Expected Value Criterion

Expected Value Criterion for Choice Under Risk

The expected value criterion, a decision rule for choosing under risk, scores every option by the probability-weighted average of its possible outcomes and takes the best score, whatever the spread of outcomes around that average. In health economics the averaged outcome is usually net monetary benefit, net health benefit or QALYs, so a cost-effectiveness model that recommends the strategy with the greatest expected net benefit is applying this criterion. The rule suits a health system that makes many similar decisions and pools risk across a large population, but it treats the decision maker as neutral to risk and needs a probability for every possible state of the world.

The rule written for a choice between health programmes

Peterson's encyclopedia entry on the St Petersburg paradox states the principle of maximising expected value. The value of an uncertain prospect is found by multiplying the value of each possible outcome by its probability and adding up the terms, and the prospect with the larger total is preferred.

$$a^{*} = \arg\max_{a \in A} \sum_{s=1}^{S} p_s \, x_{a,s}$$

where $A$ is the set of options, such as competing treatments, $s$ indexes the $S$ possible states of the world, $p_s$ is the probability of state $s$, with the probabilities summing to one, $x_{a,s}$ is the outcome of option $a$ in state $s$ in natural units such as pounds or QALYs, and $a^{*}$ is the option chosen. When outcomes are losses, the rule picks the smallest expected loss. Outcomes enter the sum untransformed, which separates the criterion from expected utility, where each outcome first passes through a utility function.

In cost-effectiveness analysis the outcome is normally net benefit at a stated value of health.

$$NMB_{a,s} = \lambda \, Q_{a,s} - C_{a,s}$$

where $NMB_{a,s}$ is the net monetary benefit of option $a$ in state $s$, $\lambda$ is the money value of one QALY, $Q_{a,s}$ is the QALYs produced and $C_{a,s}$ is the cost. Choosing the highest expected net monetary benefit applies the criterion to costs and QALYs together. The NICE manual PMG36, in section 4.2.16, asks for ICERs to be the ratio of expected additional total cost to expected additional QALYs, and for expected net health benefits to be presented at GBP 25,000 and GBP 35,000 per QALY gained. The arithmetic sits on the expected value and expected value analysis pages, and the criterion is the choice principle applied to their outputs.

The St Petersburg game and the move to expected utility

Peterson records that Nicolaus Bernoulli posed the problem in 1713 and that the paradox takes its name from the St Petersburg journal in which Daniel Bernoulli published his paper in 1738. In the standard version a fair coin is tossed until it first lands heads, and the prize is $m_n = 2^n$ units of money, where $n$ is the number of tosses. The first head on toss $n$ has probability $(1/2)^n$, so every term of the expected value equals one and the sum has no finite value, although the player wins no more than 2 units half the time. Peterson treats the game as a counterexample to the principle.

Cramér in 1728 and Daniel Bernoulli in 1738 answered that a sensible person values money by its use rather than its quantity, and Bernoulli took the utility of wealth to be its logarithm. That idea of decreasing marginal utility leads to expected utility theory, in which the expected value criterion survives as the special case of a linear utility function.

For health, Lakdawalla and Phelps show that the result that a QALY is a QALY, however it is gained, rests on expected utility being linear in quality of life, or equivalently on consumers being risk-neutral in quality of life. With a concave utility of health, the certainty equivalent of a risky health prospect lies below its expected value, and the gap reflects risk aversion.

Long-run arguments and the position of a health system

The usual defence is that a decision maker facing many similar, independent choices will end up close to the expected value across all of them. Briggs, in the Stanford Encyclopedia entry on expected utility, presents this long-run argument through the law of large numbers, under which the average gain per trial over a long run of similar gambles is overwhelmingly likely to be close to the expected value of one trial. Knight wrote in 1921 that an uncertainty which can be reduced to an objective, quantitatively determinate probability can be reduced to complete certainty by grouping cases, the logic of insurance and of risk pooling.

A national health service, funding the same treatment for thousands of patients, is close to the setting the argument needs. Lakdawalla and Phelps note that third-party payers are typically thought of as risk-neutral. Briggs also sets out the weak points. Many decisions are made only a small finite number of times, and the argument needs independent trials, which Briggs says does not hold for choices about medical treatment, citing a failed antibiotic course that makes the next failure more likely. In an appraisal the same problem appears as decision uncertainty, since uncertainty about a treatment effect is shared by every treated patient and a larger caseload does not average it away.

