Cardinal Utility

Cardinal Utility in Welfare Economics and QALY Valuation

Cardinal utility is a kind of utility whose numbers record how much better one outcome is than another, so that two gains can be compared in size as well as in order. A cardinal scale is fixed only up to its zero point and its unit, so any positive linear rescaling carries the same information. Health economists rely on this property whenever they take an expected value over uncertain health outcomes or add utility weights over time and across patients into quality-adjusted life years. In welfare economics the idea was set aside in the 1930s in favour of ranking alone and then revived in a narrower form for choice under risk, and the gap between those two forms runs through the welfarist and extra-welfarist debate. The measurement properties of QALY weights themselves are treated under cardinal scaling.

Hedonic measurement, the ordinal turn and expected utility

The early marginalists treated utility as a quantity of pleasure. Hausman, in the Stanford Encyclopedia of Philosophy entry on philosophy of economics, records that Jevons held that agents choose so as to maximise their own happiness. Brouwer and colleagues describe the classical welfarist tradition in which utilities were cardinally measured and added across individuals, with the social optimum where the sum is largest, the same arithmetic that a cost per QALY ranking uses today.

Hansson and Grüne-Yanoff, in the encyclopedia entry on preferences, explain the reaction. Pareto argued that, because no accurate procedure existed for measuring cardinal hedonic utility, social scientists should keep to ordinal comparisons, and Hicks and Allen in 1934 argued that cardinal utility should be excluded altogether. Ramsey and later von Neumann and Morgenstern then built tools that represent preference magnitudes on an interval scale, a new utility constructed to reflect the preference ordering and very different from the older hedonic concept.

Kahneman, Wakker and Sarin name Bentham's sense experienced utility and the weight of an outcome in a choice decision utility. Health valuation borrows from both. The standard gamble measures decision utility under risk, while their axioms, with cardinal instant utility, rank episodes by the integral of instant utility over time, a structure close to the QALY.

What a cardinal scale allows and what it does not

Hansson and Grüne-Yanoff call scales that represent the size of intervals, or of ratios, cardinal, and note that most work on preference measurement has aimed at an interval scale. For cardinal utility in the economic sense, an interval scale, the permitted changes are the positive linear ones.

$$v(x) = a\,u(x) + b, \quad a > 0$$

where $u(x)$ is the utility of outcome $x$ on the original scale, $v(x)$ is its utility after rescaling, $a$ is any positive constant that changes the unit and $b$ is any constant that moves the zero. An ordinal utility function may instead be replaced by any strictly increasing transformation of itself, so only the ranking survives.

What survives a positive linear rescaling is the comparison of differences. A ratio of two utility differences is the same for every choice of $a$ and $b$.

$$\frac{u(x) - u(y)}{u(z) - u(w)} = \frac{v(x) - v(y)}{v(z) - v(w)}$$

where $x$, $y$, $z$ and $w$ are any four outcomes with $u(z) \neq u(w)$. Briggs, in the encyclopedia entry on expected utility, adds that positive linear transformations of outcome utilities never change which option has the higher expected utility. Claims about curvature, such as diminishing marginal utility of income, are also claims about differences, so they hold meaning on a cardinal scale and none on an ordinal one, where an increasing transformation can turn a concave function into a convex one.

Von Neumann and Morgenstern utility and the standard gamble

Von Neumann and Morgenstern showed that if preferences over lotteries meet their axioms, a numerical utility exists and is determined up to a linear transformation. In their words, they had practically defined numerical utility as that thing for which the calculus of mathematical expectations is legitimate. This cardinal utility makes expected utility a coherent rule for comparing a risky treatment with a safe one.

The standard gamble applies the theory to health. A respondent gives the probability at which a health state is as good as a gamble between full health and immediate death.

$$u(h) = p^{*}\,u(F) + (1 - p^{*})\,u(D) = p^{*}$$

where $u(h)$ is the utility of health state $h$, $p^{*}$ is the probability of full health at which the respondent is indifferent, $u(F)$ is the utility of full health set to 1 and $u(D)$ is the utility of death set to 0. Torrance and Feeny note that such utilities are designed for individual decisions under uncertainty and need further assumptions before aggregation across people.

