Week 5: Adding up preferences: social choice and its limits

In plain words

Week 5 of the My Vote Forward seminar asks whether a group’s priorities can be read off its members’ rankings. Students study Condorcet’s paradox, May’s theorem on majority rule, Arrow’s impossibility theorem, Sen’s liberal paradox, and Gibbard’s result that voting rules can be gamed, then read Gerry Mackie’s argument that these results do not make democracy meaningless.

If three people rank three subjects differently, there may be no single “top priority” that a majority prefers to every alternative. The answer can depend on the counting method. This week explains why, and asks what that means for anyone who publishes “the public’s priorities.”

Defined terms

• Aggregation rule: a method that turns individual rankings or votes into a group result. See aggregate.
• Condorcet winner: an option that beats every other option in head-to-head majority comparisons. It does not always exist.
• Condorcet cycle: majority preferences that form a loop, A over B, B over C, and C over A.
• Independence of irrelevant alternatives: the condition that the group’s ranking of two options depends only on how individuals rank those two options.
• Strategic voting: misrepresenting one’s preferences to obtain a better outcome.
• Single-peaked preferences: preferences that can be arranged on one dimension so that each person has a favorite point and likes options less the farther they are from it.
• See also ranking, ranked-choice voting, and approval voting.

Learning objectives

By the end of week 5, a participant should be able to:

1. Construct a Condorcet cycle and explain why individual rationality does not guarantee group rationality.
2. State May’s theorem and the conditions under which majority rule is uniquely justified.
3. State Arrow’s theorem’s four conditions and its conclusion.
4. Explain Sen’s liberal paradox and Gibbard’s manipulation result in plain language.
5. Summarize Riker’s skeptical reading and Mackie’s reply, and evaluate them.
6. Explain why any published priority ranking should name its method and test its sensitivity to alternatives.

Core readings

1. Christian List, “Social Choice Theory,” Stanford Encyclopedia of Philosophy (revised 2022), sections 1 to 3.⁠¹
2. Kenneth J. Arrow, Social Choice and Individual Values (first published 1951; Yale University Press edition with a foreword by Eric Maskin), chapters 1 to 5.⁠²
3. Kenneth O. May, “A Set of Independent Necessary and Sufficient Conditions for Simple Majority Decision,” Econometrica 20(4), 1952.⁠³
4. Amartya Sen, “The Impossibility of a Paretian Liberal,” Journal of Political Economy 78(1), 1970, pp. 152 to 157.⁠⁴
5. Allan Gibbard, “Manipulation of Voting Schemes: A General Result,” Econometrica 41(4), 1973.⁠⁵
6. Gerry Mackie, Democracy Defended (Cambridge University Press, 2003), introduction and chapter 1.⁠⁶

Lecture notes

Why aggregation is a political question

Aggregation, the act of combining individual preferences into a collective result, sounds like arithmetic. Week 5 shows that it is not. Every time someone says “the public’s top priority is housing” or “most residents want the park,” they have applied a rule to individual answers. Social choice theory studies what such rules can and cannot do, and its central results are negative: no rule satisfies every condition that seems reasonable. The site’s chapter on collective priorities states the results in the context of the My Vote Forward framework. This lecture places them in a longer argument about whether democratic outcomes mean anything at all.

Condorcet’s paradox

The Stanford Encyclopedia of Philosophy entry on social choice theory, by Christian List, describes Condorcet’s second insight, “often called Condorcet’s paradox”: “majority preferences can be ‘irrational’ (specifically, intransitive) even when individual preferences are ‘rational’ (specifically, transitive).” Its example: one third of a group prefers x to y to z, a second third prefers y to z to x, and a final third prefers z to x to y. Then there are two-thirds majorities for x over y, for y over z, and for z over x.⁠¹

Hypothetical example: a neighborhood association surveys residents on three priorities, street repair, a library extension, and park maintenance. One third of residents rank streets, library, park. One third rank library, park, streets. One third rank park, streets, library. A majority prefers streets to the library, a majority prefers the library to the park, and a majority prefers the park to streets. There is no option a majority prefers to all others. Any announcement that “residents’ top priority is X” depends on how the comparisons were structured, and possibly on the order in which they were taken.

The paradox does not say cycles are common. It says they are possible, and that their possibility undermines the assumption that a group “has” a ranking the way an individual does.