Choosing on the mean while uncertainty remains

Claxton's 1999 paper gave the criterion its sharpest form in health technology assessment. He argued that rules of inference are arbitrary and irrelevant to the decisions that evaluations inform, that decisions should be based only on mean net benefits irrespective of statistical significance, and that the distribution of net benefit matters only for deciding whether more information is required. Uncertainty then belongs to the research decision, through value of information analysis, and not to the adoption decision.

NICE follows the criterion closely for estimation and more loosely for the decision. PMG36 section 4.7.12 says the committee's preferred estimate should come from a probabilistic analysis when possible unless the model is linear, and section 4.10.6 asks for expected costs and expected QALYs for each option. Section 6.2.32 then asks the committee to take into account the likelihood of decision error and its consequences for patients and the NHS, and section 6.2.33 says it will normally be more cautious about recommending a technology when it is less certain about the evidence. Where evidence is highly uncertain, section 6.2.30 allows recommendations that include managed access, data collection or research, the evidence route Claxton's argument points towards.

Rival rules when probabilities are weak or missing

The criterion needs a probability for each state. Knight separated measurable risk from a higher form of uncertainty not susceptible to measurement, and Milnor's 1951 RAND memorandum on games against nature sets out four rules that had been proposed for a table of options against unknown states.

• Laplace's criterion assumes that unknown probabilities are equal and chooses the option with the greatest average payoff, which is the expected value criterion with equal weights.
• Wald's minimax principle treats nature as an opponent, which among the options means choosing the one whose worst outcome is least bad (maximin, when outcomes are gains).
• Hurwicz's criterion weights each option's best and worst payoff by a constant measuring optimism.
• Savage's criterion measures regret as the gap between what could have been obtained had the state been known and what was obtained, and minimises the largest regret.

The Green Book, HM Treasury's 2026 appraisal guidance, allows an expected value as a central estimate but warns that estimating probabilities is often challenging and subjective. It assigns no probabilities to its climate change scenarios and asks for tail risks such as a potential pandemic to be considered proportionately.

Worked example with a pandemic vaccine stockpile

All numbers in this example are illustrative. A health department chooses between no stockpile (N), a small stockpile (S) and a large stockpile (L) of pandemic vaccine. Each cell is a net loss in GBP million, with health lost valued at a threshold and added to the costs, so smaller is better.

1. Expected losses. For N, 0.70 × 0 + 0.25 × 200 + 0.05 × 2,000 = 150. For S, 0.70 × 40 + 0.25 × 60 + 0.05 × 1,200 = 103. For L, 0.70 × 150 + 0.25 × 160 + 0.05 × 300 = 160. The criterion chooses the small stockpile, GBP 57 million ahead of the large one.

2. Spread. The variance of the loss under S is 0.70 × 1,600 + 0.25 × 3,600 + 0.05 × 1,440,000 − 103² = 63,411, a standard deviation of about GBP 251.8 million. Under L it is 0.70 × 22,500 + 0.25 × 25,600 + 0.05 × 90,000 − 25,600 = 1,050, about GBP 32.4 million.

3. Rules that set the probabilities aside. The worst losses are 2,000, 1,200 and 300, so Wald's rule chooses L. Laplace's equal weights give (0 + 200 + 2,000) ÷ 3 = 733.3 for N, (40 + 60 + 1,200) ÷ 3 = 433.3 for S and (150 + 160 + 300) ÷ 3 = 203.3 for L, so L again. The smallest losses in the three states are 0, 60 and 300, so the largest regrets are 1,700 for N, 900 for S and 150 for L, and Savage's rule also chooses L.

4. Switching value. Holding the moderate pandemic at 0.25 and writing $p$ for the probability of a severe one, the expected loss is $L_S = 45 + 1160p$ for S and $L_L = 152.5 + 150p$ for L. They are equal when p = 107.5 ÷ 1,010 = 0.106, so the choice of S holds only while the chance of a severe pandemic stays below about 0.11.

