Robust Go Or No-Go Decisions Under Probability Uncertainty

Why Test a Decision Across Probability Ranges?

A go/no-go recommendation often depends on probabilities of moving through development transitions: for example, the chance of clinical success followed by the chance of approval conditional on clinical success. A single set of best estimates can produce a clear expected-value choice, yet modestly different probabilities may reverse it. Before committing resources, decision-makers need to know whether the preferred action holds across a defensible range of those probabilities, or whether it relies on a narrow assumption.

The question here is not how likely each probability value is. Instead, specify a set of plausible values and ask whether one action is preferred throughout that set. The conclusion is always conditional on the range, the payoff model, and the transition structure used to define the decision.

Definition: Let $\theta$ denote the vector of uncertain transition probabilities, and let $\Theta$ be the set of probability vectors regarded as plausible. Write $EV(\text{go}\mid\theta)$ and $EV(\text{no-go}\mid\theta)$ for the expected values of the two actions at a fixed $\theta$. Define the difference $$ \Delta(\theta)=EV(\text{go}\mid\theta)-EV(\text{no-go}\mid\theta). $$ Go is weakly preferred throughout $\Theta$ if $\Delta(\theta)\ge 0$ for every $\theta\in\Theta$, equivalently if $\inf_{\theta\in\Theta}\Delta(\theta)\ge 0$. No-go is weakly preferred throughout $\Theta$ if $\Delta(\theta)\le 0$ for every $\theta\in\Theta$, equivalently if $\sup_{\theta\in\Theta}\Delta(\theta)\le 0$. Go is strictly preferred pointwise if $\Delta(\theta)>0$ at every $\theta\in\Theta$; no-go is strictly preferred pointwise if $\Delta(\theta)<0$ throughout.

An infimum is the greatest lower bound, whether or not that bound is attained at a point in $\Theta$. Thus the infimum formulation avoids assuming that a smallest value exists. Pointwise strict preference also differs from a uniform positive margin: $\Delta(\theta)>0$ at every plausible value can hold even when the infimum is zero. A uniform positive margin requires $\inf_{\theta\in\Theta}\Delta(\theta)>0$. If $\Theta$ is compact and $\Delta$ is continuous, a minimum and maximum are attained; otherwise, use infimum and supremum or establish the extrema by other means.

If $\Delta$ is positive for some plausible values and negative for others, neither action is robustly preferred. If it is zero at some values and never has the opposite sign, the action with nonnegative difference is weakly preferred, but not strictly preferred everywhere. These distinctions matter when the decision maker requires a margin rather than merely no expected-value disadvantage.

Deriving a Transition-Probability Decision Rule

Suppose a program must pass two sequential transitions. Let $p$ be the probability of passing the first transition and $q$ the probability of passing the second conditional on passing the first. If the program receives payoff $R$ only after both transitions and incurs a go cost $C$, while no-go has value zero, the probability of receiving the payoff is $pq$. This multiplication follows from the definition of conditional probability: $P(\text{both pass})=P(\text{first passes})P(\text{second passes}\mid\text{first passes})$.

EV(\text{go}\mid p,q)=Rpq-C,\qquad EV(\text{no-go}\mid p,q)=0,\qquad \Delta(p,q)=Rpq-C.

For positive $R$ and $C$, go is preferred at a particular $(p,q)$ exactly when $pq>C/R$. This break-even condition can also be written $q>C/(Rp)$ for $p>0$. It is a threshold statement, not a claim that either probability is known precisely.

If $p$ and $q$ vary independently over a rectangular plausible set, and all values are nonnegative, $Rpq-C$ increases with either probability. Its lowest and highest values are therefore at the lower-lower and upper-upper corners. This shortcut is valid only when those corners are feasible and the stated monotonicity holds. With dependent probabilities or a non-rectangular set, evaluate the difference over the actual feasible set; combining separately chosen bounds could describe an impossible scenario.

Worked Example: A Range That Does Not Support One Robust Choice

Suppose a hypothetical program has a payoff of $R=\$150$ million if both transitions succeed and a go cost of $C=\$40$ million. No-go has value zero. The clinical transition probability is regarded as plausible from 0.45 to 0.60; the approval probability conditional on clinical success is regarded as plausible from 0.50 to 0.70. Assume every combination in this rectangle is feasible.