May’s theorem: when majority rule is the answer

The first positive result is narrower and important. List’s entry explains that May (1952) introduced four procedural requirements for a rule deciding between two alternatives: universal domain (the rule must accept any combination of votes), anonymity (it treats all voters equally), neutrality (it treats both alternatives equally), and positive responsiveness (a shift in votes toward an alternative helps that alternative). May proved that majority rule is the only aggregation rule satisfying all four.⁠^1,3

May’s theorem is the strongest formal case for majority rule, and it is limited to two options. For a yes or no ballot measure, it says majority rule uniquely satisfies conditions most people accept. For priorities among ten subjects, it says nothing, because there are more than two alternatives.

Arrow’s theorem

Kenneth Arrow’s Social Choice and Individual Values, first published in 1951, introduced “Arrow’s Impossibility Theorem” and, in the publisher’s words, “founded the field of social choice theory in economics and political science”.⁠² As stated in List’s entry and summarized in the site’s chapter on collective priorities, the theorem says no rule for combining individual rankings over three or more options into a group ranking can satisfy all of the following at once: universal domain; weak Pareto (if everyone prefers x to y, so does the group); independence of irrelevant alternatives; and non-dictatorship, while also producing a complete and consistent group ranking.⁠¹

Each condition has a plain meaning worth stating. Universal domain: the rule cannot refuse to process some combination of opinions. Weak Pareto: unanimous preferences are respected. Independence of irrelevant alternatives: adding or removing a third option should not flip the group’s ranking of the first two. Non-dictatorship: no single person’s ranking always becomes the group’s. Arrow showed these cannot all hold together when there are three or more options.

The theorem’s practical message is that every method gives something up. List’s entry notes that the Borda count, which scores options by their positions in each ranking, “avoids Condorcet’s paradox but violates one of Arrow’s conditions, the independence of irrelevant alternatives,” so adding an unrelated option can reorder the others. Plurality rule also avoids the paradox, but an option that a majority ranks below every other option can still win under plurality: the entry’s example has 34 percent ranking x over y over z, 33 percent y over z over x, and 33 percent z over y over x, so x wins on first choices although two thirds rank it last.⁠¹ Pairwise majority comparison respects independence but can cycle. Ranked-choice voting, studied in the site’s Maine case, and approval voting each make their own tradeoffs. No method is neutral; each is a choice.

Escaping the theorem by restricting inputs

List’s entry describes the best-known escape. If inputs are required to satisfy certain “cohesion” conditions, “aggregation rules such as pairwise majority voting will produce complete and transitive social preferences.” The best-known condition is single-peakedness (Black 1948): alternatives can be aligned on one dimension so each person has a most preferred position and likes alternatives less as they move away from it.⁠¹ Many political choices along a single left and right scale may be close to single-peaked. Priorities among unrelated subjects, such as transport, health, and defense, have no obvious single dimension, which is one reason a priority ranking is more vulnerable to cycles than a choice among tax rates.

Sen: rights and majorities

Amartya Sen’s 1970 paper gives a second impossibility result with a moral edge. His abstract notes “a common objection to the method of majority decision is that it is illiberal”: if you prefer pink walls to white, society should permit it “even if a majority of the community would like to see your walls white,” and whether you sleep on your back or belly “is a matter in which the society should permit you absolute freedom, even if a majority of the community is nosey enough to feel that you must sleep on your back.” Sen formalizes “this concept of individual liberty in an extremely weak form” and shows it conflicts with the Pareto principle under universal domain.⁠⁴

For this course, Sen’s result supports a design principle: some questions should not be aggregated at all. The site’s chapters on tyranny of the majority and does a majority decide rights? develop the point. A priorities channel that asked people to rank whether others’ rights should be protected would be aggregating something that liberal democracy deliberately places outside aggregation.

Gibbard: every rule can be gamed

Allan Gibbard proved a conjecture “that no system of voting can preclude strategic voting,” the securing by a voter of “an outcome he prefers through misrepresentation of his preferences.” His result: “any non-dictatorial voting scheme with at least three possible outcomes is subject to individual manipulation”.⁠⁵ (The result is usually called the Gibbard and Satterthwaite theorem, after Gibbard and Satterthwaite; List’s entry discusses it under that name.⁠¹)

Strategic behavior matters greatly for a priorities channel. If people know a published ranking will influence budgets, organized groups have an incentive to rank insincerely: to push a rival’s subject to the bottom rather than express true priorities. Gibbard’s theorem says no method can make this impossible. Design can make it harder or less rewarding, and the site’s chapters on agenda capture and single-issue mobilization discuss the risk.

