Arrow's Impossibility Theorem

Arrow's Impossibility Theorem and Health Priority Setting

Arrow's impossibility theorem is a theorem of welfare economics about collective choice. It shows that once there are at least two people and more than two alternatives, such as three health programmes competing for one budget, every method of building a social ranking out of individual rankings breaks at least one of five conditions that each look reasonable on its own. Health economists meet the result whenever they try to say what society prefers, because a commissioning panel or appraisal committee that pools its members' rankings cannot be guaranteed a consistent answer. The theorem also explains why welfare economics leaned so much on the Pareto principle, and why ranking options by total QALYs escapes the impossibility only by adding a value judgement about comparing health between people.

The question Arrow asked about pooling preferences

The puzzle is older than Arrow. Morreau, in the Stanford Encyclopedia of Philosophy, describes the paradox of voting discovered by Condorcet in 1785. With three people and three alternatives, pairwise majority voting can prefer the first alternative to the second, the second to the third and the third to the first, leaving a cycle in place of a social ordering.

Arrow started with the properties an aggregation procedure ought to have and asked which procedures have them, in place of studying one voting method at a time. Morreau reports that Arrow did this while still a graduate student, that the original proof appeared in his 1951 book Social Choice and Individual Values, and that he restated the conditions in the 1963 second edition. List, in the same encyclopedia, records that Arrow won the Nobel Memorial Prize in Economics in 1972 and that the result prompted much work in social choice theory and welfare economics.

In health the alternatives can be programmes, coverage decisions or budget allocations. Each person's preferences enter only as a ranking. Morreau explains that Arrow confined the inputs to ordinal information because he held that the utility of one person cannot meaningfully be added to the utility of another. An aggregation rule is then written as follows.

$$R = F(R_1, R_2, \ldots, R_n)$$

where $R$ is the social ordering of the alternatives, $F$ is the aggregation rule, which Arrow called a social welfare function, $R_i$ is the ranking of the alternatives by person $i$, and $n$ is the number of people whose preferences count.

The five conditions and the result

Each condition expresses a demand that most people would make of a fair procedure. Morreau notes that Arrow himself regarded them as value judgements expressing citizens' sovereignty and rationality, not as truths beyond dispute.

List states the theorem plainly. If there are more than two alternatives, no aggregation rule satisfies all five conditions. He notes that pairwise majority voting satisfies every condition except social ordering, which is what the Condorcet cycle shows. With two alternatives there is no impossibility, and List describes May's 1952 theorem showing that majority rule is the only two-option rule meeting four procedural conditions, among them equal treatment of voters and of alternatives. The weak Pareto condition is the unanimity form of the principle behind a Pareto improvement, so the theorem shows that respecting unanimity cannot be combined with the other four demands.

Worked example with three commissioners and three programmes

All rankings and QALY figures in this example are illustrative. Three members of a local commissioning panel must rank a catch-up HPV vaccination campaign (A), a community mental health team (B) and a diabetes prevention programme (C). Member 1 ranks them A, B, C. Member 2 ranks them B, C, A. Member 3 ranks them C, A, B.

1. Pairwise majority votes. Members 1 and 3 prefer A to B, so A beats B by 2 votes to 1. Members 1 and 2 prefer B to C, so B beats C by 2 to 1. Members 2 and 3 prefer C to A, so C beats A by 2 to 1.

2. The cycle and the agenda. A beats B, B beats C and C beats A, so majority voting names no best programme. If the panel votes A against B first, A wins and then loses to C. If it votes B against C first, B wins and then loses to A. If it votes A against C first, C wins and then loses to B. Whoever sets the order of votes picks the winner.

3. A points system. A Borda count gives each programme points for its place in each member's ranking.

$$S(x) = \sum_{i=1}^{n} \left( m + 1 - r_i(x) \right)$$

where $S(x)$ is the Borda score of programme $x$, $n$ is the number of members, $m$ is the number of programmes and $r_i(x)$ is the place, counting 1 for first, that member $i$ gives to $x$. With $m = 3$, A scores 3 + 1 + 2 = 6, B scores 2 + 3 + 1 = 6 and C scores 1 + 2 + 3 = 6, a three-way tie.