Here $\Delta(p,q)=150pq-40$, in millions of dollars. Since the expression increases with both probabilities, its lower and upper values occur at the corresponding corners:

\inf_{\Theta}\Delta=150(0.45)(0.50)-40 =150(0.225)-40 =33.75-40=-\$6.25\text{ million},

\sup_{\Theta}\Delta=150(0.60)(0.70)-40 =150(0.42)-40 =63-40=\$23.0\text{ million}.

The negative lower value means go is not weakly preferred throughout the range. The positive upper value means no-go is not weakly preferred throughout it. The preferred action changes within the plausible set. For a given $p$, the break-even $q$ is $q^*=40/(150p)$. At $p=0.45$, $q^*=0.5926$; at $p=0.60$, $q^*=0.4444$. Both thresholds fall within the stated $q$ range, consistent with the endpoint comparison. These calculations establish sensitivity to the joint transition assumptions, not the probability that either action is best.

Robust Preference With a Positive Margin

A decision can remain preferred over a range even when the probability estimates are imprecise. The size of the worst-case difference is informative: a small positive infimum indicates a fragile advantage, while a larger positive infimum indicates more protection against the specified probability uncertainty. A positive result is still only as credible as the bounds and the payoff assumptions.

Worked Example: Go Remains Preferred Across a Plausible Rectangle

Consider a second hypothetical program. A successful passage through both a clinical transition and a subsequent approval transition yields $\$220$ million; the go cost is $\$58$ million, and no-go has value zero. Suppose $p$ ranges from 0.60 to 0.70 and $q$, conditional on clinical success, ranges from 0.45 to 0.55. Assume all combinations in the rectangle are feasible.

The incremental expected value is $\Delta(p,q)=220pq-58$, in millions of dollars. It increases with both probabilities, so its lowest value is at $p=0.60,q=0.45$:

\inf_{\Theta}\Delta =220(0.60)(0.45)-58 =220(0.27)-58 =59.4-58 =\$1.4\text{ million}.

Because this infimum is positive, go is strictly preferred at every plausible pair and has a uniform positive margin of at least $\$1.4$ million under this model. For comparison, the upper value is $220(0.70)(0.55)-58=84.7-58=\$26.7$ million. The robust conclusion is not that the program will succeed or that the payoff is certain; it is that go has the greater modeled expected value throughout the stated probability set.

The smallest margin is modest relative to the scale of the program. A decision-maker who requires a larger minimum advantage may not regard this as sufficient support for action, even though go is mathematically robustly preferred under the stated criterion. Robust preference and practical adequacy of the margin are related but distinct judgments.

Finding a Decision That Depends on One Probability

A useful way to diagnose a probability-sensitive recommendation is to solve for the value at which the action changes. If holding other assumptions fixed reveals a threshold inside the plausible range, that probability can reverse the choice. This identifies a decision-relevant assumption; it does not by itself show how likely the threshold will be crossed or whether new data would be worth collecting.

Worked Example: An Approval Probability That Reverses the Choice

Suppose a hypothetical program has a $\$200$ million payoff upon clinical success and subsequent approval, with a $\$50$ million go cost. For illustration, hold the clinical transition probability at $p=0.60$ and vary the conditional approval probability $q$ from 0.35 to 0.50. No-go is worth zero.

At these fixed assumptions, $\Delta=200(0.60)q-50=120q-50$, in millions of dollars. The break-even value solves $120q-50=0$, giving $q^*=50/120=0.4167$. At the low endpoint, $120(0.35)-50=42-50=-\$8.0$ million; at the high endpoint, $120(0.50)-50=60-50=\$10.0$ million. Thus this range crosses the decision threshold: no-go has the greater expected value below $q^*$, and go has the greater expected value above $q^*$, with a tie at the threshold.

Now allow the clinical probability to range from 0.58 to 0.62. The threshold becomes $q^*=50/(200p)$, which ranges from $50/124=0.4032$ at $p=0.62$ to $50/116=0.4310$ at $p=0.58$. The plausible approval range still includes these thresholds. For example, at $q=0.40$, even the larger clinical probability gives $200(0.62)(0.40)-50=-\$0.40$ million, while at $q=0.44$, even the smaller clinical probability gives $200(0.58)(0.44)-50=\$1.04$ million. The sign switches across the approval range despite the comparatively narrow clinical range. This makes the approval probability a decision-critical assumption in this scenario; it does not establish that it is the only important uncertainty in a broader model.

