Social Choice Theory

Social Choice Theory in Health Economics

Social choice theory is a body of formal models and theorems about aggregation rules, the procedures that turn many individual rankings, votes, welfare levels or expert judgements into a single social ranking or decision. Its questions arise whenever a health system ranks options for a whole population, whether by adding QALYs across patients or by a committee voting on a recommendation. Arrow showed that no rule for combining purely ordinal rankings over three or more options meets a short list of reasonable conditions, and Sen argued that the way forward is richer information, such as welfare measures that can be compared between people. That argument is one root of the extra-welfarist position in health economics, while welfarist welfare economics has mostly relied on the Pareto principle.

Profiles and aggregation rules

The basic framework, which Christian List's entry in the Stanford Encyclopedia of Philosophy describes as still standard, was introduced by Arrow in 1951. Each individual holds a complete and transitive preference ordering over a set of social alternatives, the combination of everyone's orderings is a profile, and an aggregation rule assigns a social ranking to each admissible profile.

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

where $R_i$ is the preference ordering of individual $i$ over the set of alternatives $X$, $n$ is the number of individuals, $F$ is the aggregation rule and $R$ is the resulting social ranking.

In a health setting the alternatives might be three uses of the same addition to an NHS budget, and the individuals patients, the public or committee members. Arrow called these rules social welfare functions, but as procedures for ranking any profile they differ from the Bergson-Samuelson social welfare function used in health economics, which is one chosen function of individual utilities or health. Sen later generalised Arrow's framework to take individual welfare numbers as inputs, an object List calls a social welfare functional, and it links the two traditions.

Majority rule, Condorcet's paradox and Arrow's conditions

Choices between two options have a clean answer. May proved in 1952 that a rule for a two-way choice satisfies universal domain, anonymity (all voters treated equally), neutrality (both options treated equally) and positive responsiveness if and only if it is majority rule. A weighted majority rule breaks anonymity, and a supermajority rule that keeps a default option unless, say, two thirds vote against it breaks neutrality.

With three or more options majority voting can fail. In Condorcet's paradox one third of a group ranks x above y above z, a second third ranks y above z above x and the last third ranks z above x above y, so two-thirds majorities prefer x to y, y to z and z to x, and no option beats every other. As List states Arrow's theorem, no aggregation rule over more than two alternatives satisfies universal domain, ordering (a complete and transitive social ranking), the weak Pareto principle, independence of irrelevant alternatives and non-dictatorship. Pairwise majority voting meets every condition except ordering.

The Gibbard-Satterthwaite theorem, published in 1973 and 1975, adds an incentive problem. No rule that accepts every profile, always picks a single winner, can produce at least three outcomes and is not a dictatorship is strategy-proof, so on a priority-setting panel some member can sometimes gain by misreporting a ranking.

Escape routes and the price of each

Most responses to the theorem amount to choosing which condition to give up. The table summarises the routes List describes and their form in health economics.

Black proved in 1948 that majority cycles cannot occur if every profile is single-peaked, so that options line up on one dimension and each person's preference falls away on either side of a favourite. List reports evidence from deliberative opinion polls consistent with the idea that group deliberation can induce single-peaked preferences, while noting that more empirical work is needed. If that holds, a deliberative process such as a citizen jury before a vote makes a majority cycle less likely. List also notes that Arrow's theorem carries over to several criteria combined by one decision maker into an all-things-considered ordering, which is the structure of multi-criteria decision analysis in HTA.

Interpersonal comparisons and the QALY sum

Sen's response, which List describes as the route that has received most attention, is to enrich the inputs with numerical welfare for each person and alternative, plus an assumption about which features of those numbers can be compared. Under Arrow's ordinalist assumption the impossibility returns, but once welfare levels, units or zero points can be compared across people, rules satisfying all of Arrow's conditions exist. Classical utilitarianism needs comparable units and the maximin rule needs comparable levels.

$$x \, R \, y \iff \sum_{i=1}^{n} W_i(x) \ge \sum_{i=1}^{n} W_i(y)$$

where $W_i(x)$ and $W_i(y)$ are the welfare of individual $i$ under alternatives $x$ and $y$, $n$ is the number of individuals and $x \, R \, y$ means that $x$ is ranked at least as high as $y$. This is the utilitarian rule.

Cost-utility analysis applies this rule with health in place of welfare. The NICE manual (PMG36, section 4.2.15) justifies cost-effectiveness analysis by NICE's focus on maximising health gains from a fixed NHS and personal social services budget, and its reference case (Table 4.1) gives an additional QALY the same weight regardless of the other characteristics of the people who receive it, except in specific circumstances. Equal weighting is anonymity in May's sense, and adding QALYs presumes the unit comparability that utilitarianism requires.

Departures from equal weights turn the sum into a weighted rule. PMG36 (section 6.2.10 and Table 6.1) lets committees apply QALY weights of 1.2 or 1.7 for severity, based on proportional or absolute QALY shortfall, and asks that modifiers be supported by reason, coherence and available evidence. PMG36 (section 4.3.4) also settles whose rankings count, since health-related quality of life should be valued from public preferences of a representative sample of the UK population using a choice-based method. In money-based appraisal the HM Treasury Green Book (2026) allows distributional weights that give a higher social value to benefits for lower-income households, using an elasticity of marginal utility of consumption of 1.25. Equity-weighted analysis is the health version of the same move.

