What is Arrow’s Impossibility Theorem (short answer)
Arrow’s Impossibility Theorem shows that when a group tries to convert individual ranked preferences (each person orders the options from best to worst) into one collective ranking, no rule can satisfy a small set of plausible fairness conditions simultaneously once there are three or more options. Put differently: any voting rule that produces a complete group preference ordering must violate at least one intuitive fairness requirement.
Why this matters (one-sentence)
The theorem reveals a fundamental tension in designing “perfect” ranked-vote systems and explains why all voting rules have trade‑offs.
Key concepts and definitions
– Social choice theory: the study of methods for aggregating individual preferences into collective decisions.
– Social ordering (social welfare ordering): a complete ranking of the available alternatives that represents the group’s preference.
– Nondictatorship: no single individual’s preferences always determine the group ranking regardless of others.
– Pareto efficiency (Pareto unanimity): if every voter strictly prefers option X to option Y, then the group ranking should place X ahead of Y.
– Independence of irrelevant alternatives (IIA): the group’s relative ranking of two options X and Y should depend only on individual rankings of X versus Y, not on how people rank other irrelevant options.
– Unrestricted domain (universal domain): the aggregation rule must accept any possible combination of individual preference orderings as input.
The five conditions Arrow used (concise checklist)
1. Unrestricted domain: accept any set of individual rankings.
2. Social ordering: produce a complete, transitive group ranking.
3. Pareto efficiency: unanimous individual preference → group reflects it.
4. Independence of irrelevant alternatives (IIA): pairwise relative order unaffected by unrelated options.
5. Nondictatorship: no single voter always determines outcomes.
Arrow’s conclusion (clear statement)
If there are at least three options, no aggregation rule can satisfy all five of the above conditions at once. Any rule that satisfies four must violate the fifth.
Worked numeric example (voter-cycle paradox)
Setup:
– Three projects A, B, C.
– 99 voters split into three equal blocs of 33 each, with these preference orders:
– Bloc 1 (33 voters): A > B > C
– Bloc 2 (33 voters): B > C > A
– Bloc 3 (33 voters): C > A > B
Pairwise majority comparisons:
– A vs B: voters in Bloc 1 and Bloc 3 prefer A over B → 33 + 33 = 66 prefer A; 33 prefer B. So A beats B (66–33).
– B vs C: Bloc 1 and Bloc 2 prefer B over C → 33 + 33 = 66 prefer B; 33 prefer C. So B beats C (66–33).
– C vs A: Bloc 2 and Bloc 3 prefer C over A → 33 + 33 = 66 prefer C; 33 prefer A. So C beats A (66–33).
Result:
Majorities produce a cycle: A > B, B > C, and C > A. That cycle violates transitivity, so you cannot form a single coherent social ordering that respects all pairwise majority preferences. This illustrates why any aggregation rule that tries to be fair in the Arrow sense runs into contradictions.
Implications and practical takeaways
– No ranked voting method can be simultaneously “perfect” under Arrow
—that is, satisfying all of Arrow’s fairness criteria—unless you accept trivial or dictatorial outcomes. In practice this means designers of voting rules must trade off among desirable properties. Below are the most important consequences, practical choices, and a short checklist to evaluate or pick a voting method.
Key trade-offs (what you can relax)
– Independence of irrelevant alternatives (IIA). IIA requires that society’s preference between X and Y depend only on individual preferences between X and Y. Many practical methods (Borda count, plurality, approval) violate IIA; relaxing IIA often makes methods more usable.
– Unrestricted domain (universality). Requiring the system to accept every possible set of individual rankings forces impossibility. Many systems work well if preferences are structured (e.g., mostly single-peaked).
– Non-dictatorship. Allowing a dictator (one voter whose preferences always decide) trivially satisfies the other conditions but is politically unacceptable.
– Pareto efficiency. This is rarely relaxed; it says if everyone prefers X to Y, the social ranking must too.