Assumptions, Diagnostics, and Alternatives

Before using a robust comparison, check that the decision model and the plausible set are coherent. In particular:

Define transition events conditionally. A later-stage probability may apply only among programs that reached that stage. Multiplying probabilities is justified for sequential conditional probabilities, not for quantities with mismatched denominators.
Use feasible combinations. If probabilities share data, mechanisms, or constraints, their joint plausible set may not be a rectangle. Do not assume every combination of marginal endpoints can occur.
Check the payoff calculation. Include costs and payoffs at the stages at which they occur. A cost paid regardless of later failure is not multiplied by the probability of final success. State whether values are incremental and use consistent units.
Verify extrema. Endpoint evaluation requires the relevant monotonicity and feasible endpoints. For a nonlinear model with interior extrema, or a constrained set, inspect the full function and feasible region rather than relying on corners.
Inspect the boundary. A threshold close to a bound means small changes in the plausible range or value assumptions may change the robustness result. Ties should be reported as ties, not as strict preference.

A robust range analysis differs from choosing the action at a single point estimate: a point estimate answers which action is preferred at one specified probability vector, while robustness asks whether the sign is preserved across a set. It also differs from assigning a probability distribution over uncertain inputs and averaging over that distribution. That probability-weighted approach requires a defensible distribution and can favor an action even when it is worse in some plausible scenarios. Robust analysis instead makes no claim about how probability mass is distributed within the range.

Value-of-information analysis addresses a different question: whether information that could change the action is worth its cost. A range crossing a decision threshold can motivate such an analysis, but it does not establish that a study is valuable; study cost, timing, decision consequences, and how information could change beliefs all matter. Conversely, a robust action over the specified range does not rule out value in learning about uncertainties omitted from that range or about other model assumptions.

Common pitfall: Saying “the minimum difference is positive” without checking that a minimum exists, or treating strict preference at every point as equivalent to a strictly positive infimum. Use $\inf_{\theta\in\Theta}\Delta(\theta)\ge0$ for weak go preference. For strict pointwise go preference, verify $\Delta(\theta)>0$ for every plausible value. A positive infimum establishes the stronger claim of a uniform positive margin; it is not necessary for pointwise strict preference on every possible set.

Robustness is always relative to a stated plausible set. Bounds that are too narrow can create false confidence; bounds that include incoherent combinations can make a decision appear less robust than it is. Report the probability ranges, their rationale, the payoff assumptions, the worst-case difference, and any threshold that changes the preferred action. This makes clear both what the analysis supports and what it leaves unresolved.

Check Your Understanding

Try each problem, then compare your reasoning with the solutions below.

A go action has incremental expected value $\Delta(p)=100p-30$, in millions of dollars, and $p\in[0.40,0.60]$. Is go robustly preferred across this range? What is the worst-case difference?
Suppose $\Theta=(0,1)$ and $\Delta(\theta)=\theta$. Is go strictly preferred at every plausible value? Does it have a uniform positive margin?
A program has $\Delta(p,q)=120pq-36$, in millions of dollars, with feasible rectangular ranges $p\in[0.50,0.70]$ and $q\in[0.40,0.60]$. Is either action weakly preferred throughout?
Why can marginal probability ranges be insufficient for a robust decision calculation when the transition probabilities are dependent?

Solutions. (1) The minimum occurs at $p=0.40$: $\inf\Delta=100(0.40)-30=40-30=\$10$ million. Since it is positive, go is strictly preferred throughout with a uniform positive margin. (2) For every $\theta\in(0,1)$, $\Delta(\theta)=\theta>0$, so go is strictly preferred pointwise. But $\inf_{\theta\in\Theta}\Delta(\theta)=0$, so there is no uniform positive margin; the infimum is not attained. (3) The minimum is $120(0.50)(0.40)-36=24-36=-\$12$ million, and the maximum is $120(0.70)(0.60)-36=50.4-36=\$14.4$ million. Both signs occur, so neither action is weakly preferred throughout. (4) The marginal ranges do not specify which pairs of probabilities can occur together. Combining separate endpoints could create impossible pairs and an incorrect worst-case value; the joint feasible set must be used.