Devlin, Parkin and Janssen explain that EQ-5D value sets turn a descriptive profile into a single cardinal number. The NICE manual for technology appraisal and highly specialised technologies (PMG36) asks for utilities based on public preferences using a choice-based method and states that the time spent in each health state is multiplied by the utility, an operation that assumes at least an interval scale.

The vNM scale carries attitude to risk as well as the value of the state. Bleichrodt shows that using QALYs to represent von Neumann and Morgenstern preferences imposes mutual utility independence between life years and health status, a constant proportional trade-off of life years for health status and risk neutrality over life years. He also cites risk aversion over life years among patients with operable lung cancer, which breaks the last condition.

Cardinal measurement without interpersonal comparison

Cardinality concerns one person's scale and says nothing about how units compare between people. List, in the encyclopedia entry on social choice theory, reports Sen's proof that Arrow's impossibility result still holds when each person's utility is cardinal but not comparable across people.

Classical utilitarianism needs a common unit across people. List gives it as a possible rule when every individual function shares the same scaling constant.

$$W(x) = \sum_{i=1}^{n} u_i(x)$$

where $W(x)$ is social welfare in social state $x$, $u_i(x)$ is the cardinal utility of person $i$ in that state and $n$ is the number of people. Hausman reports Harsanyi's proof that if individual preferences and social evaluations both satisfy the expected utility axioms and society follows unanimous individual preferences, social evaluations are a weighted sum of individual utilities. The weights still carry the interpersonal judgement, and the social welfare function page covers the wider family of rules.

Health economics meets this problem every time it adds QALYs across patients. Brouwer and colleagues note that welfarist economics normally deems interpersonal comparisons of utility impossible or meaningless, while extra-welfarism compares health and capabilities instead of utility. They add that applied cost-benefit analysis makes an interpersonal comparison by giving equal weight to each unit of money gained or lost.

Diminishing marginal utility of income in UK appraisal

The Green Book (2026), HM Treasury's guidance on appraisal, states that an additional GBP 1,000 of income is more valuable to a low-income household than to a high-income one and calls this the diminishing marginal utility of income. That statement compares utility differences, so it presupposes a cardinal scale. The guidance allows costs and benefits to be weighted by the income of the group that bears them, using an elasticity of marginal utility of consumption of 1.25, and the weights themselves are covered under diminishing marginal utility. The Green Book treats health effects differently, uprating the value of future health effects by 2% a year because diminishing marginal utility of consumption does not apply to them.

Worked example of a risky operation and two patients

All numbers in this example are illustrative. A patient with severe angina can stay on medical treatment, a state with a standard gamble utility of 0.80, or have surgery. Surgery gives full health (utility 1) with probability 0.75, a disabling stroke (utility 0.40) with probability 0.15 and operative death (utility 0) with probability 0.10.

Step 1 with the original scale

Expected utility weights each outcome by its probability. The original scale gives this result.

$$EU_{S} = 0.75 \times 1 + 0.15 \times 0.40 + 0.10 \times 0 = 0.81$$

where $EU_{S}$ is the expected utility of surgery. Medical treatment gives 0.80 for certain, so surgery is preferred by 0.01.

Step 2 with a positive linear rescaling

Using $v = 10u + 5$ the four states score 15, 13, 9 and 5. Surgery then scores 0.75 times 15 plus 0.15 times 9 plus 0.10 times 5, which is 13.1, against 13 for medical treatment. The margin of 0.1 is 10 times the original 0.01, and the ratio of the gap between full health and angina to the gap between angina and stroke is 0.5 on both scales (0.20 over 0.40, and 2 over 4).

Step 3 with a transformation that keeps only the order

Square roots keep the ranking but are not linear, giving scores of 1, about 0.894, about 0.632 and 0. Surgery now gives 0.75 plus 0.15 times 0.632, about 0.845, while medical treatment gives about 0.894, so the choice reverses, and the ratio of gaps falls to about 0.403 (0.1056 over 0.2620).

