionally incomplete. When required inequalities fail, the method may leave two prospects unordered rather than forcing a choice with hidden weights. The important discipline is to name the object being assessed, the boundary around it and the kind of conclusion the evidence can support. Without those three choices, technical vocabulary creates confidence without creating a checkable claim.
Method. Report crossings and the utility functions or thresholds that drive disagreement. Add assumptions only when they are substantively justified and accepted by the decision process. Write the procedure before looking at the preferred result, retain intermediate evidence and record who can challenge each assumption. A method becomes reusable when another reader can repeat the steps, locate disagreements and understand why a different answer emerged.
Worked example. A offers better low outcomes but B offers a larger upper tail. Some increasing utilities prefer A and others B, so first-order dominance correctly returns no order. The example is deliberately small enough to inspect. Its purpose is not to imitate every real case but to expose the decision rule, the evidence it consumes and the point at which judgement enters.
Changed-context transfer. A safety regulator may impose a catastrophe constraint that resolves the choice, while an investor may use a stated utility or mandate. The added rule should be explicit. Keep the underlying principle while changing the actors, incentives or data. If the conclusion survives only because the original labels were familiar, the learner has memorised a case rather than acquired a transferable method.
Limits. Decision makers often require action despite incomparability. Use robust options, deliberation or value-sensitive criteria rather than pretending dominance exists. A responsible conclusion therefore states what remains unknown, what evidence could reverse it and when the analysis must be repeated. Precision about limits is part of the result, not a disclaimer added after it.
Misconception to avoid. An incomplete result means the method failed. It often reveals genuine value conflict or insufficient evidence. Test the tempting shortcut against a counterexample and ask which stakeholder bears the cost when it fails. The correction should alter the workflow, not merely add cautious wording.
Practical evidence check
- Find two admissible utility functions that rank a crossing pair differently and explain the policy implication.
- Return to chapter contents
Worked discrete comparison
| Outcome | Probability A | Probability B | Cumulative A / B |
|---|---|---|---|
| 0 | 0.10 | 0.25 | 0.10 / 0.25 |
| 5 | 0.40 | 0.25 | 0.50 / 0.50 |
| 10 | 0.50 | 0.50 | 1.00 / 1.00 |
A has no larger cumulative probability at any threshold and is strictly lower at 0, so A first-order dominates B when higher outcomes are preferred.
CHAPTER 6 OF 10 · Test evidence and alternatives
6. Estimate dominance under sampling uncertainty
Core idea. Empirical CDFs are estimates. Apparent crossings or gaps can reflect sampling variation, dependence, unequal sample sizes or survey design. Statistical inference asks whether observed violations are compatible with the tested order. The important discipline is to name the object being assessed, the boundary around it and the kind of conclusion the evidence can support. Without those three choices, technical vocabulary creates confidence without creating a checkable claim.
Method. Declare the null and alternative, sampling unit, dependence structure, test statistic, contact-set treatment and bootstrap. Use methods designed for inequality restrictions rather than pointwise confidence alone. Write the procedure before looking at the preferred result, retain intermediate evidence and record who can challenge each assumption. A method becomes reusable when another reader can repeat the steps, locate disagreements and understand why a different answer emerged.
Worked example. Two programmes have empirical CDFs separated at most thresholds. A bootstrap test evaluates whether the supremum violation is too large for the dominance null under the sampling design. The example is deliberately small enough to inspect. Its purpose is not to imitate every real case but to expose the decision rule, the evidence it consumes and the point at which judgement enters.
Changed-context transfer. With clustered outcomes by school, resampling individuals independently understates uncertainty. Cluster-aware resampling changes the conclusion. Keep the underlying principle while changing the actors, incentives or data. If the conclusion survives only because the original labels were familiar, the learner has memorised a case rather than acquired a transferable method.
Limits. Dominance tests can have low power near contact points and are sensitive to tuning, support truncation and dependence. A responsible conclusion therefore states what remains unknown, what evidence could reverse it and when the analysis must be repeated. Precision about limits is part of the result, not a disclaimer added after it.
Misconception to avoid. Failure to reject proves dominance. It only means the test did not find sufficient evidence against its null. Test the tempting shortcut against a counterexample and ask which stakeholder bears the cost when it fails. The correction should alter the workflow, not merely add cautious wording.
Practical evidence check
- Write the null in symbols and prose, then verify that the software rejection direction matches the reported sentence.