4. Borda breaks independence. Morreau gives a four-option case, adapted here with D, extended GP opening hours. Member 1 ranks A, B, C, D and members 2 and 3 rank B, A, C, D. With $m = 4$, A scores 4 + 3 + 3 = 10 and B scores 3 + 4 + 4 = 11, so B is ranked above A. If member 1 moves B to the bottom, giving A, C, D, B, then A still scores 10 but B scores 1 + 4 + 4 = 9, and A now ranks above B. No member changed a view of A against B, yet the social order of A and B reversed. Member 1 could also misreport a ranking to lift a favourite programme, which Morreau notes is a known weakness of Borda counting.

5. Adding cardinal health information. Suppose each member speaks for a population group and the panel has QALY estimates. Group 1 gains 30 QALYs from A, 20 from B and 10 from C. Group 2 gains 5 from A, 40 from B and 25 from C. Group 3 gains 12 from A, 8 from B and 15 from C. Each group's ranking matches step 1, so the ordinal profile still cycles. Adding QALYs gives A 30 + 5 + 12 = 47, B 20 + 40 + 8 = 68 and C 10 + 25 + 15 = 50, a transitive ranking of B, then C, then A.

6. Interpretation. Steps 1 to 4 show the theorem at work, since majority voting fails social ordering and the Borda count fails independence. Step 5 escapes only because it uses information Arrow excluded, the size of each group's gain and the judgement that a QALY counts the same in every group.

Ways out and what each costs a health economist

Morreau describes two broad routes out of the impossibility, doing less or using more information. Both have counterparts in health economics.

Restricting the domain. Morreau reports Black's 1948 result that with an odd number of voters and single-peaked preferences, pairwise majority voting always yields an ordering whose top is the median voter's favourite. He also reports arguments that deliberation can move preferences towards this kind of structure, a reason to value deliberative processes in priority setting over simple polling of members.

Using cardinal and comparable information. List explains that Sen extended Arrow's framework to welfare measures, and that once welfare can be compared between people in suitable ways, aggregation rules satisfying analogues of all five conditions exist. Classical utilitarianism, which ranks options by the sum of welfare, works with comparable units, and Rawlsian maximin works with comparable levels. List concludes that in a welfare-aggregation setting the impossibility can be traced to a lack of interpersonal comparability.

Health as the measure. Dolan and Tsuchiya argue that a social welfare function defined over health is useful for policy because health is more readily compared between people than utility is. Rawlins and Culyer, then chair and former vice chair of the National Institute for Clinical Excellence, explain that most of its appraisals assumed that a QALY is a QALY, giving the same weight to a QALY whoever gains it, and that this was itself a social value judgement. Read through Sen's framework, ranking options by total QALYs in cost-utility analysis is a utilitarian rule over a cardinal, interpersonally comparable health measure. Arrow's ordinal theorem leaves such a rule untouched, and the escape rests on the comparability judgement. Equity-weighted analysis changes the weights while keeping comparability.

Why the theorem matters for welfarism and extra-welfarism

Brouwer, Culyer, van Exel and Rutten (2008) write that since Arrow's work on the impossibility of constructing a non-dictatorial social welfare function, economists' thinking about increases in social welfare has still been dominated by the Pareto improvement, which has little cutting power outside voluntary trading in well-regulated markets. They identify Sen's work as one important starting point for extra-welfarism, which in their account permits outcomes other than utility and interpersonal comparisons of wellbeing that allow analysis beyond Paretian economics.