Welfarism, extra-welfarism and the informational basis

The informational route meets the main methodological divide in health economics. Brouwer, Culyer, van Exel and Rutten write that since Arrow's work on the impossibility of a non-dictatorial social welfare function, economists' thinking on increases in social welfare has been dominated by the Pareto improvement, and that it was primarily Sen who began to release economists from that limited scope. They treat Sen's work as one starting point for extra-welfarism, which permits sources of valuation other than the affected individuals and interpersonal comparisons of wellbeing in several dimensions. They note that applied cost-benefit analysis already compares people, by weighting each unit of money equally whoever receives it. Both schools aggregate, and the dispute is over whether the rule may use individual utility only or also health and capabilities valued by a decision maker.

Committees, expert panels and judgement aggregation

HTA committees often judge clinical benefit and cost-effectiveness separately before reaching a recommendation. Judgement aggregation, a newer branch of the field, studies groups voting on such linked propositions. The doctrinal paradox, which List credits to Kornhauser and Sager (1986), shows majorities accepting each of two premises that jointly imply a conclusion while a majority rejects the conclusion. In an illustrative appraisal version, one member accepts both premises, a second accepts only clinical benefit and a third accepts only cost-effectiveness, so majorities accept each premise yet two of the three oppose a positive recommendation.

Probabilistic judgements behave differently. List reports results by Aczél and Wagner and by McConway showing that, under universal domain, collective coherence, zero preservation and independence, the admissible pooling rules are exactly the weighted linear averages of individual probabilities, known in health economics as the linear pool. The reference case for structured expert elicitation proposed by Bojke and colleagues recommends fitting a distribution to each expert's judgements and then summarising them by linear pooling with equal weights, an anonymous rule. List adds that linear pooling, except in its dictatorial form, does not generally preserve conditional independence judgements that every expert holds.

Worked example with one budget and four aggregation rules

All numbers in this example are illustrative. An integrated care board can fund one of three programmes of equal cost, a cancer drug (A), a vaccination catch-up campaign (B) or a community mental health service (C). Each of three population groups ranks the programmes by its own expected QALY gain per person, and a local panel gives each group votes in proportion to its size.

1. Plurality. First preferences give A 5,000 votes, B 4,000 and C 3,000, so plurality rule picks A.

2. Pairwise majority. B beats A by 7,000 votes to 5,000, B beats C by 9,000 to 3,000 and C beats A by 7,000 to 5,000. B is the Condorcet winner and the social ranking is B, C, A.

3. Borda count. With 2 points for a first place, 1 for a second and 0 for a third, A scores 5,000 × 2 = 10,000, B scores 5,000 × 1 + 4,000 × 2 + 3,000 × 1 = 16,000 and C scores 4,000 × 1 + 3,000 × 2 = 10,000. B wins and A ties with C.

4. QALY sum. Programme A yields 5,000 × 0.04 = 200 QALYs. Programme B yields 5,000 × 0.01 plus 4,000 × 0.035 plus 3,000 × 0.01, which is 50 + 140 + 30 = 220 QALYs. Programme C yields 4,000 × 0.005 plus 3,000 × 0.08, which is 20 + 240 = 260 QALYs, so a cost-utility comparison of these equal-cost options picks C.

5. Interpretation. Four defensible rules pick three different programmes from the same data. Only the QALY sum uses group 3's large gain per person from C, an interpersonal comparison of health that Arrow's framework excludes and Sen's admits.

6. A cycle. If group 3 ranked A second, majorities would prefer A to B, B to C and C to A, and the order of votes would decide the outcome.

What the theory can and cannot settle in HTA

Social choice theory leaves the choice of rule to the committee and shows which conditions can hold together and what each rule assumes about the information available, and List observes that Arrow's axiomatic method, which identifies the conditions that single out a rule, is now more influential than the impossibility theorem itself. For economic evaluation the consequence is that the aggregation rule and its informational basis are value judgements to be stated, not technical defaults. Reporting results under more than one rule makes the judgement visible, as the Green Book does by requiring weighted estimates alongside unweighted ones.

Common misreadings of social choice results

Most errors come from citing an impossibility result without the conditions behind it. Two recur in health policy debate.

1. Reading Arrow as proof that collective choice is impossible. Riker read the theorem as a proof of the impossibility of populist democracy, but List reports that most social choice theorists have moved beyond negative readings to the possibilities opened by relaxing assumptions, a reading Sen promoted in his Nobel lecture.
2. Treating QALY maximisation as value-free because it avoids Arrow's theorem. It avoids the theorem by assuming that health units are comparable across people and by weighting them equally, and NICE departs from equal weights for severity.

Sources

• Bojke L, Soares MO, Claxton K, Colson A, Fox A, Jackson C, Jankovic D, Morton A, Sharples L, Taylor A. Reference case methods for expert elicitation in health care decision making. Medical Decision Making. 2022;42(2):182-193. https://pmc.ncbi.nlm.nih.gov/articles/PMC8777312/
• Brouwer WBF, Culyer AJ, van Exel NJA, Rutten FFH. Welfarism vs. extra-welfarism. Journal of Health Economics. 2008;27(2):325-338. Author manuscript: https://eprints.whiterose.ac.uk/id/eprint/3789/
• HM Treasury. The Green Book (2026). Updated 28 October 2026. Distributional weighting. https://www.gov.uk/government/publications/the-green-book-appraisal-and-evaluation-in-central-government/the-green-book-2026
• List C. Social choice theory. In: Zalta EN, Nodelman U, editors. The Stanford Encyclopedia of Philosophy. First published 18 December 2013; 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. Sections 4.2.15, 4.3.4 and 6.2.10, Tables 4.1 and 6.1. https://www.nice.org.uk/process/pmg36