Ordinal Utility

Ordinal Utility in Welfare Economics and Health Appraisal

Ordinal utility is the kind of utility that carries information about order and nothing more, so a higher number marks a preferred option while the size of the gap between numbers has no meaning. A patient who prefers day surgery to an overnight stay, and an overnight stay to a long wait, can be described by 3, 2 and 1 or equally well by 100, 7 and 0. For health economists the idea matters because it decides which welfare judgements can be drawn from preferences alone. It explains why welfarist cost-benefit analysis measures gains in money and why cost-utility analysis needs cardinal QALY weights that can be added across patients.

What an ordinal utility number records

Hausman, in the Stanford Encyclopedia of Philosophy entry on philosophy of economics, states that when preferences are complete and transitive and satisfy a further continuity condition they can be represented by an ordinal utility function. Such a function represents the ranking and contains no information beyond it. Hansson and Grüne-Yanoff, in the entry on preferences, write the representation as follows.

$$A \succ B \iff u(A) > u(B)$$

where $A$ and $B$ are two options, such as two ways of treating the same condition, $\succ$ means strictly preferred by the person, and $u$ is a utility function that gives each option a number.

Any function that gives the preferred option the larger number works as a representation. Osborne and Rubinstein, in Models in Microeconomic Theory, prove that an increasing function of a utility function is itself a utility function for the same preferences (their Proposition 1.3).

$$w(x) = f\big(u(x)\big), \qquad f(a) > f(b) \text{ whenever } a > b$$

where $x$ is any option, $u$ is one utility function for the person, $f$ is any strictly increasing function and $w$ is the new utility function it produces. Hansson and Grüne-Yanoff call this transformation property the defining characteristic of ordinal scales. A statement about ordinal utility is therefore meaningful only if it stays true under every such $f$, so "the patient prefers day surgery" survives while "day surgery is worth three times the long wait" does not.

How economics came to treat utility as ordinal

Hansson and Grüne-Yanoff report that Pareto argued for confining social science to ordinal comparisons, because no accurate way of measuring cardinal hedonic utility was available. In their account Hicks and Allen, in 1934, went further and argued that cardinal utility should be excluded.

Brouwer, Culyer, van Exel and Rutten describe the result for health economics. In the classical tradition utilities were cardinally measured and could be added across individuals. In the Paretian tradition of the new welfare economics, utility was usually ordinally measured and interpersonal comparisons were held to be impossible or meaningless, and social judgements rested on the Pareto principle. That tradition still frames the welfarist side of the debate over how health care should be evaluated, and its wider structure is set out under welfare economics.

Which welfare judgements survive a change of utility numbers

Since any increasing $f$ is allowed, a welfare judgement built on ordinal utility must give the same answer whichever representation is used. Three results used in health economics pass this condition and one familiar operation fails it.

The Pareto test

A move is a Pareto improvement when at least one person prefers it and nobody prefers the status quo. Hausman notes that this conclusion can be drawn without any interpersonal comparison. Each person's ranking of the two options is unchanged by any increasing transformation of that person's utility, so the verdict holds under every representation. Its reach is small, since almost every health policy, such as moving NHS budget between services, has losers as well as winners.

The marginal rate of substitution

The rate at which a person will trade one good for another also survives. For a person whose utility depends on two goods, the marginal rate of substitution under the transformed function is as follows.

$$\text{MRS}_w = \frac{\partial w / \partial x_1}{\partial w / \partial x_2} = \frac{f'(u)\, \partial u / \partial x_1}{f'(u)\, \partial u / \partial x_2} = \text{MRS}_u$$

where $x_1$ and $x_2$ are quantities of the two goods, $u$ and $w = f(u)$ are two utility functions for the same person, $f'(u)$ is the slope of the transformation, which is positive and the same in numerator and denominator, and $\text{MRS}_u$ and $\text{MRS}_w$ are the rates computed from each function. The factor $f'(u)$ cancels, so the trade-off between, for example, health care visits and other consumption is a property of the preferences and not of the numbers chosen.

