Paradoxes

Social choice & voting-rule paradoxes #

  • Condorcet cycle (Condorcet’s paradox) — pairwise majorities can be cyclic (A>B, B>C, C>A). References: Stanford Encyclopedia overview; original tradition. ( Stanford Encyclopedia of Philosophy)
  • Condorcet winner/loser paradoxes — some rules can ignore a Condorcet winner or elect a Condorcet loser. Reference: SEP survey of method criteria and paradoxes. ( Stanford Encyclopedia of Philosophy)
  • Arrow’s impossibility — no rank-ordering rule can satisfy unrestricted domain, Pareto, IIA, and non-dictatorship simultaneously. References: SEP overview; Arrow’s book. ( Stanford Encyclopedia of Philosophy, Wikipedia)
  • Gibbard–Satterthwaite — every “reasonable” single-winner rule is manipulable when ≥3 options. References: Gibbard (1973); Satterthwaite (1975). ( ScienceDirect, Stanford Encyclopedia of Philosophy)
  • No-show (participation) paradox — a voter can make their preferred outcome lose by turning out. Reference: Moulin (1988). ( ScienceDirect)
  • Monotonicity paradox (IRV/STV) — raising a candidate on some ballots can make them lose (and lowering can make them win). References: Woodall (1997); short explainer. ( ScienceDirect, ia802809.us.archive.org)
  • IIA/Spoiler effect — adding or removing a non-winning option can flip the winner (failure of Independence of Irrelevant Alternatives). Reference: Arrow overview. ( Stanford Encyclopedia of Philosophy)
  • Ostrogorski paradox — majority agrees with Party A on most issues yet, via party voting, elects Party B (or vice-versa). References: Nurmi’s treatment; aggregation-paradox taxonomy. ( SpringerLink, arXiv)
  • Discursive dilemma / doctrinal paradox — majority voting on logically related propositions can yield collectively inconsistent judgments. References: List & Pettit (2002); Mongin (2012) review. ( Stanford Encyclopedia of Philosophy, SpringerLink)

Apportionment & seat-allocation paradoxes #

  • Alabama paradox — increasing the total number of seats can make a state/party lose a seat under largest remainders. References: Balinski & Young (book); overview. ( Michigan Law Scholarship Repository, Wikipedia)
  • Population paradox — a faster-growing unit can lose a seat to a slower-growing one under some methods. References: Balinski & Young; overview. ( Michigan Law Scholarship Repository, Wikipedia)
  • New-states paradox — adding a new state and seats can reshuffle others’ seats counterintuitively. References: Balinski & Young; overview. ( Michigan Law Scholarship Repository, Wikipedia)
  • Balinski–Young impossibility — no apportionment method can satisfy both quota and population monotonicity once there are enough units. References: classic exposition; lecture notes statement. ( IIASA PURE, dominik-peters.de)

Turnout & collective-action paradoxes #

  • Paradox of voting (Downs/Riker–Ordeshook) — for an individual, expected instrumental benefits rarely exceed costs, yet people vote. References: Downs (1957/1957 article excerpt); Riker & Ordeshook (1968). ( rochelleterman.com, Cambridge University Press & Assessment)
  • Tullock’s paradox of revolution — mass participation in risky revolt should be rare under standard rational-choice assumptions, yet revolutions happen. References: Tullock (1971); journal record. ( Cooperative Individualism, SpringerLink)

Two-tier systems & voting-power paradoxes #

  • Referendum (two-tier) paradox — a national majority can lose when outcomes are aggregated by districts/units. References: Nurmi (book); Lahrach & Merlin (2012). ( SpringerLink)
  • Bloc/weight paradoxes in power indices — merging voters or increasing nominal weight can reduce measured power (and vice-versa). References: Felsenthal & Machover (monograph); “product paradox” note. ( Elgar Online, JSTOR)
  • Penrose square-root insight — in two-tier systems, equalizing individual influence suggests weighting representatives ≈√population (counter-intuitive at first glance). References: Kirsch (survey); Zyczkowski et al. (EU applications). ( fernuni-hagen.de, London School of Economics)

Legislative choice, agendas & cycling #

  • Majority cycles / chaos theorem — with ≥2 policy dimensions, majority rule can reach (almost) any point by agenda setting; no stable Condorcet point generically. References: McKelvey/Schofield overview; Plott’s equilibrium conditions. ( Wikipedia, JSTOR)
  • Agenda-setter/status-quo paradoxes — the same voters can be led to radically different outcomes by sequencing and status-quo control. Reference: Romer & Rosenthal (1978). ( Ed Egan)

Data & inference paradoxes that bite politics #

  • Simpson’s paradox — associations reverse when data are aggregated vs. stratified (classic in voting/inequality analysis). References: SEP overview; Simpson (1951). ( Stanford Encyclopedia of Philosophy, math.bme.hu)
  • Ecological fallacy — correlations across areas need not reflect individual behavior (e.g., inferring individual vote from precinct patterns). Reference: Robinson (1950). ( digamo.free.fr)

