Concept Architecture
Concept
Theoretically, Folding Back is the systematic evaluation procedure used in decision tree analysis to determine the optimal sequence of decisions under uncertainty. Beginning at the terminal nodes of a decision tree, expected values are calculated at chance nodes and optimal alternatives are selected at decision nodes by working backwards towards the root of the tree. The method is grounded in expected utility theory and dynamic programming and exists to identify the decision that maximises expected value or expected utility.
Mathematically, Folding Back is performed using backward induction. Expected values are calculated at each chance node as the probability-weighted average of downstream outcomes, while decision nodes are resolved by selecting the branch with the preferred objective function, such as the lowest expected cost, greatest expected quality-adjusted life-years or highest expected net benefit. The process continues recursively until the initial decision node is reached.
In practice, Folding Back is implemented after all probabilities, costs and health outcomes have been assigned within a decision tree. Health economists use the procedure to evaluate diagnostic strategies, treatment pathways, screening programmes and other interventions involving sequential decisions under uncertainty. Most decision-analytic software performs Folding Back automatically once model parameters have been specified.
Purpose
Used to solve decision trees by calculating expected values through backward induction and identifying the optimal intervention according to predefined economic or clinical decision criteria.
Mathematical Formulae
Primary Formula
Expected value at a chance node:
EV = ????� P?V?
Decision node:
V? = max(EV?, EV?, ?, EV?)
or, for cost minimisation,
V? = min(EV?, EV?, ?, EV?)
Supporting Formulae
Recursive backward induction:
V? = f(V???)
where f(�) denotes either expectation at a chance node or optimisation at a decision node.
Related Mathematical Methods
- Decision tree analysis
- Backward induction
- Dynamic programming
- Expected value analysis
- Expected utility theory
- Decision analysis
Example
A treatment decision has two possible outcomes.
| Outcome | Probability | Cost (�) |
|---|---|---|
| Successful treatment | 0.80 | 3,000 |
| Treatment failure | 0.20 | 9,000 |
The chance node is evaluated as:
EV = 0.80(3,000) + 0.20(9,000) = �4,200
A competing treatment has an expected cost of �4,800. During Folding Back, the decision node selects the first treatment because it has the lower expected cost. This process is repeated for every decision node until the optimal initial strategy is identified.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| SUMPRODUCT | =SUMPRODUCT(B2:B3,C2:C3) | Calculate expected value at each chance node. |
| MIN | =MIN(E2:E4) | Select the least costly strategy at a decision node. |
| MAX | =MAX(E2:E4) | Select the strategy with the greatest expected benefit or net benefit. |
| IF | =IF(E2=MIN($E$2:$E$4),""Optimal"","""") | Identify the preferred intervention after Folding Back. |
VBA (Optional)
Automate backward induction by recursively evaluating decision trees and identifying the optimal strategy across multiple scenarios.
Sources
- Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes. 4th ed.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
- TreeAge Software. TreeAge Pro User Guide.
- Raiffa H. Decision Analysis: Introductory Lectures on Choices Under Uncertainty.
- Hunink MGM, Weinstein MC, Wittenberg E, et al. Decision Making in Health and Medicine.
Related Concepts (3)
Library
Publications
1
A Taxonomy of Model Structures for Economic Evaluation of Health Technologies — Brennan, Chick & Davies, Vol. 15, No. 12 ed., 2006 (Health Economics)
An influential paper classifying decision-analytic model structures along axes of expected value vs randomness, entity heterogeneity, and Markovian vs non-Markovian structure — providing a framework for choosing between decision trees, Markov cohort models, microsimulation, discrete event simulation and system dynamics.
Journal ArticleView source →
Frequently Asked Questions (6)
What is folding back in a decision tree?
The process of calculating a decision tree's expected values by working backward from final outcomes toward the initial decision node.
Source: Weinstein & Fineberg 1980
What does folding back compute at each node?
As folding back moves from right to left through a tree, it replaces each node with a single value summarising everything beyond it. At a chance node it takes the probability-weighted average of the values on its branches, giving the expected value of reaching that point, while at a decision node it takes the best value among the branches, since the decision-maker would choose it. Repeating this collapses the whole tree to one value per initial strategy. Hunink and colleagues (2014) describe these node calculations.
Source: Hunink et al. 2014
How does folding back work?
Folding back works by starting at the terminal nodes, where outcomes are known, and moving backward through the tree. At each chance node, the expected value is calculated as the sum of its branches' values weighted by their probabilities. At each decision node, the option with the best expected value is selected, and that value is carried back. Repeating this toward the root gives the expected value of each initial strategy, combining all the probabilities and outcomes along the paths.
Source: Weinstein & Fineberg 1980
Why is folding back done backward?
Folding back is done backward because the value of a decision depends on what follows it, so the consequences of later events and choices must be evaluated before an earlier choice can be assessed. Starting from the known terminal outcomes and working back, each node's value can be computed from the values of the nodes that follow it, which have already been determined. This backward order ensures that when a decision node is reached, the expected values of all its options are available for comparison.
Source: Weinstein & Fineberg 1980
How does folding back handle chance and decision nodes differently?
Folding back handles the two node types differently because they represent different things. At a chance node, where outcomes are uncertain and not chosen, it averages, computing the probability-weighted mean of the branches' values. At a decision node, where the option is chosen, it maximises, selecting the branch with the best expected value rather than averaging. This reflects that uncertainty is averaged over while choices are optimised, so folding back correctly combines the two to find the best strategy.
Source: Weinstein & Fineberg 1980
What does folding back yield?
Folding back yields the expected value of each strategy at the initial decision node, and identifies the option with the best expected value, which is the recommended choice. For economic evaluation, it produces the expected cost and expected effect of each strategy, from which cost-effectiveness is calculated. By combining all the probabilities and outcomes in the tree into these expected values, folding back turns the structure of the decision problem into the numbers needed to compare and choose among the options.
Source: Weinstein & Fineberg 1980
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 30 Sep 2025
Content version: 1.0.0
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- Persistent URI
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- Term code
- HE-EM-DM-035
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