Money measures of benefit

A person's willingness to pay for a programme is the sum of money that leaves the person indifferent between having the programme and paying, and not having it and keeping the money. Indifference is a statement about ranking, so the sum is the same under every representation. A money measure therefore turns ordinal information into a number with a common unit, one reason welfarist cost-benefit analysis values health effects in money and not in utility.

What fails, adding utilities

Adding utility numbers across people does not survive, because transforming one person's numbers can change which option has the largest total, as the example below shows. Hausman notes that, because identifying welfare with preference satisfaction makes interpersonal comparisons questionable, few economists defend a utilitarian view of policy as maximising total or average welfare.

Worked example with three uses of a local health budget (illustrative)

The numbers below are illustrative. Two residents, Asha and Ben, rank three uses of the same local health budget. Option X keeps current clinic hours, option Y adds evening clinics and option Z funds a home visiting service. Asha's utility numbers, in her first representation, are 1, 4 and 9. Ben's are 3, 8 and 6.

Step 1, Pareto test of Y against X. Asha has 4 for Y against 1 for X and Ben has 8 against 3, so both prefer Y. Under the square root representation Asha has 2 against 1, so the verdict is unchanged. Y is a Pareto improvement on X under any representation.

Step 2, Pareto test of Z against Y. Asha prefers Z (9 against 4) and Ben prefers Y (8 against 6). Neither option is a Pareto improvement on the other, and no transformation of either person's numbers can change that.

Step 3, adding utilities. With the first representation the totals are 1 + 3 = 4 for X, 4 + 8 = 12 for Y and 9 + 6 = 15 for Z, so Z has the largest total.

Step 4, the same preferences on another scale. The square root is strictly increasing, so $\sqrt{1} = 1$, $\sqrt{4} = 2$ and $\sqrt{9} = 3$ describe Asha's ranking exactly as well. The totals become 1 + 3 = 4 for X, 2 + 8 = 10 for Y and 3 + 6 = 9 for Z, so Y now has the largest total.

The two representations carry identical information about both residents, yet the summed ranking of Y and Z reverses. With ordinal utility alone, the analyst can say that Y beats X and that the choice between Y and Z needs further information, such as money valuations or an explicit judgement about whose gain counts for more.

Why cost-benefit analysis prices outcomes in money

Cost-benefit analysis extends the Pareto test to policies with losers. Hausman describes a potential Pareto improvement as a change in which the winners would be willing to pay more than the losers would need in compensation, with the compensation entirely hypothetical. Every input to this test is a money amount derived from individual rankings, so it stays consistent with ordinal utility. Brouwer and colleagues point out that applied cost-benefit analysis still makes interpersonal comparisons, since it compares uncompensated money gains and losses across people by giving each unit of money an equal shadow weight, whoever receives it.

Going further requires assumptions about utility that ordinal theory does not supply. The Green Book (2026), HM Treasury guidance on appraisal, states that an extra GBP 1,000 of income is more valuable to a low-income household than to a high-income one, a principle it calls the diminishing marginal utility of income (paragraph 7.8). Its distributional weights use an exponent of 1.25, described as an estimate of the elasticity of marginal utility of consumption (paragraph 7.15). A weight of that kind compares the size of utility gains across households, which is information an ordinal ranking cannot provide.

Ordinal utility, QALYs and the extra-welfarist alternative

Cost-utility analysis rests on different foundations. The NICE technology appraisal and highly specialised technologies guidance manual (PMG36, sections 4.2.15 and 4.3.1) asks for health effects to be expressed in QALYs and justifies cost-effectiveness analysis by NICE's focus on maximising health gains from a fixed NHS and personal social services budget. Its reference case gives an additional QALY the same weight regardless of the other characteristics of the people who receive it, except in specific circumstances. Summing QALYs across patients and treating one person's gain as equal to another's are the operations that ordinal utility rules out.

The health state weights behind a QALY are therefore built to be cardinal. Hausman notes that preferences meeting further conditions, including independence, can be represented by an expected utility function that is unique up to a positive affine transformation, so comparisons of differences between values carry meaning for one person. The standard gamble draws on that theory, and the requirement that equal gaps mean equal gains is discussed under cardinal scaling.