5. Value of perfect information. Knowing the state in advance, the department would pick the smallest loss in each state, giving 0.70 × 0 + 0.25 × 60 + 0.05 × 300 = 30. The expected value of perfect information is 103 − 30 = GBP 73 million, the most that perfect foresight could add.

6. Interpretation. The criterion and the three other rules disagree because of a 5% tail. A payer confident in the probabilities would follow the criterion, while one that doubts the tail estimate, or is particularly risk-averse in the Green Book's sense, has grounds to prefer the large stockpile.

Welfarist and extra-welfarist readings of the criterion

In welfare economics, Brouwer and colleagues distinguish extra-welfarism by its rejection of the welfarist tenet that only individual utility counts. In a welfarist cost-benefit analysis the outcomes that count are individual utilities, so an average of money amounts is a fair stand-in for welfare when those affected are close to risk-neutral or each bears a small share of the risk. Brouwer and colleagues regard reading extra-welfarism as health maximisation as mistaking one valid application for the whole, but where a health system does take health as its maximand, maximising expected QALYs is the expected value criterion applied to health, risk-neutral over health by construction.

The criterion also bears on the choice of a social welfare function. Freeman's encyclopedia entry on the original position describes Harsanyi's argument that, under complete uncertainty, a person should assume an equal probability of occupying any position in society, which leads to the principle of average utility. That is Laplace's equal-weight form of the criterion applied behind a veil of ignorance. Rawls argued instead for the maximin rule, given the gravity of a choice that is not repeatable or renegotiable. In health priority setting the same contrast lies between maximising total QALYs and giving priority to those in the worst health.

Misreadings of the expected value criterion

One misreading treats the expected value as the most likely outcome. The Green Book defines a central estimate as the most likely value of a cost or benefit, yet allows an expected value to serve as one, and in the example the most likely state is no pandemic, in which having no stockpile costs nothing, but that option has an expected loss of GBP 150 million. Another reads Laplace's equal weights as a neutral absence of probabilities, although in the example they give the severe pandemic a probability of one third.

Sources

• Briggs RA. Normative theories of rational choice: expected utility. Stanford Encyclopedia of Philosophy. First published 8 August 2014, substantive revision 18 September 2023. https://plato.stanford.edu/entries/rationality-normative-utility/
• Brouwer WBF, Culyer AJ, van Exel NJA, Rutten FFH. Welfarism vs. extra-welfarism. Journal of Health Economics. 2008;27(2):325-338. https://doi.org/10.1016/j.jhealeco.2007.07.003
• Claxton K. The irrelevance of inference: a decision-making approach to the stochastic evaluation of health care technologies. Journal of Health Economics. 1999;18(3):341-364. https://doi.org/10.1016/S0167-6296(98)00039-3
• Freeman S. Original position. Stanford Encyclopedia of Philosophy. First published 20 December 2008, substantive revision 24 October 2023. https://plato.stanford.edu/entries/original-position/
• HM Treasury. The Green Book (2026). Updated 28 October 2026. https://www.gov.uk/government/publications/the-green-book--2/the-green-book-2026
• Knight FH. Risk, Uncertainty, and Profit. Boston: Houghton Mifflin; 1921. Part III, Chapter VII. https://www.econlib.org/library/Knight/knRUP.html
• Lakdawalla DN, Phelps CE. Evaluation of medical technologies with uncertain benefits. NBER Working Paper 26058. National Bureau of Economic Research; 2019, revised 2020. https://www.nber.org/papers/w26058
• Milnor JW. Games against nature. Research Memorandum RM-679-PR. Santa Monica: RAND Corporation; 1951. https://www.rand.org/pubs/research_memoranda/RM0679.html
• National Institute for Health and Care Excellence. NICE technology appraisal and highly specialised technologies guidance: the manual (PMG36). Published 31 January 2022, last updated 13 October 2026. Sections 4.2.16, 4.7.12, 4.10.6, 6.2.30, 6.2.32 and 6.2.33. https://www.nice.org.uk/process/pmg36
• Peterson M. The St. Petersburg paradox. Stanford Encyclopedia of Philosophy. First published 30 July 2019, substantive revision 1 August 2023. https://plato.stanford.edu/entries/paradox-stpetersburg/