- Return to chapter contents
CHAPTER 7 OF 10 · Test evidence and alternatives
7. Use restricted and almost-dominance carefully
Core idea. Restricted dominance narrows the utility class or support; almost dominance tolerates bounded violations. These approaches can be useful when strict global dominance is too demanding, but thresholds introduce additional value choices. The important discipline is to name the object being assessed, the boundary around it and the kind of conclusion the evidence can support. Without those three choices, technical vocabulary creates confidence without creating a checkable claim.
Method. Justify derivative bounds, outcome ranges or violation ratios before analysis. Report sensitivity across plausible settings and show where excluded preferences would disagree. Write the procedure before looking at the preferred result, retain intermediate evidence and record who can challenge each assumption. A method becomes reusable when another reader can repeat the steps, locate disagreements and understand why a different answer emerged.
Worked example. A tiny CDF crossing occurs in a measurement-noisy extreme tail. A pre-specified violation bound may support an almost-dominance statement while preserving the strict result as no dominance. The example is deliberately small enough to inspect. Its purpose is not to imitate every real case but to expose the decision rule, the evidence it consumes and the point at which judgement enters.
Changed-context transfer. If the crossing occurs at a catastrophic threshold, its small probability mass may still be morally decisive. Tail importance cannot be inferred from size alone. Keep the underlying principle while changing the actors, incentives or data. If the conclusion survives only because the original labels were familiar, the learner has memorised a case rather than acquired a transferable method.
Limits. Flexible restrictions can be chosen after seeing data to manufacture an order. Independent specification and sensitivity checks are essential. A responsible conclusion therefore states what remains unknown, what evidence could reverse it and when the analysis must be repeated. Precision about limits is part of the result, not a disclaimer added after it.
Misconception to avoid. Almost dominance is nearly the same as dominance. It is a different, parameter-dependent claim. Test the tempting shortcut against a counterexample and ask which stakeholder bears the cost when it fails. The correction should alter the workflow, not merely add cautious wording.
Practical evidence check
- Vary the restriction over a justified range and identify the exact stakeholder preferences excluded at each setting.
- Return to chapter contents
CHAPTER 8 OF 10 · Handle limits and disagreement
8. Embed dominance constraints in optimization
Core idea. Optimization can require a chosen portfolio or policy distribution to dominate a benchmark. This converts distributional preference into constraints rather than one expected-value objective. The important discipline is to name the object being assessed, the boundary around it and the kind of conclusion the evidence can support. Without those three choices, technical vocabulary creates confidence without creating a checkable claim.
Method. Represent the distribution under each decision, impose dominance inequalities on support points or scenarios, and inspect dual values, feasibility and approximation error. Write the procedure before looking at the preferred result, retain intermediate evidence and record who can challenge each assumption. A method becomes reusable when another reader can repeat the steps, locate disagreements and understand why a different answer emerged.
Worked example. A pension portfolio maximizes expected return subject to second-order dominance over a benchmark, protecting a broad class of risk-averse beneficiaries. The example is deliberately small enough to inspect. Its purpose is not to imitate every real case but to expose the decision rule, the evidence it consumes and the point at which judgement enters.
Changed-context transfer. Add transaction costs and estimation uncertainty. A formerly feasible dominance constraint may become brittle, motivating robust or softened formulations. Keep the underlying principle while changing the actors, incentives or data. If the conclusion survives only because the original labels were familiar, the learner has memorised a case rather than acquired a transferable method.
Limits. Scenario discretization can miss tail events and make constraints large. Model risk remains even when the solver reports optimality. A responsible conclusion therefore states what remains unknown, what evidence could reverse it and when the analysis must be repeated. Precision about limits is part of the result, not a disclaimer added after it.
Misconception to avoid. A mathematically feasible dominance constraint guarantees social legitimacy. The benchmark and utility class are normative choices. Test the tempting shortcut against a counterexample and ask which stakeholder bears the cost when it fails. The correction should alter the workflow, not merely add cautious wording.
Practical evidence check
- Re-solve after perturbing probabilities and support points; report whether the recommended decision and dominance margin remain stable.
- Return to chapter contents
Inference decision table
| Finding | Interpretation | Next step |
|---|---|---|
| Clear sample dominance with robust test | Evidence supports dominance under the test design | Check practical size and assumptions |
| Curves cross materially | First-order dominance fails | Consider second order or report incomparability |
| Small local violation | Could be noise or substantively important | Use restricted or almost-dominance analysis transparently |
| Test inconclusive | Data do not resolve the order | Collect evidence or retain partial preference |
Report the null orientation, sampling design, contact sets, bootstrap procedure and sensitivity choices.
