Paradox of Voting

Paradox of Voting in Health Priority Setting

The paradox of voting, also called Condorcet's paradox, is a problem in collective choice where a group whose members each rank the options consistently produces, through pairwise majority votes, a group ranking that goes round in a circle. A funding panel could prefer service A to B, B to C and C to A, so that no option survives every vote. For health economics it matters because priority setting is often done by committees, and because it is the simplest case of the difficulty that Arrow's impossibility theorem describes for any social welfare function built from individual rankings. A separate puzzle with the same name, why rational citizens vote at all, is set aside here.

Condorcet's cycle with three health programmes

The Marquis de Condorcet described the cycle in 1785. Christian List, in the Stanford Encyclopedia of Philosophy, sums up his finding that majority preferences can be intransitive even when every individual's preferences are transitive. A transitive ranking is one where preferring A to B and B to C implies preferring A to C. Michael Morreau's entry on Arrow's theorem gives the standard example, read here as three panel members choosing between three health programmes.

Members 1 and 3 rank A above B, members 1 and 2 rank B above C, and members 2 and 3 rank C above A. Each pairwise contest is won two votes to one, and the results form a loop. Morreau notes that each person's ranking has a best option while the majority preference has none. Eric Pacuit's entry on voting methods states the result as the absence of a Condorcet winner, an option that beats every other one in head-to-head votes.

The pairwise majority rule can be written as a condition on vote counts.

$$x \succ_M y \iff N(x \succ y) > N(y \succ x)$$

where $x$ and $y$ are any two options, $x \succ_M y$ means the group majority prefers $x$ to $y$, and $N(x \succ y)$ is the number of members who rank $x$ above $y$. The paradox is the case where $A \succ_M B$, $B \succ_M C$ and $C \succ_M A$ hold together, so the group relation fails transitivity although each member's ranking passes it.

Why the paradox matters for welfare economics and health

Morreau explains that Arrow turned the question around. Instead of testing one voting rule, he set out conditions that any method of combining individual rankings should meet, including respect for unanimous preferences, no dictator and a social ordering that never cycles, and showed that no method meets them all except in the very simplest cases. Morreau concludes that Condorcet's paradox is a sign of a wider problem with combining many individual preferences into one, and more than an anomaly of a single voting method.

This result sits at the root of the welfarist and extra-welfarist debate in health economics. Brouwer, Culyer, van Exel and Rutten (2008) write that ever 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, which they say has little cutting power beyond voluntary trade in well-regulated markets. Welfare economics without interpersonal comparisons can therefore say little about a choice between health programmes in which some people gain and others lose.

List traces the impossibility, in a welfare-aggregation setting, to a lack of interpersonal comparability. Once comparisons of welfare levels or units between people are allowed, aggregation rules exist that meet analogues of Arrow's conditions. Arrow himself had held that interpersonal comparison of utilities has no meaning. List notes that ordinal preferences are easier to justify as inputs for a voting method than for a planner ranking social states, and reports Sen's argument that such a planner may need comparable welfare information.

Health economics supplies such information through the QALY. Brouwer and colleagues write that measures such as the QALY allow individuals to be compared within a health domain. A ranking of programmes by QALYs gained for a given budget is a ranking of numbers, so it cannot cycle. The ranking then rests on a chosen outcome measure and on whoever chose it, and Brouwer and colleagues say this requires an authority such as decision-makers, an elected or appointed committee or a citizens' jury. They link this to the decision-making approach of Sugden and Williams (1978), in which decision-makers are the source of values and weights. Extra-welfarism thus moves the weighting of outcomes onto committees, and a committee that settles disagreements by majority vote over three or more options meets Condorcet's problem again.

Committees, agendas and the order of votes

Many appraisal decisions come down to two options, to recommend a technology or not, and with two options a majority vote cannot cycle. NICE's standing orders for its technology appraisal and highly specialised technologies committees, dated September 2023, say decisions will normally be reached by consensus, with an anonymous majority vote only at the chair's discretion. An appraisal committee that reaches consensus avoids a mechanical rule that could cycle.

Cycles matter when a panel compares three or more uses of a fixed budget, or three or more possible recommendations such as full funding, funding for a subgroup and no funding. Morreau shows that an intransitive majority makes the choice depend on the path taken, so that dividing the options into smaller sets in different ways gives different final choices. List notes that similar agenda effects arise in sequential pairwise majority voting. Whoever orders the votes can then decide the result without changing anyone's preferences, which bears on the transparency that accountability for reasonableness asks of priority-setting bodies.

Majority voting also ignores how much each person stands to gain or lose. List reports a result of Brighouse and Fleurbaey that in a two-option vote where stakes differ between voters, total utility is maximised by a majority rule weighted in proportion to each voter's stake. A vote among patient groups counts a group facing a small gain the same as one facing a large one, while a cost-per-QALY comparison weighs the size of each gain.

Conditions under which majority rule gives a clear answer

Whether cycles occur in a given committee depends on how its members' preferences are structured. Morreau describes Duncan Black's 1948 result that when the number of voters is odd and every voter's preferences are single peaked along one common ordering of the options, pairwise majority voting always produces an ordering, with the median voter's preferred option at the top. A preference is single peaked when each voter has a most preferred point on the ordering and likes options less the further they lie from it.

Some health budget decisions have this shape. A panel setting spending on a screening programme at a low, middle or high level may well have single-peaked preferences, and the median member's preferred level then beats each of the others. Choices among unlike services, such as a vaccination campaign, a hospice expansion and a diagnostic test, need not line up along one dimension, so the condition gives less protection there. Morreau reports arguments that deliberation can move preferences towards single-peakedness, and empirical work by List and colleagues showing that it sometimes does, one reason for a deliberative process or a citizen jury before any vote.

