Concept Architecture
Concept
Theoretically, Expected Utility is the probability-weighted average utility associated with uncertain outcomes. It forms the foundation of expected utility theory, which assumes that rational decision makers choose the alternative that maximises expected utility rather than expected monetary value alone. In health economics, expected utility provides the theoretical basis for evaluating healthcare interventions under uncertainty by combining the utility of health outcomes with their probabilities of occurrence.
Mathematically, Expected Utility is represented as the sum of the utilities associated with all possible outcomes, each weighted by its probability. The resulting value represents the average utility that would be obtained if the uncertain decision were repeated many times. Expected utility underpins decision tree analysis, Bayesian decision analysis and cost-utility analysis, where expected quality-adjusted life-years (QALYs) are calculated by weighting health outcomes according to their probabilities.
In practice, Expected Utility is estimated using probabilities derived from clinical evidence and utility values obtained from preference-based measures such as the EQ-5D, Health Utilities Index (HUI) or SF-6D. Health economic models calculate expected utility for alternative interventions and combine these estimates with expected costs to determine incremental cost effectiveness. Expected utility is routinely applied in decision trees, Markov models, microsimulation and probabilistic sensitivity analysis.
Purpose
Used to quantify the average utility associated with uncertain health outcomes, enabling healthcare interventions to be compared according to their expected health benefits under uncertainty.
Mathematical Formulae
Primary Formula
Expected utility:
EU = ????� p?U?
where:
- EU is the expected utility
- p? is the probability of outcome i
- U? is the utility associated with outcome i
- ?p? = 1.
Supporting Formulae
Expected quality-adjusted life-years:
E(QALY) = ????� p?(U? ? T?)
where:
- U? is the health-state utility
- T? is the time spent in the health state.
Expected net benefit:
E(NMB) = ?E(QALY) ? E(C)
where:
- ? is the willingness-to-pay threshold
- E(C) is the expected cost.
Related Mathematical Methods
- Expected utility theory
- Decision tree analysis
- Bayesian decision analysis
- Cost-utility analysis
- Markov modelling
- Monte Carlo simulation
- Probabilistic sensitivity analysis
Example
A treatment has two possible outcomes.
| Outcome | Probability | Utility |
|---|---|---|
| Successful treatment | 0.80 | 0.90 |
| Treatment failure | 0.20 | 0.45 |
The expected utility is:
EU = (0.80 ? 0.90) + (0.20 ? 0.45)
EU = 0.72 + 0.09 = 0.81
The treatment therefore has an expected utility of 0.81, which can be compared with competing interventions when estimating expected quality-adjusted life-years.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| SUMPRODUCT | =SUMPRODUCT(ProbabilityRange,UtilityRange) | Calculate expected utility from outcome probabilities and utilities. |
| SUMPRODUCT | =SUMPRODUCT(ProbabilityRange,UtilityRange,TimeRange) | Calculate expected quality-adjusted life-years. |
| SUM | =SUM(ProbabilityRange) | Verify that probabilities sum to one. |
| IF | =IF(SUM(ProbabilityRange)=1,""Valid"",""Check"") | Validate probability inputs before analysis. |
| XLOOKUP | =XLOOKUP(State,UtilityTable[State],UtilityTable[Utility]) | Retrieve health-state utility values within economic models. |
VBA (Optional)
Automate calculation of expected utility and expected quality-adjusted life-years across multiple intervention strategies within decision-analytic models.
Sources
- von Neumann J, Morgenstern O. Theory of Games and Economic Behavior. Princeton University Press; 1944.
- Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes. 4th ed. Oxford University Press; 2015.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press; 2006.
- Keeney RL, Raiffa H. Decisions with Multiple Objectives: Preferences and Value Trade-Offs. Cambridge University Press; 1993.
- National Institute for Health and Care Excellence (NICE). Health Technology Evaluation Manual. Latest edition.
Related Concepts (3)
Library
Publications
2
Decision Modelling for Health Economic Evaluation — Briggs, Claxton & Sculpher, 1st Edition ed., 2006 (Oxford University Press)
Foundational textbook on decision-analytic modelling for economic evaluation, covering decision trees, Markov models, handling parameter and structural uncertainty, probabilistic sensitivity analysis, and value of information. Volume 1 in the Handbooks in Health Economic Evaluation series.
BookView source →Bayesian Theory — José M. Bernardo & Adrian F. M. Smith, 1st Edition ed., 1994 (John Wiley & Sons)
A comprehensive theoretical account of Bayesian inference, prior and posterior distributions, probability, information and statistical decision theory.
BookView source →
Frequently Asked Questions (6)
What is expected utility?
The probability-weighted average of the utility of each possible outcome of a decision under uncertainty, used to identify a rational preferred option.
Source: von Neumann & Morgenstern 1944
How does expected utility account for attitudes to risk?
Expected value weights monetary or health outcomes by their probabilities but treats a pound or a life-year as equally valuable however much is already held, which does not match how people actually regard risk. Expected utility instead weights the utility of each outcome, and because utility usually rises less steeply as outcomes improve, it captures a preference for certainty over an equally valued gamble. This lets it represent risk-averse or risk-seeking behaviour that expected value cannot. Von Neumann and Morgenstern (1944) established the framework.
Source: von Neumann & Morgenstern 1944
How is expected utility calculated?
Expected utility is calculated by assigning a utility to each possible outcome, weighting each utility by the probability of the outcome, and summing these products. This gives the average utility expected from the uncertain prospect. The option with the highest expected utility is preferred. Because utilities rather than raw outcomes are averaged, the calculation incorporates the decision maker's valuation of outcomes, including how they weigh gains, losses, and risk, so it captures preferences that expected value alone would not.
Source: von Neumann & Morgenstern 1944
How does expected utility differ from expected value?
Expected utility averages the utility of outcomes, while expected value averages the outcomes themselves in their natural units, such as money. Expected utility incorporates attitudes to risk through the shape of the utility function, so a risk-averse decision maker values a certain outcome more than a risky one with the same expected value. When the utility function is linear, the two coincide, but where risk aversion matters, expected utility, not expected value, is the appropriate criterion, since it reflects preferences over risk.
Source: von Neumann & Morgenstern 1944
What is the expected utility theorem?
The expected utility theorem, due to von Neumann and Morgenstern, states that if a decision maker's preferences over uncertain prospects satisfy certain axioms, such as completeness, transitivity, continuity, and independence, then those preferences can be represented by an expected utility function, so the decision maker acts as if maximising expected utility. The theorem provides the foundation for expected utility as the criterion of rational choice under uncertainty, deriving it from reasonable conditions on preferences rather than assuming it directly.
Source: Raiffa & Schlaifer 1961
How is expected utility used in decision making?
Expected utility is used to choose among options under uncertainty by selecting the one with the highest expected utility, incorporating the decision maker's valuation of outcomes and risk. It underlies formal decision analysis, where outcomes are assigned utilities and options compared by expected utility. In health, it connects to the use of utilities for health states, and it provides the theoretical basis for evaluating uncertain prospects, though its application requires eliciting utilities that capture preferences, which can be demanding.
Source: Raiffa & Schlaifer 1961
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 9 Oct 2025
Content version: 1.0.0
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- Persistent URI
- https://healtheconomics.wiki/concept/expected-utility
- Term code
- HE-EM-MP-014
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