CHAPTER 9 OF 10 · Handle limits and disagreement
9. Report distributions, tests and practical magnitude
Core idea. A complete result includes curves, inequality margins, uncertainty, practical thresholds and the preference class. A binary label hides whether dominance is strong, local or fragile. The important discipline is to name the object being assessed, the boundary around it and the kind of conclusion the evidence can support. Without those three choices, technical vocabulary creates confidence without creating a checkable claim.
Method. Publish the empirical distributions, test design, effect sizes, sensitivity analyses and code or pseudocode. Distinguish population, sample and model-based claims. Write the procedure before looking at the preferred result, retain intermediate evidence and record who can challenge each assumption. A method becomes reusable when another reader can repeat the steps, locate disagreements and understand why a different answer emerged.
Worked example. Programme A passes a second-order test but gains are concentrated at outcomes with little policy relevance. The report shows both formal order and decision-threshold effects. The example is deliberately small enough to inspect. Its purpose is not to imitate every real case but to expose the decision rule, the evidence it consumes and the point at which judgement enters.
Changed-context transfer. A small formal violation affects a critical poverty line. Practical importance may exceed its graphical size. Keep the underlying principle while changing the actors, incentives or data. If the conclusion survives only because the original labels were familiar, the learner has memorised a case rather than acquired a transferable method.
Limits. Multiple comparisons, selective outcomes and data cleaning choices can distort findings. Pre-specification and audit trails help. A responsible conclusion therefore states what remains unknown, what evidence could reverse it and when the analysis must be repeated. Precision about limits is part of the result, not a disclaimer added after it.
Misconception to avoid. The p-value is the dominance result. It is one uncertainty measure, not the welfare interpretation or practical magnitude. Test the tempting shortcut against a counterexample and ask which stakeholder bears the cost when it fails. The correction should alter the workflow, not merely add cautious wording.
Practical evidence check
- Ask a reader to reproduce the sign, order and inference conclusion from the report without private notes.
- Return to chapter contents
CHAPTER 10 OF 10 · Transfer and govern
10. Govern the normative boundary and revisit it
Core idea. Stochastic dominance clarifies consequences of a preference class but does not decide whose utility, which outcomes or what distributional ethics should govern. Those choices require legitimate deliberation. The important discipline is to name the object being assessed, the boundary around it and the kind of conclusion the evidence can support. Without those three choices, technical vocabulary creates confidence without creating a checkable claim.
Method. Document stakeholder scope, outcome construction, excluded attributes, benchmark, utility restrictions and remedy for data error. Review when populations, risks or mandates change. Write the procedure before looking at the preferred result, retain intermediate evidence and record who can challenge each assumption. A method becomes reusable when another reader can repeat the steps, locate disagreements and understand why a different answer emerged.
Worked example. A health programme is ranked on patient outcomes but ignores caregiver burden. Governance adds a second outcome rather than stretching one scalar dominance claim beyond its scope. The example is deliberately small enough to inspect. Its purpose is not to imitate every real case but to expose the decision rule, the evidence it consumes and the point at which judgement enters.
Changed-context transfer. A regulator uses a legally mandated safety floor before applying dominance among feasible options. The institutional rule and statistical order remain distinct. Keep the underlying principle while changing the actors, incentives or data. If the conclusion survives only because the original labels were familiar, the learner has memorised a case rather than acquired a transferable method.
Limits. Interpersonal welfare comparisons and rights may resist reduction to one distribution. A responsible conclusion therefore states what remains unknown, what evidence could reverse it and when the analysis must be repeated. Precision about limits is part of the result, not a disclaimer added after it.
Misconception to avoid. A distributional method is value-neutral because it uses mathematics. Its input and preference class encode substantive choices. Test the tempting shortcut against a counterexample and ask which stakeholder bears the cost when it fails. The correction should alter the workflow, not merely add cautious wording.
Practical evidence check
- Hold a review in which affected groups can challenge the outcome variable, utility class, data quality and decision use.
- Return to chapter contents
Build a stochastic-dominance evidence worksheet
State the decision, unit of analysis, time horizon and affected people before collecting evidence. Separate observations from interpretations and interpretations from recommendations. Record data provenance, missingness and incentives that could distort reporting.
