Why Represent Development as a Tree?
A development program unfolds through decisions made at different times. A team may decide to begin a trial, observe its outcome, and then choose whether to continue, stop, or pursue an alternative. A single expected-value calculation can obscure this sequence: it may not show which action is best after each possible result. A decision tree makes the timing explicit, so the analysis can distinguish what is uncertain from what is chosen.
The tree is evaluated from its terminal outcomes back toward its starting point. At a chance node, calculate the probability-weighted value of the possible outcomes. At a decision node, compare the values of the available actions and retain the best one for the decision-maker’s objective. This backward evaluation is called rollback.
Values at Chance and Decision Nodes
Suppose the analysis maximizes expected net monetary value, with all payoffs measured from the same perspective and at a common valuation date. If a chance node has mutually exclusive, exhaustive outcomes \(i=1,\ldots,k\), with branch probabilities \(p_i\) and downstream values \(V_i\), its value is the probability-weighted average:
At a decision node with available actions \(a\), let \(V(a)\) be the value of the branch that follows action \(a\). The rollback value is the greatest available value:
These rules assume the probabilities and payoffs are appropriate inputs for the decision, that the analysis uses a consistent objective, and that all cash flows have been placed on a comparable basis. Maximizing expected monetary value implicitly treats the decision-maker as risk-neutral with respect to those monetary payoffs. If the analysis instead uses a defined utility scale, the same tree structure can be evaluated using expected utility rather than expected money. Costs should be represented on the branches where they are incurred. When timing matters, future cash flows need to be expressed at the common valuation date before comparison.
The chance-node formula follows by partitioning the possible outcomes into branches: the value on each branch contributes in proportion to the probability of that branch. The decision-node formula follows because the decision-maker selects one action, not a probability-weighted mixture of actions, so the preferred action is the one with the highest value under the stated objective. Applying these rules from the leaves toward the root gives the value of the complete sequential plan.
Worked Example: Rolling Back a Two-Phase Program
Worked Example: Rolling Back a Two-Phase Program
Consider a hypothetical development program with a Phase I trial followed, if the team chooses to continue, by Phase II. All amounts below are in millions of dollars, are expressed at a common valuation date, and are hypothetical. The Phase I cost is $12 million. The Phase I outcome is favorable with probability 0.70 and unfavorable with probability 0.30. After a favorable Phase I outcome, Phase II costs $24 million; the probability of Phase II success, conditional on this outcome and reaching Phase II, is 0.55. After an unfavorable Phase I outcome, the corresponding Phase II success probability is 0.10. A Phase II success yields a payoff of $150 million; failure yields $0. The team may stop or continue after observing the Phase I outcome.
Construct the tree. Start with the Phase I chance node: favorable (0.70) or unfavorable (0.30). Each outcome leads to a decision node with the actions stop and continue. A continue branch incurs the Phase II cost and leads to a Phase II chance node with success and failure outcomes. A stop branch has no further development cost or payoff. The Phase I cost is incurred before its outcome is observed.
Roll back after a favorable Phase I outcome. If the team continues, its value from that decision point is the expected Phase II payoff less the Phase II cost:
Stopping has value $0 million from this point onward. Since $58.5 million exceeds $0, the preferred action after a favorable outcome is continue.
Roll back after an unfavorable Phase I outcome.
The value of stopping is $0 million. Since continuing has a lower value, the preferred action after an unfavorable outcome is stop.
Evaluate the starting decision. The selected actions are therefore continue after a favorable Phase I outcome and stop after an unfavorable one. From the start, the expected net value is
The $12 million Phase I cost is included once, at the start of the tree. Under these assumptions, entering the program has an expected net value of $28.95 million compared with an available no-development option worth $0. The preferred path is to run Phase I, continue to Phase II only after a favorable result, and otherwise stop.
How to Construct and Roll Back the Tree
A useful construction begins with the decision whose value is being assessed and proceeds forward in time. It then rolls backward after every branch and terminal payoff has been specified. The sequence below is a practical check that the tree captures the decision structure rather than merely listing possible outcomes.
Specify who is making the decision, whether the objective is expected net value or another defined measure, and the date and units for comparing payoffs.
Use decision nodes for actions the team controls and chance nodes for uncertain outcomes. Include the stop option wherever it is genuinely available.
Give each chance branch a probability conditional on reaching that node. Check that probabilities from a node sum to one, and state where each cost and payoff is counted.
Calculate expected values at chance nodes and retain the best available action at decision nodes. Continue until the value and preferred action at the starting point are determined.
Worked Example: A Sunk Cost at a Continuation Decision
Worked Example: A Sunk Cost at a Continuation Decision
In a hypothetical program, the team has already spent $9 million on Phase I. The observed outcome is favorable. Continuing now would cost a further $20 million. Conditional on this result and proceeding, Phase II has a 0.35 probability of success and a 0.65 probability of failure. Success yields $80 million; failure yields $0. The team can stop now for no further cost or payoff.
