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A minimax allocation is only as credible as the uncertainty set it protects. Instead of guessing one stakeholder-weight vector, elicit ranges, trade-offs and exclusions that teams can defend. Then choose a simultaneous-error allocation whose worst credible decision loss is acceptable without pretending that every imaginable priority is equally plausible.
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Chapter contents
Choose the route closest to the decision, or read from Chapter 1. The chapters remain open; these grouped routes are a navigable index.
Route 1 · Define the decision
Route 2 · Work the ledgers
Route 3 · Transfer the method
Route 4 · Check limits and evidence
Route 5 · Ask and continue
CHAPTER 1 OF 12 · DEFINE THE DECISION
1. A weight interval represents governance uncertainty
A queue threshold weight is a decision input, not a fact discovered from waiting-time data. Customer-service, clinical-safety and staffing teams may value the same delay differently. An interval such as 0.30 to 0.50 records that unresolved priority honestly.
Statistical uncertainty and preference uncertainty are distinct. More queue observations can reduce an estimated tail-probability error, but they cannot decide how costly a missed 60-minute breach should be.
CHAPTER 2 OF 12 · DEFINE THE DECISION
2. Elicit constraints, not four unrelated ranges
Begin with threshold definitions, affected decisions and consequence questions. Ask which threshold can outweigh another, what priority reversals are unacceptable and whether any weight needs a minimum floor. Convert those answers into bounds and linear trade-off constraints.
All weights must remain non-negative and sum to one. Separate interviews can produce marginal intervals that cannot coexist; a feasibility check turns the elicitation into one joint ambiguity set rather than four disconnected opinions.
CHAPTER 3 OF 12 · WORK THE LEDGERS
3. A stakeholder-interval ledger
Suppose the protected queue curve uses four delay thresholds. The final column records a governance reason rather than a statistical estimate.
| Threshold | Weight interval | Illustrative centre | Reason for bounds |
|---|---|---|---|
| 15 min | 0.05–0.15 | 0.10 | Early-service signal |
| 30 min | 0.30–0.50 | 0.40 | Public promise |
| 60 min | 0.25–0.45 | 0.35 | Escalation risk |
| 120 min | 0.05–0.20 | 0.15 | Rare severe delay |
The additional joint constraint expresses a real priority: most protection must remain around the promise and escalation thresholds. Without it, the box of marginal intervals admits combinations nobody endorsed.
CHAPTER 4 OF 12 · WORK THE LEDGERS
4. Worked distributionally robust comparison
Simulation evaluates three alpha allocations over every feasible weight vector. Loss combines interval width and the cost of crossing a wrong operational boundary.
| Design | Nominal loss | Worst feasible loss | Where the worst case lands |
|---|---|---|---|
| Promise-heavy | 4.9 | 10.8 | High 60-minute weight |
| Balanced robust | 6.1 | 8.2 | Upper 30- and 60-minute bounds |
| Rare-delay-heavy | 5.6 | 11.3 | Low 120-minute weight |
Choose the robust design only if its worst-case protection and nominal cost both meet the decision standard. Robustness is a trade, not a free improvement.
CHAPTER 5 OF 12 · WORK THE LEDGERS
5. An ambiguity set needs a declared geometry
A polytope is natural for interval and trade-off constraints. A moment-based or Wasserstein ambiguity set is useful when weights are treated as a distribution learned from repeated elicitation, but its radius and reference sample still require justification.
Larger sets protect more possibilities and usually widen intervals or reduce power. Plot the solution path as the set expands. A sudden allocation change often reveals which stakeholder assumption controls the design.
CHAPTER 6 OF 12 · TRANSFER THE METHOD
6. Separate preference ambiguity from queue-model ambiguity
The weight set says what matters. Arrival dependence, service-time tails and importance-sampling stability say what the queue model can estimate. Cross the two only where their interaction changes the decision.
A distributionally robust alpha design cannot repair missing tail support or a broken simulator. Keep diagnostics for estimation, simultaneous coverage and stakeholder weights in separate audit columns.
CHAPTER 7 OF 12 · TRANSFER THE METHOD
7. Changed-context transfer: emergency triage waits
Transfer the method to triage thresholds where clinicians, patients and capacity planners define different consequences. A minimum clinical-safety weight may be non-negotiable even when the public target receives the largest centre weight.
The transfer check redefines the loss, verifies that thresholds refer to comparable patient groups and excludes combinations that violate clinical governance. It does not copy the service-queue intervals mechanically.
CHAPTER 8 OF 12 · TRANSFER THE METHOD
8. Connect the error-budget branch
The weight-misspecification guide introduces minimax regret. The threshold-importance guide explains unequal protection.
Use the simultaneous-band guide to define the protected family before optimising within it.
CHAPTER 9 OF 12 · CHECK LIMITS AND EVIDENCE
9. Limits and common misconceptions
A worst case is only worst inside the declared ambiguity set. It offers no protection against omitted stakeholders, new thresholds or losses outside the model.
Weight intervals are not confidence intervals unless they arise from a specified sampling model. Elicited ranges are governance statements and should be reported as such.
CHAPTER 10 OF 12 · CHECK LIMITS AND EVIDENCE
10. Practical evidence checklist
Archive interview prompts, decision owners, interval bounds, joint constraints, feasibility results, ambiguity-set radius, candidate alpha allocations, queue scenarios and loss calculations.
Report nominal and worst-case performance, active constraints, sensitivity to set expansion and the first evidence that would justify revising a bound. Version changes rather than silently replacing them.
CHAPTER 11 OF 12 · ASK AND CONTINUE
11. Frequently asked questions
Should every stakeholder get one weight vector?
Not necessarily. Ranges and trade-off statements can be more honest than forced precision, provided the joint set remains feasible.
Is the largest ambiguity set safest?
No. An excessively broad set can produce an unusable design and hide the decisions that are genuinely plausible.
Can probabilities be placed on weight scenarios?
Yes, but that changes the target toward expected loss. Distributional robustness can then protect a neighbourhood of that probability model.
CHAPTER 12 OF 12 · ASK AND CONTINUE
12. Sources and next reading
Bretz and colleagues develop weighted graphical multiple-comparison strategies. Rahimian and Mehrotra review distributionally robust optimisation and ambiguity-set choices. Delage and Ye study moment-based distributional uncertainty with data-driven confidence regions. The queue ledgers above are original teaching examples.
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