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A multistage capacity plan must be robust to demand uncertainty without using tomorrow’s information today, so the distance and policy must respect the same filtration.
Open the chapter contents · Queueing Theory root · Gold Standard of Education · How X Works
Chapter contents
Choose the route closest to the question, or read from Chapter 1 for the complete argument. The chapters remain open on the page; these grouped routes are an index.
Route 1 · Define information flow
Route 2 · Work the scenario tree
Route 3 · Calibrate and transfer
Route 4 · Check limits and evidence
Route 5 · Ask and continue
CHAPTER 1 OF 12 · DEFINE INFORMATION FLOW
1. Why a one-stage ambiguity ball is not enough
A one-stage Wasserstein ball asks how far a demand distribution may move before a capacity decision changes. A multistage queue adds another constraint: a decision at time t may use only information revealed by time t. If a distributional comparison pairs whole future paths without respecting that information flow, a policy can appear robust because it quietly uses tomorrow’s demand today.
The right question is therefore twofold. How different are the demand processes, and can their histories be coupled without sending future information backwards? Multistage distances and non-anticipative policies keep those questions together.
CHAPTER 2 OF 12 · DEFINE INFORMATION FLOW
2. Filtrations and non-anticipativity
A filtration is the growing record of what has been observed. At stage 1 it may contain today’s arrivals and current staffing; at stage 2 it also contains the next observation. A policy yₜ must be measurable with respect to the information available at stage t.
In a scenario tree, paths that share the same history occupy the same decision node. Their actions must be identical until they branch. Writing one capacity variable per complete path without equality constraints breaks this rule and creates perfect foresight.
CHAPTER 3 OF 12 · WORK THE SCENARIO TREE
3. A four-path information tree
Suppose period-1 demand is either 8 or 14. Each value has two possible period-2 continuations, and all four paths have probability 0.25.
| Path | Period 1 | Period 2 | Stage-1 history group |
|---|---|---|---|
| A | 8 | 8 | Low node |
| B | 8 | 16 | Low node |
| C | 14 | 10 | High node |
| D | 14 | 18 | High node |
The table is more than a list of paths. It declares which futures are indistinguishable when the first action is chosen. Any ambiguity model that changes probabilities must preserve that decision structure or explain a different observation process.
CHAPTER 4 OF 12 · WORK THE SCENARIO TREE
4. The perfect-foresight trap
At the low node, base period-2 capacity is 10. Reserving six extra places costs 9 units; unmet jobs cost 4 each. If no reserve is made, A costs 0 and B costs 24. Conditional on the low node, the expected shortage cost is 12, so the non-anticipative action is to reserve and pay 9 for either future.
| Policy at the low node | A cost | B cost | Conditional expected cost | Feasible at stage 1? |
|---|---|---|---|---|
| Never reserve | 0 | 24 | 12 | Yes |
| Always reserve | 9 | 9 | 9 | Yes |
| Reserve only on B | 0 | 9 | 4.5 | No |
A pathwise optimiser without non-anticipativity constraints will choose the last row and report a false benefit. Robustness cannot repair an information leak in the policy model.
CHAPTER 5 OF 12 · WORK THE SCENARIO TREE
5. Nested and causal transport
Ordinary Wasserstein distance compares complete outcome vectors. In a multistage model, an optimal transport plan may match paths using their later coordinates even when earlier decisions cannot see those coordinates. Nested distance instead compares conditional future distributions recursively along histories.
Pflug and Pichler’s nested distributions place scenario values and information structure in one object. The distance generalises the two-stage Wasserstein idea while respecting the branching process. Related causal transport formulations express the same principle as a restriction on admissible couplings.
CHAPTER 6 OF 12 · CALIBRATE AND TRANSFER
6. Calibrating uncertainty at each history
A single global radius can hide where uncertainty enters. One alternative is to calibrate conditional distributions at each tree node, with enough observations and regularisation to avoid unstable tiny-node estimates. Another is to construct a process-level ball under nested distance.
Record the units and weights at every stage. Moving one job today may be more consequential than moving one job next week because setup lead times differ. Radius choices must not be tuned after seeing which capacity plan looks attractive.
CHAPTER 7 OF 12 · CALIBRATE AND TRANSFER
7. Changed-context transfer: rolling forecasts
Now suppose the queue is replanned every morning. Tomorrow’s capacity decision may use tomorrow’s forecast update, but today’s setup order cannot. A static robust plan may be unnecessarily conservative because it ignores future recourse; a clairvoyant plan may be impossibly optimistic because it anticipates the update.
The transfer check is to write the decision timeline explicitly: observation, action, lead time, next observation. Evaluate the policy on full paths while enforcing common actions for common histories. This connects Wasserstein radius calibration to joint quantile scenarios and feasible capacity paths.
CHAPTER 8 OF 12 · CALIBRATE AND TRANSFER
8. Policy stability rather than path stability
Two scenario trees can look close yet produce different policies near a decision boundary. Compare the whole policy: node actions, trigger thresholds, reserve starts and cancellation rules. Then evaluate each policy on held-out paths under the same observation timing.
Report where policies first diverge. If only a rare branch changes, show its probability and cost. If the root action changes, treat that as a major instability even when the objective difference is small.
CHAPTER 9 OF 12 · CHECK LIMITS AND EVIDENCE
9. Limits and common misconceptions
Nested distance does not create data for poorly observed branches. A detailed tree can increase estimation error when each history contains few samples. Merging nodes improves sample size but may erase decision-relevant information.
Non-anticipativity is not the same as ignoring forecasts. A valid policy may use every forecast available at that time. It simply cannot use later observations. Nor does a causal coupling prove that the forecast process itself is unbiased.
CHAPTER 10 OF 12 · CHECK LIMITS AND EVIDENCE
10. Practical evidence checklist
Keep the scenario-tree topology, probabilities, history labels, observation times and action lead times. Save every non-anticipativity constraint or policy parameterisation. Verify with a small tree whose optimal action can be calculated by hand, including an intentionally clairvoyant benchmark that should be rejected.
For robustness, retain the process metric, stage weights, radius calibration and conditional sample sizes. Test the final policy on original and held-out paths, and report regret against the best policy with the same information—not against a perfect-foresight oracle.
CHAPTER 11 OF 12 · ASK AND CONTINUE
11. Frequently asked questions
Can one capacity variable be used for every stage?
Only if capacity is truly fixed. Adaptive capacity requires stage-indexed actions tied to histories; complete-path variables need equality constraints across shared histories.
Is nested distance always necessary?
No. A two-stage problem with one here-and-now decision may be adequately represented by ordinary Wasserstein distance. The need grows when decisions and observations alternate.
Why keep a perfect-foresight result?
As a diagnostic lower bound. It shows how much apparent value comes from illegal future information.
CHAPTER 12 OF 12 · ASK AND CONTINUE
12. Sources and next reading
Pflug and Pichler introduce a distance for multistage stochastic optimisation models, building scenario values and information structure into nested distributions. Their earlier work on version-independence and nested distributions explains why a multistage model should not depend on an arbitrary probability-space representation. The tree and cost ledger above are original teaching examples.
Continue through the Queueing Theory root, Wasserstein ambiguity sets, scenario reduction and joint quantile capacity paths. The Gold Standard of Education and How X Works provide the wider transfer route.
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