Normative-political paradoxes #

  • Sen’s “Paretian liberal” paradox — minimal individual rights can conflict with Pareto efficiency under social choice aggregation. References: Sen (1970); Harvard DASH copy. ( Chicago Journals, Dash)
  • Popper’s paradox of tolerance — unbounded tolerance can enable intolerance to destroy tolerance itself. Reference: The Open Society and Its Enemies. ( cdn.oujdalibrary.com)

Social choice & voting rules #

Paradox Setup / Assumptions Striking Result Classic Example Typical Fix / Mitigation
Arrow’s “Impossibility” Rank aggregation; unrestricted domain; Pareto; IIA; non-dictatorship; ≥3 options No rule satisfies all criteria Textbook 3-candidate profiles Relax IIA; restrict domains (single-peaked); use scoring rules
Condorcet (Voting) Paradox Majority rule; ≥3 options Collective preferences can be cyclic A≻B, B≻C, C≻A cycles Agenda control; Condorcet methods; restrict to single-peaked
Condorcet Winner Paradox Some voting rules (e.g., Borda) Candidate beating all others head-to-head can lose CW loses under Borda Use Condorcet-consistent rules
Condorcet Loser Paradox Some rules Candidate losing to all others can win CL wins under certain profiles Condorcet-loser elimination
Gibbard–Satterthwaite Onto, non-dictatorial rule; ≥3 options Every such rule is manipulable Strategic misreporting Restricted domains; randomized mechanisms (Gibbard 1977)
No-Show (Participation) Paradox Certain rules (e.g., STV) A voter can help their preference by abstaining STV counterexamples Use participation-respecting rules (trade-offs elsewhere)
Monotonicity Paradox Certain rules (e.g., STV) Raising a candidate can make them lose Classic STV profiles Choose monotone rules (but may forfeit other properties)
Spoiler / IIA Failure Plurality & many rules Non-winning “clone” entry/removal flips the winner Vote-splitting Runoffs, IRV, Condorcet, fusion, primaries
Ostrogorski Paradox Party vs issue aggregation Party winner opposed by issue-by-issue majorities Two-party issue bundles Issue-based referenda; PR with multidimensional parties
Discursive Dilemma / Doctrinal Judgment aggregation Proposition-wise vs platform-wise yields inconsistency Court panels, committees Impose logical constraints; use premise- or conclusion-based rules

Apportionment & seat allocation #

Paradox Setup / Assumptions Striking Result Classic Example Typical Fix / Mitigation
Alabama Paradox House size increases A state loses a seat when total seats rise 19th-c. US apportionment Divisor methods (Hill/Huntington, Webster)
Population Paradox Differential growth Faster-growing state loses to slower-growing US historical cases Use population-monotone methods (but see impossibility)
New States Paradox Add state + seats Existing states’ seats reshuffle unexpectedly Admission scenarios Adopt divisor methods; stability targets
Quota Violations Some methods Allocations fall outside lower/upper quota Hamilton’s method Prefer divisor methods; accept mild violations
Balinski–Young “Impossibility” Desire quota + population monotonicity No method satisfies both for all profiles General theorem Pick which axiom to relax; be transparent

Turnout & collective action #

Paradox Setup / Assumptions Striking Result Classic Example Typical Fix / Mitigation
Paradox of Voting (Downs) Tiny pivotality; positive cost Rational turnout ≈ 0, yet people vote Mass elections Civic duty/expressive benefits; selective incentives
Tullock’s Paradox of Revolution High social benefit; high risk Under-participation despite big stakes Revolts & protests Coordination, focal points, selective incentives
Olson’s Collective Action Large groups; public goods Bigger groups do worse absent incentives Lobbying vs. mass goods Entrepreneurs; selective/club goods; institutions

Two-tier systems & power indices #

Paradox Setup / Assumptions Striking Result Classic Example Typical Fix / Mitigation
Referendum (Two-Tier) Paradox District vs national majority National majority loses under district rule (or vice versa) US Electoral College-like cases Weighting schemes; runoffs; proportional aggregation
Voting-Power / Bloc Paradoxes Weighted voting; coalitions Adding members/weight can reduce power; blocs can backfire EU Council, shareholder votes Square-root weighting; power-aware design

Legislative choice & agenda #

Paradox Setup / Assumptions Striking Result Classic Example Typical Fix / Mitigation
Agenda / Path Dependence Majority rule with cycles Order of votes changes the winner Pairwise sequencing Restrict agendas; germaneness; closed rules
McKelvey “Chaos” Phenomenon ≥2 policy dimensions; open agenda Majority can reach “almost anywhere” Continuous policy spaces Structure-induced equilibrium (committees, jurisdictions)

Data & inference #

Paradox Setup / Assumptions Striking Result Classic Example Typical Fix / Mitigation
Simpson’s Paradox Heterogeneous subgroups Aggregates reverse subgroup trends Turnout, admissions, policing Stratify; causal models; beware ecological fallacy

Normative–political #

Paradox Setup / Assumptions Striking Result Classic Example Typical Fix / Mitigation
Sen’s Liberal Paradox Minimal rights + Pareto + unrestricted prefs Rights can conflict with Pareto efficiency “Prude” vs “Lewd” toy model Restrict domains; redefine rights/efficiency
Paradox of Tolerance (Popper) Unlimited tolerance Tolerance empowers intolerant to end tolerance Extremist speech cases Conditional tolerance; militant democracy tools