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Rollback, expected value, and risk

Solve the tree from right to left, preserve the conditional policy, and interpret expected value alongside the full consequence distribution.

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Rollback, expected value, and risk

Rollback evaluates the tree from the terminal nodes back to the root. It produces a conditional policy: what to choose at each decision node if that node is reached. The calculation must include branch increments exactly once and follow the declared objective—maximize value or minimize cost.

Calculation rules

For a chance node with outcomes (i), the rolled-back value is:

EV(chance) = Σ pᵢ × [branch incrementᵢ + downstream valueᵢ]

For a decision node, compare included branches using branch increment + downstream value and select the maximum or minimum according to the value model. Version 1 does not parse free-text constraints or automatically exclude an infeasible alternative: the modeler must omit or separately quarantine infeasible branches, and the reported preference remains provisional until that review is complete. Record the selected branch and retain viable alternatives; deleting merely lower-valued branches destroys the audit trail and sensitivity context.

Compute only after structural validation. Every reachable chance node needs probabilities totaling 1, every reachable terminal needs a value, and each branch needs a valid destination. When these conditions fail, show an incomplete result—not a plausible-looking zero.

Expected value is not a forecast

Expected value describes a probability-weighted average under the model. A project with a 60% path to 100 and a 40% path to −50 has an EV of 40, but the realized consequence will not be 40. For one-off, safety-critical, insolvent, irreversible, or strongly asymmetric decisions, the distribution and thresholds may matter more than the mean.

If decision-maker risk preferences are material, use a defensible expected-utility model or show risk constraints separately. Do not silently label money as “utility,” infer a utility curve from chat, or treat a generic score as welfare. The chosen unit, optimization direction, unacceptable threshold, and basis must be visible.

Risk profile to review

  • probability and magnitude of consequences below the unacceptable threshold;
  • best and worst reachable terminal outcomes under the policy;
  • concentration of value in a low-probability high-impact branch;
  • outcomes affecting safety, environment, rights, accessibility, communities, or compliance that cannot be adequately offset by gains elsewhere;
  • options that are infeasible because of budget, authority, schedule, or mandatory requirements;
  • divergence between expected monetary value and expected utility, where both have been approved.

Reproduce and challenge

The report should identify the model version, value unit, objective, probability tolerance, calculation time, selected policy, unresolved warnings, and input bases. Recalculate whenever structure, probabilities, values, costs, or optimization direction change. Compare the result with a simple hand calculation on at least one path and investigate surprising recommendations rather than rationalizing them after the fact.

References

NASA Systems Engineering Handbook: Decision Analysis

NASA guidance on framing, alternatives, uncertainty, assumptions, sensitivity, recommendations, and decision-analysis reporting.

https://www.nasa.gov/reference/6-8-decision-analysis/

HM Treasury, The Green Book (2026)

UK government guidance on appraisal, decision trees, risk and uncertainty, sensitivity, switching values, optimism bias, and distributional impacts.

https://www.gov.uk/government/publications/the-green-book-appraisal-and-evaluation-in-central-government/the-green-book-2026