The $9 million has already been spent and cannot be recovered by either current action. Compare the future value of continuing with the future value of stopping:
Continuing is preferred at this decision node, with an expected incremental value of $8 million from this point onward. If the question instead asks for the value of the entire program from before Phase I began, its expected value along this path would be \(-9+8=-1\) million. Those are different quantities answering different questions. The negative whole-program value does not change the current comparison: the spent $9 million is not an additional cost of continuing.
Worked Example: Choosing Different Actions After Different Results
Worked Example: Choosing Different Actions After Different Results
Suppose a hypothetical Phase II trial costs $8 million and has a favorable readout with probability 0.60 and an unfavorable readout with probability 0.40. After seeing the result, the team can choose a standard Phase III program, an adapted program, or stop. The following figures are hypothetical. They give each program’s success probability conditional on the observed Phase II result and the relevant program being undertaken. A standard-program success yields $100 million and costs $30 million. An adapted-program success yields $90 million and costs $20 million. Either program yields $0 on failure. Stopping has value $0 after the Phase II readout.
After a favorable result: the standard program has value \(0.60(100)-30=30\) million. The adapted program has value \(0.55(90)-20=29.5\) million. Stopping is worth $0 million. The preferred action is the standard program, worth $30 million.
After an unfavorable result: the standard program has value \(0.25(100)-30=-5\) million. The adapted program has value \(0.45(90)-20=20.5\) million. Stopping is worth $0 million. The preferred action is the adapted program, worth $20.5 million.
The expected value at the Phase II chance node, before paying its cost, is
After subtracting the $8 million Phase II cost, the value from before that trial is \(26.2-8=18.2\) million. The selected plan is not one action applied regardless of the readout: it is standard development after a favorable result, adapted development after an unfavorable result, and stopping only if its value exceeds the alternatives. This example relies on the supplied probabilities and payoffs; it does not establish how those inputs should be estimated.
Diagnostics, Comparisons, and Limitations
A tree is useful only if its branches correctly represent the decisions and uncertainty in the program. Check that every chance node has a complete set of mutually exclusive outcomes, that the probabilities on its branches sum to one, and that each probability is defined for the node it labels. Check that the available actions are genuinely available at that point in time. A tree that places a decision before the result that informs it, or omits a feasible stop action, can recommend a misleading path.
Payoffs need similar scrutiny. A common error is to subtract a cost both at the time it is incurred and again from a later branch. Another is to compare values that use different time points, currencies, or perspectives. Label costs explicitly and check a complete path from start to finish. In the two-phase example, the Phase I cost is paid once for any path that enters Phase I, while the Phase II cost is incurred only on branches where the team continues.
A decision table can be adequate when there is only one decision and a small set of outcomes. A tree is more informative when the action available later depends on what is observed earlier, because it displays that conditional sequence. For a large program with many uncertain quantities and repeated decisions, a tree can become too wide to inspect by hand; a spreadsheet or computational model may be more practical, provided it preserves the same timing, branch logic, and rollback calculations.
Rollback produces a preferred action only relative to the specified probabilities, payoffs, and objective. It does not show that those inputs are certain or that expected monetary value captures every relevant consideration. Development decisions may also depend on safety, patient benefit, operational constraints, or other criteria not included in the example payoffs. If plausible changes to an input alter the selected path, that is a reason to investigate the input’s influence and communicate the recommendation’s dependence on assumptions, not to present the selected path as unconditional.
Check Your Understanding
Work each problem by identifying the node type and rolling back from the available outcomes. Solutions follow each question.
- A chance node has outcomes with probabilities 0.30 and 0.70 and values of $40 million and \(-$5 million, respectively. What is its value?
Solution: \(0.30(40)+0.70(-5)=12-3.5=8.5\). The chance-node value is $8.5 million. - A trial costs $6 million. Success has probability 0.25 and yields $30 million; failure yields $0. Stopping is worth $0. Should the team run the trial?
Solution: The trial value is \(0.25(30)+0.75(0)-6=7.5-6=1.5\) million. Since $1.5 million exceeds $0, running the trial is preferred under these assumptions. - A team has already spent $12 million. Continuing costs a further $18 million; success has probability 0.50 and yields $50 million, while failure yields $0. What is the current continuation value, and should the team continue rather than stop for $0?
Solution: From the current decision point, the value is \(0.50(50)+0.50(0)-18=25-18=7\) million. Continue, because $7 million exceeds $0. The $12 million already spent is not included in this current comparison. - At a chance node, three outcomes have probabilities 0.20, 0.50, and 0.25. What should be checked before calculating its expected value?
Solution: The probabilities sum to \(0.20+0.50+0.25=0.95\), not 1. The outcome branches are incomplete or the probabilities are inconsistent. Resolve that issue before rolling back the node.