Riker, Mackie, and what the theorems mean

The theorems were turned into a political argument by William Riker. List’s entry records that Riker, who inspired the Rochester school in political science, “interpreted it as a mathematical proof of the impossibility of populist democracy,” citing Riker’s Liberalism Against Populism (1982). The same entry notes a different response: Amartya Sen and others “took it to show that ordinal preferences are insufficient for making satisfactory social choices and that social decisions require a richer informational basis”.⁠¹ Students who want Riker’s position in his own words should read the book. This seminar reads the most sustained reply.

Gerry Mackie’s Democracy Defended takes on the skeptical tradition directly. Its description states that the view that democracy is “unavoidably chaotic, arbitrary, meaningless, and impossible” began with Condorcet and “continued most notably with Arrow and Riker in the twentieth century.” Mackie argues that “problems of cycling, agenda control, strategic voting, and dimensional manipulation are not sufficiently harmful, frequent, or irremediable” to be of normative concern. He examines “every serious empirical illustration of cycling and instability, including Riker’s famous argument that the US Civil War was due to arbitrary dimensional manipulation,” and concludes that “almost every empirical claim is erroneous, and none is normatively troubling”.⁠⁶

The two positions share the mathematics and disagree about its significance. Riker’s side holds that a possibility which cannot be ruled out undermines any claim to have found the popular will. Mackie’s side holds that what matters is how often such problems occur and whether institutions can manage them. The question is partly empirical, which is why Mackie’s book spends most of its length on cases.

What a responsible priority report would do

Three practical rules follow from the week. First, name the method: a published ranking should state exactly how individual answers were combined. Second, test sensitivity: report whether the top items change under alternative reasonable methods, such as Borda, pairwise majority, or the share of participants listing an item in their top three. Third, show the disagreement: when results depend on the method, say so, rather than presenting a single ranking as fact. These rules do not escape Arrow. They make the dependence on method visible, which is what honesty requires.

Discussion questions

1. Construct your own Condorcet cycle using three local issues. Would a cycle like yours be likely in your community, or is it only a possibility?
2. May’s theorem justifies majority rule for two options. Should a priorities channel be redesigned as a series of two-option choices? What would be lost?
3. Which of Arrow’s four conditions would you give up for a priority ranking, and why?
4. Sen’s paradox suggests some questions should not be aggregated. Which subjects or questions should a priorities channel refuse to ask?
5. Gibbard shows every rule can be gamed. How would you detect strategic ranking in published data?
6. Is Mackie’s empirical defense sufficient, or does Riker’s point about possibility still undermine claims to know “the public’s priorities”?

Assignment

Method sensitivity exercise (1,200 words plus tables). Using a small synthetic dataset you construct (label it “synthetic”), of at least 30 respondents ranking five subjects, compute the group ranking under three methods: plurality of first choices, Borda count, and pairwise majority comparison. Report whether a Condorcet winner exists. Then add a sixth subject that is ranked low by most respondents and recompute. Write up which results changed, which of Arrow’s conditions explains each change, and how a published release should present the results. Do not use real survey data.

Evidence

Statements of Condorcet’s paradox, May’s theorem, Arrow’s theorem, single-peakedness, and the Gibbard and Satterthwaite theorem are taken from the Stanford Encyclopedia of Philosophy entry by Christian List and from the cited papers’ abstracts. Sen’s and Mackie’s arguments are quoted from their abstracts or publisher descriptions. Riker’s interpretation is stated as reported in the Stanford Encyclopedia entry. The neighborhood example is hypothetical. All sources were opened on the review date.

Competing interpretations

Three readings of the impossibility results remain in play. The skeptical reading, associated with Riker, treats Arrow’s theorem as a proof that populist democracy, government by a discoverable popular will, is impossible. The defensive reading, Mackie’s, holds that the problems are not sufficiently harmful, frequent, or irremediable to be of normative concern. Sen’s informational reading holds that rankings alone carry too little information, and that better social choices need richer inputs. A third, institutional reading holds that the theorems are design guidance: they tell us which conditions each method gives up, so institutions should choose methods suited to their purpose and disclose them. These readings are not fully exclusive.

Application to this framework

My Vote Forward proposes collecting priorities as rankings. This week shows that any published aggregate of those rankings depends on the method, can in principle cycle, and can be gamed. The framework’s collective priorities chapter should be judged on whether it names its method, reports sensitivity, and declines to aggregate questions about others’ rights.

Unresolved questions

• How often do real priority rankings produce cycles or method-dependent winners? Little public data exists for priority lists specifically.
• Could a ratings-based approach, rather than rankings, reduce some problems while introducing others? List’s entry discusses ratings; the empirical tradeoffs for public priorities are untested here.
• What level of strategic ranking would make a priorities channel misleading, and how would anyone know?