Expected Value Analysis

An approach to decision-making under uncertainty that evaluates options based on the probability-weighted average of their possible outcomes.

Source: von Neumann & Morgenstern 1944

Each possible outcome is assigned a value and a probability, the two are multiplied for every outcome, and the products are summed to give the probability-weighted average. Options are then compared on that single figure. The approach rests on the axioms of expected utility theory, which show that a decision maker whose preferences satisfy certain consistency conditions behaves as though maximising an expectation of this kind. Where outcomes are described on a natural scale such as life years, the expectation is in those units; where they are described on a preference scale, the expectation is in utility.

Source: von Neumann & Morgenstern 1944

Because decisions about services are repeated across many patients, so the average outcome is what actually materialises rather than a hypothetical. A treatment with a small chance of a very good outcome and a large chance of a modest one will, applied across a population, deliver close to its expected value. This is why a decision maker allocating resources across many patients is on firmer ground maximising expectations than an individual facing a single irreversible choice. The argument does not extend to a single patient facing an irreversible choice, which is why the framework suits allocation decisions better than individual clinical ones.

Source: Briggs, Claxton & Sculpher 2006

That probabilities can be assigned to the outcomes, that values can be attached to them on a scale permitting multiplication and addition, and that the decision maker is indifferent between prospects with the same expectation. The last assumption is the contested one, since it implies neutrality towards risk, and both individuals and institutions frequently prefer a certain outcome to an uncertain one with the same average. Violations of these conditions are well documented in experimental work, including systematic departures in how people weigh small probabilities, which the framework does not accommodate.

Source: von Neumann & Morgenstern 1944

By valuing outcomes on a utility scale that is not linear in the underlying quantity, so that additional gains are worth progressively less and losses weigh more heavily. Maximising expected utility on such a scale then reproduces risk-averse behaviour while retaining the expectation framework. Standard economic evaluation generally works with expected values rather than expected utility, on the reasoning that a system pooling risk across many patients need not be risk averse in the way an individual is. The position is contested, since a health system can face outcomes large enough relative to its budget that treating them as poolable is questionable.

Source: Drummond et al. 2015

It requires probabilities that are frequently not available, so figures are elicited or assumed and the result inherits that uncertainty without displaying it. It reduces a distribution to a single number, so two options with the same expectation but very different spreads appear equivalent. And where an outcome is catastrophic and irreversible, maximising an expectation may not describe how a decision maker should reason, since averaging assumes the situation will recur. Sensitivity analysis on the assumed probabilities is therefore standard, since a result that reverses across the plausible range rests on the assumption rather than the evidence.

Source: Briggs, Claxton & Sculpher 2006