For every important claim, add a rival explanation and one observation that would favour it. Give uncertainty a visible place instead of hiding it inside a single score. Assign an owner and review date to assumptions that may change.
Run the worksheet with someone who did not design the analysis. Ask them to reconstruct the conclusion and identify where judgement entered. Revise any step that depends on private intuition or undefined terminology.
Finish with an action table: act now, monitor, investigate or stop. Link each action to evidence and a reversal condition so the method supports learning rather than defending the first answer.
Design a changed-context transfer check
Rewrite the case with a different scale, institution and distribution of power while preserving the underlying structure. Predict which parts of the method should remain stable and which require new evidence.
Remove one convenient data source and add one conflicting stakeholder account. Re-run the reasoning. If the answer changes, explain whether the change reflects better evidence, a different value choice or an unstable method.
Switch the time horizon. A decision that looks efficient this week may weaken resilience over years, while a long-run benefit may impose an unacceptable immediate burden. Report both rather than averaging them away.
Ask a reviewer to produce a counterexample. A transfer check passes when the method reveals why the new case differs, not when it mechanically produces the same conclusion.
Audit the reasoning and the record
Preserve source versions, transformations, thresholds, exclusions and decision notes. A final chart without its lineage cannot be independently checked and may conceal silent changes.
Trace one favourable and one unfavourable case from raw evidence to recommendation. Confirm that identical rules were applied and that exceptions were authorised, explained and reviewable.
Interview affected users about how the process works in practice. Formal policy may differ from operational reality, especially when time pressure, inaccessible interfaces or asymmetric power limit challenge.
Publish a proportionate account of method, limits, ownership and remedy. Protect sensitive data while ensuring that confidentiality is not used to block legitimate scrutiny.
Turn findings into accountable action
Assign every recommendation to a named role with authority, resources and a deadline. Distinguish mitigation from remedy: reducing future risk does not repair harm already caused.
Choose leading indicators that reveal drift early and outcome indicators that show whether people actually benefited. Disaggregate results where averages can hide concentrated burden.
Create an escalation rule for new evidence, missed thresholds and recurring complaints. Independent review should be available when the original decision maker has an incentive to defend the system.
Close the loop by publishing what changed, what did not and why. Re-open the analysis after material changes instead of treating one successful review as permanent certification.
Questions readers ask
Does a higher mean imply first-order dominance?
- No. CDFs can cross even when one mean is higher.
What does first-order dominance assume?
- Expected utility with every increasing utility function over the defined scalar outcome.
What does second-order dominance add?
- Concavity, representing risk aversion, alongside monotonicity under the stated convention.
Is no dominance indecision?
- It is a valid partial order showing that more assumptions are needed to rank the prospects.
Can samples prove population dominance?
- They provide statistical evidence under a sampling model; uncertainty and test design must be reported.
Why use almost dominance?
- To study bounded violations transparently when strict global inequalities are too demanding, while acknowledging the extra threshold choice.
Practise, verify and continue
Construct two discrete distributions, calculate their CDFs, test first-order dominance and integrate for second-order dominance. Then add one small tail probability that reverses the strict result and discuss whether a restricted claim is justified.
Apply the method to income, treatment outcomes and waiting times. For each, define whether higher or lower is better, the population, utility class, critical thresholds and sampling design.
Related eduKate reading
Expected utility, risk aversion and certainty equivalents
Spectral risk measures and quantile weighting
Chance-constrained optimisation and reliability targets
Distributionally robust optimisation and ambiguity
Authoritative sources and scope
Davidson, Statistical Inference in the Presence of Stochastic Dominance
Davidson, Testing for Restricted Stochastic Dominance: working paper
Linton, Maasoumi and Whang, Consistent Testing for Stochastic Dominance
Dentcheva and Ruszczyński, Optimization with Stochastic Dominance Constraints
PySDTest: a Python package for stochastic dominance tests
This article is educational, not investment, actuarial, clinical or public-policy advice. The examples use simplified distributions to reveal the logic. Production analysis should verify sign conventions, outcome construction, population, dependence, survey or experimental design, missingness, support, contact sets, bootstrap choices and multiple testing. Dominance conclusions apply only to the declared utility class and scalar outcome. Rights, catastrophic thresholds, ambiguity, multi-attribute welfare and interpersonal justice may require additional constraints or deliberation. A test result should be accompanied by curves, margins, uncertainty and sensitivity, and a solver result should be stress-tested for estimation and discretization error. Incomparability is often the honest answer.