Brouwer and colleagues describe extra-welfarism as permitting interpersonal comparisons of well-being in several dimensions, which moves analysis beyond Paretian economics. On that view, adding QALYs across people rests on an explicit judgement that health gains are comparable, a judgement welfarist economics would not draw from preference rankings alone.

Richer information and the impossibility result

List, in the Stanford Encyclopedia entry on social choice theory, explains that Arrow's framework assumes preferences are ordinal and not interpersonally comparable, which Sen formalised as ordinal measurability with no interpersonal comparability. Under that assumption no rule for ranking social options with more than two alternatives meets all of Arrow's conditions, a result known as Arrow's impossibility theorem. List reports that Sen took the theorem to show that ordinal preferences are insufficient for satisfactory social choices.

Small additions to ordinal information change the result. Under ordinal measurability with interpersonal level comparability, the same increasing transformation must apply to everyone, so it becomes meaningful to say who is worse off, and List gives maximin as a social ranking that works under that assumption. Maximin ranks options by the welfare of the worst-off person alone. A health policy that gives priority to the sickest patients needs only this kind of level comparison, while a rule that adds gains, such as total QALY maximisation, needs comparable units. These choices sit behind any social welfare function used in health appraisal.

Common misreadings of ordinal utility

Several errors recur when ordinal ideas meet health data. Each treats order information as if it measured size.

The first is to read the gap between ranks as strength of preference. EQ-5D level codes are ordinal, and Devlin, Parkin and Janssen show that the level sum score, which treats the level labels as numbers and adds them, cannot even be treated as ordinal, as discussed under ordinal scaling.

The second is to assume that diminishing marginal utility follows from ordinal theory. With one good $x$, the functions $u = \sqrt{x}$ and $w = u^4 = x^2$ rank every quantity identically, yet the first has falling marginal utility and the second rising marginal utility, so the curvature carries no meaning on an ordinal reading. Diminishing marginal utility is a cardinal claim, so the Green Book weights rest on an elasticity estimate and not on rankings.

The third is to judge a utility number by its label. A utility function in a consumer model is usually ordinal, while a health state utility produced by a choice-based valuation is meant to be cardinal, so the meaning comes from the method behind the number.

Sources

• Brouwer WBF, Culyer AJ, van Exel NJA, Rutten FFH. Welfarism vs. extra-welfarism. Journal of Health Economics. 2008;27(2):325-338. https://eprints.whiterose.ac.uk/id/eprint/3789/
• Devlin N, Parkin D, Janssen B. Methods for Analysing and Reporting EQ-5D Data. Cham: Springer; 2020. Chapter 2, Analysis of EQ-5D profiles. https://link.springer.com/content/pdf/10.1007/978-3-030-47622-9_2.pdf
• Hansson SO, Grüne-Yanoff T. Preferences. In: Zalta EN, ed. Stanford Encyclopedia of Philosophy. Substantive revision 14 March 2022. https://plato.stanford.edu/entries/preferences/
• Hausman DM. Philosophy of economics. In: Zalta EN, ed. Stanford Encyclopedia of Philosophy. Substantive revision 4 September 2018. https://plato.stanford.edu/entries/economics/
• HM Treasury. The Green Book: UK government guidance on appraisal. 2026 edition, updated 28 October 2026. Chapter 7, Distributional analysis and place-based analysis. https://www.gov.uk/government/publications/the-green-book--2
• List C. Social choice theory. In: Zalta EN, ed. Stanford Encyclopedia of Philosophy. Substantive revision 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). Published 31 January 2022, last updated 13 October 2026. Chapter 4, Economic evaluation. https://www.nice.org.uk/process/pmg36/chapter/economic-evaluation-2
• Osborne MJ, Rubinstein A. Models in Microeconomic Theory. Expanded second edition. Cambridge: Open Book Publishers; 2023. Chapter 1, Preferences and utility. https://doi.org/10.11647/obp.0362