List reports that the share of possible preference combinations producing a majority cycle grows with the number of voters and options, so that if every combination were equally likely, cycles would be probable in large electorates. Small systematic departures from that scenario can make cycles much less likely.

Worked example with a commissioning panel and three services

All numbers in this example are illustrative. A 13-member commissioning panel has GBP 2 million to spend on one of three services, a weight management programme (A), a smoking cessation service (B) or a community mental health service (C). Members form three groups.

1. Pairwise votes. A against B gives A the votes of groups 1 and 3, so 6 + 3 = 9 against 4. B against C gives B the votes of groups 1 and 2, so 6 + 4 = 10 against 3. C against A gives C the votes of groups 2 and 3, so 4 + 3 = 7 against 6. The majority relation is a cycle, and there is no Condorcet winner.

2. The agenda decides. If A meets B first, A wins and then loses to C, so C is chosen. If B meets C first, B wins and then loses to A, so A is chosen. If A meets C first, C wins and then loses to B, so B is chosen. Each service can be made the winner by the order of votes.

3. A positional count. The Borda count, described by Pacuit, gives each option points for its place in each ranking.

$$BS(x) = \sum_{k=1}^{m} (m-k) \, n_k(x)$$

where $BS(x)$ is the Borda score of option $x$, $m$ is the number of options, $k$ is a position in a ranking with 1 the top place, and $n_k(x)$ is the number of members who put $x$ in position $k$. With three options, first place earns 2 points, second 1 and last 0. A scores 6 × 2 + 4 × 0 + 3 × 1 = 15. B scores 6 × 1 + 4 × 2 + 3 × 0 = 14. C scores 6 × 0 + 4 × 1 + 3 × 2 = 10. The count picks A by one point.

4. A cost per QALY ranking. Suppose the panel instead agrees to fund the service that buys most health, and the evidence gives 80 QALYs for A, 95 for B and 70 for C. The cost per QALY is GBP 2,000,000 / 80 = 25,000 for A, GBP 2,000,000 / 95 = 21,053 for B and GBP 2,000,000 / 70 = 28,571 for C. The order is B, then A, then C, a ranking of single numbers that is transitive and needs no agenda.

5. Interpretation. The rankings stayed the same throughout, yet pairwise voting gave no answer, the agenda could produce any answer and the Borda count gave A by a single point. The cost-per-QALY rule gave a stable answer, B, because it replaced the members' rankings with one cardinal measure comparable across patients. That is an extra-welfarist choice of what counts, and the panel still has to agree to it.

How the paradox relates to neighbouring ideas

Arrow's theorem generalises the paradox to every rule that meets his conditions. The puzzle that shares the name is a different matter. Jason Brennan's encyclopedia entry describes it as the observation, which it attributes to Downs in 1957, that the expected costs of voting, including its opportunity cost, appear to exceed the expected benefits, so it is surprising that anyone votes. That puzzle concerns whether an individual act is rational, while Condorcet's paradox concerns whether a group's combined preference is coherent.

A social welfare function that weighs total health against its distribution works with measures comparable across people, which places it in the setting where List reports that the impossibility can be avoided. Multi-criteria decision analysis gives each option a weighted score, which ranks options transitively but leaves open how the weights are chosen.

Common misreadings and limits

Most mistakes treat the paradox as a curiosity or as proof that collective health decisions are impossible. Neither reading fits the sources.

1. Every vote is suspect. A choice between two options cannot cycle, and single-peaked preferences guarantee a majority ordering when the number of voters is odd.
2. Voters must be irrational for a cycle to occur. Every member in the example ranks consistently, and the inconsistency appears only in the majority relation.
3. Changing the voting rule removes the problem. With three or more options, Arrow's theorem shows that every rule built only on individual rankings breaks at least one of his conditions. List notes that the Borda count avoids Condorcet's paradox but breaks the independence of irrelevant alternatives.
4. A cost-effectiveness ranking solves the paradox. Ranking by cost per QALY avoids cycles by using comparable cardinal information, but it rests on a value judgement about what counts and who decides.

Sources

• Brennan J. The ethics and rationality of voting. In: Stanford Encyclopedia of Philosophy. First published 28 July 2016, substantive revision 21 February 2025. Section 1 introduction. https://plato.stanford.edu/entries/voting/
• 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
• List C. Social choice theory. In: Stanford Encyclopedia of Philosophy. First published 18 December 2013, substantive revision 14 October 2022. Sections 1.1 to 1.3, 2.4, 3.1, 4.3 and 5.4. https://plato.stanford.edu/entries/social-choice/
• Morreau M. Arrow's theorem. In: Stanford Encyclopedia of Philosophy. First published 13 October 2014, substantive revision 7 December 2025. Sections 1, 4.2 and 5.1. https://plato.stanford.edu/entries/arrows-theorem/
• National Institute for Health and Care Excellence. Technology Appraisals and Highly Specialised Technologies Evaluation Committees. Terms of reference and standing orders. September 2023. Items 26 to 28. https://www.nice.org.uk/Media/Default/Get-involved/Meetings-In-Public/Technology-appraisal-committee/TAC-terms-of-reference-standing-orders.pdf
• Pacuit E. Voting methods. In: Stanford Encyclopedia of Philosophy. First published 3 August 2011, substantive revision 24 June 2019. Sections 1.1, 2.1 and 3.1. https://plato.stanford.edu/entries/voting-methods/