A welfarist evaluation that refuses interpersonal comparisons of utility cannot use the Pareto principle to rank policies with losers as well as gainers, and once three or more such policies are compared, any rule built from ordinal preferences alone runs into Arrow's problem. Extra-welfarist evaluation, with health as the outcome and comparisons between people allowed, works in the richer informational setting where social rankings are possible. Brouwer and colleagues also describe the decision-making approach of Sugden and Williams (1978), in which decision-makers are the source of values and weights. A ranking that follows a legitimate decision-maker's stated objectives does not pool individual rankings, so the aggregation question moves to how those objectives are chosen and held to account. Rawlins and Culyer wrote that the institute had no particular legitimacy to determine society's values and set up a Citizens Council so that its social value judgements would resonate with the public.

Multiple criteria and other aggregation problems

The same logic applies when criteria replace people. Morreau explains that this brings multi-criteria problems within the Arrow and Sen framework, where the theorem rules out an overall ordering if analogues of its conditions are appropriate. List adds that the result carries over to the multiple criteria a single decision maker uses, and records applications to evidence amalgamation.

Health care multi-criteria decision analysis mostly avoids the ordinal version of the problem. Thokala and colleagues, writing for the ISPOR task force in 2016, describe value measurement models as by far the most widely used approach in health care, usually additive models that multiply each option's score on a criterion by the criterion's relative weight. These models use information about how much better one option is than another and how criteria trade off, the cardinal and comparable information that Arrow's framework leaves out. The scores and weights then need the justification that the theorem had demanded of the procedure.

How the theorem relates to neighbouring ideas

Arrow's social welfare function is a rule mapping every possible profile of rankings to a social ranking. The Bergson and Samuelson social welfare function ranks social states for one given set of preferences, and Morreau explains that Arrow's theorem does not limit this single-profile approach, because either the unrestricted domain or the independence condition is out of place there.

Common misreadings of the impossibility

Five errors recur when the theorem is cited. Each stretches what Arrow proved.

1. Democracy is proved incoherent. Morreau records that some, following Riker, read the theorem this way, while others argue that procedures cannot reasonably be required to meet all of Arrow's conditions.
2. Every choice is affected. The impossibility needs more than two alternatives, and for a choice between two options majority rule works, as May showed.
3. Any workable rule must have a dictator. A dictator is forced only if all the other conditions are kept. Relaxing the domain, the independence condition or the ordinal input opens the possibilities that Morreau and List describe.
4. QALY maximisation falls foul of Arrow. Adding QALYs uses cardinal, interpersonally comparable information outside Arrow's framework, so the ordinal theorem does not apply to it. The open question is whether that comparison is justified.
5. Independence of irrelevant alternatives is welfarism. Morreau shows that the independence condition leaves room for non-welfare features such as equality, and that welfarism corresponds to a stronger neutrality condition.

Sources

• 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
• Dolan P, Tsuchiya A. The social welfare function and individual responsibility: some theoretical issues and empirical evidence from health. HEDS Discussion Paper 03/01. Sheffield: University of Sheffield; 2003. Published in revised form in Journal of Health Economics. 2009;28(1):210-220. https://eprints.whiterose.ac.uk/id/eprint/11035/
• List C. Social choice theory. In: Stanford Encyclopedia of Philosophy. Substantive revision 14 October 2022. Sections 1.2, 2.2, 3.2 and 4.1 to 4.4. https://plato.stanford.edu/entries/social-choice/
• Morreau M. Arrow's theorem. In: Stanford Encyclopedia of Philosophy. Substantive revision 7 December 2025. Sections 1 to 6. https://plato.stanford.edu/entries/arrows-theorem/
• Rawlins MD, Culyer AJ. National Institute for Clinical Excellence and its value judgments. BMJ. 2004;329(7459):224-227. https://doi.org/10.1136/bmj.329.7459.224
• Thokala P, Devlin N, Marsh K, et al. Multiple criteria decision analysis for health care decision making, an introduction. Report 1 of the ISPOR MCDA Emerging Good Practices Task Force. Value in Health. 2016;19(1):1-13. https://doi.org/10.1016/j.jval.2015.12.003