Step 4 with two patients and one budget

Programme X raises patient A from 0.60 to 0.70, a gain of 0.10, and programme Y raises patient B from 0.50 to 0.58, a gain of 0.08, so summing utilities favours X. Without interpersonal comparability A's scale may legitimately be multiplied by 0.5, which makes A's gain 0.05 and favours Y. The QALY removes this freedom by anchoring everyone at 1 for full health and 0 for death, and that common unit is a value judgement that measurement alone cannot supply.

Boundaries with neighbouring utility concepts

Cardinal scaling and interval scaling deal with the measurement properties of health state values, while this page deals with the economic theory behind them. Ordinal scaling covers ranked health data, and ranking-only preference theory belongs to ordinal utility. The time trade-off yields cardinal values through a trade of life years instead of a gamble, so its values do not come from expected utility theory even when they are interval-scaled.

Common misreadings of cardinal utility

The commonest error treats vNM utility as a measure of how intensely a state is felt. Hansson and Grüne-Yanoff stress that the vNM function was constructed to reflect a preference ordering, and Bleichrodt's risk neutrality condition shows that attitude to risk enters the scale, so a low standard gamble value can reflect aversion to the gamble as well as dislike of the state. A second error reads cardinal as comparable across people, although cardinality within each person leaves Arrow's problem in place and summing QALYs rests on an added equity judgement.

A third error assumes that any number on a 0 to 1 scale is a cardinal utility. Devlin, Parkin and Janssen describe valuation methods as meant to give a scale with cardinal properties, an aim open to testing, and Richardson argued that the time trade-off and person trade-off give more satisfactory units than the standard gamble or rating scale.

Sources

• Bleichrodt H. QALYs and HYEs: under what conditions are they equivalent? Journal of Health Economics. 1995;14(1):17-37. https://doi.org/10.1016/0167-6296(94)00033-Z
• Briggs RA. Normative theories of rational choice: expected utility. The Stanford Encyclopedia of Philosophy. Revised 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
• Devlin N, Parkin D, Janssen B. Methods for Analysing and Reporting EQ-5D Data. Cham: Springer; 2020. Chapters 1 and 4. https://doi.org/10.1007/978-3-030-47622-9
• Hansson SO, Grüne-Yanoff T. Preferences. The Stanford Encyclopedia of Philosophy. Revised 14 March 2022. https://plato.stanford.edu/entries/preferences/
• Hausman DM. Philosophy of economics. The Stanford Encyclopedia of Philosophy. Revised 4 September 2018. https://plato.stanford.edu/entries/economics/
• HM Treasury. The Green Book (2026). London: HM Treasury; updated 28 October 2026. https://www.gov.uk/government/publications/the-green-book--2/the-green-book-2026
• Kahneman D, Wakker PP, Sarin R. Back to Bentham? Explorations of experienced utility. Quarterly Journal of Economics. 1997;112(2):375-406. https://doi.org/10.1162/003355397555235
• List C. Social choice theory. The Stanford Encyclopedia of Philosophy. Revised 14 October 2022. https://plato.stanford.edu/entries/social-choice/
• National Institute for Health and Care Excellence. NICE technology appraisal and highly specialised technologies guidance: the manual (PMG36). London: NICE; published 31 January 2022, last updated 13 October 2026. Sections 4.3.1 and 4.3.2. https://www.nice.org.uk/process/pmg36
• Richardson J. Cost utility analysis: what should be measured? Social Science and Medicine. 1994;39(1):7-21. https://doi.org/10.1016/0277-9536(94)90162-7
• Torrance GW, Feeny D. Utilities and quality-adjusted life years. International Journal of Technology Assessment in Health Care. 1989;5(4):559-575. https://doi.org/10.1017/S0266462300008461
• von Neumann J, Morgenstern O. Theory of Games and Economic Behavior. Princeton: Princeton University Press; 1944. Third edition 1953, sections 3.5 to 3.7. https://archive.org/details/in.ernet.dli.2015.215284