– Transitivity/completeness. Some aggregation rules output intransitive or incomplete social “preferences” (e.g., pairwise-majority cycles). Allowing non-transitive outcomes avoids Arrow’s impossibility but makes social choice harder to interpret.
Common voting alternatives and the main trade-offs
– Plurality (first-past-the-post): simple, violates IIA and Condorcet consistency (may not elect the candidate who would win all head-to-head matchups). Vulnerable to vote splitting.
– Borda count: uses rank-based cardinal scores (for n options, assign n−1 points to top, n−2 to second, etc.). Can reduce vote-splitting but is susceptible to strategic ranking and violates IIA.
– Condorcet methods (e.g., Schulze, Ranked Pairs): elect the candidate who wins all pairwise contests if one exists (Condorcet winner). Good for pairwise consistency, but may need tie-breaking when cycles exist and can be complex.
– Instant-runoff voting (IRV; ranked-choice): eliminates the lowest first-preference candidate and redistributes votes. Simple to implement in single-winner races; fails Condorcet consistency and can be non-monotonic.
– Approval voting: voters mark all candidates they approve of. Simple and less sensitive to spoilers; loses information about intensity of preference.
No method is universally best; pick one depending on whether you value simplicity, Condorcet consistency, resistance to strategic voting, or other properties.
Worked numeric example (continuing the 99-voter cycle)
Preferences in three equal blocs (33 voters each):
– Bloc 1: A > B > C
– Bloc 2: B > C > A
– Bloc 3: C > A > B
Pairwise majorities produce a cycle (A beats B, B beats C, C beats A). Compare methods:
– Borda count (3 candidates): points = 2 for top, 1 for second, 0 for last.
– A: 33*2 + 33*0 + 33*1 = 66 + 0 + 33 = 99
– B: 33*1 + 33*2 + 33*0 = 33 + 66 + 0 = 99
– C: 33*0 + 33*1 + 33*2 = 0 + 33 + 66 = 99
– Result: three-way tie — Borda does not resolve the cycle here.
– Instant-runoff (IRV):
– First preferences: A 33, B 33, C 33. No majority; elimination tie-breaker required. Outcome depends on tie-breaking rule — shows IRV can be indeterminate under symmetric cycles.
– Cond
– Condorcet method (pairwise): compare every pair of candidates head-to-head; a Condorcet winner is a candidate who would beat every other candidate in those one-on-one contests. In the 3-group, 99-voter example with group preferences:
– Group 1 (33): A > B > C
– Group 2 (33): B > C > A
– Group 3 (33): C > A > B
Pairwise tallies:
– A vs B: groups 1 and 3 prefer A → A 66 vs B 33
– B vs C: groups 1 and 2 prefer B → B 66 vs C 33
– C vs A: groups 2 and 3 prefer C → C 66 vs A 33
No Condorcet winner exists because the preferences cycle (A beats B, B beats C, C beats A). This is the Condorcet paradox (intransitive social preference).
Condorcet-completion methods
– When no Condorcet winner exists, many systems use a completion method (a rule to pick a winner from the cyclic results). Two widely used completion algorithms:
1. Ranked Pairs (Tideman): order all pairwise victories by margin (largest to smallest). “Lock in” the strongest wins one-by-one provided locking that win does not create a cycle in the directed graph. The graph that remains yields the ranking and winner.
2. Schulze method: compute strongest paths between each pair (the strength of the best route from X to Y, where a route’s strength is the smallest margin on that route). The candidate whose strongest paths beat all opponents is the winner.
– Both methods are Condorcet-consistent: if a true Condorcet winner exists, they pick it. They differ only in how they break cycles and in some edge cases.
Practical checklist: how to evaluate and select a voting method
1. Decide your priorities (examples):
– Require a Condorcet winner if one exists → choose a Condorcet-consistent method.
– Prioritize simplicity for voters and counters → consider plurality or IRV (instant-runoff).
– Minimize strategic voting incentives → consider score voting or approval, but note different trade-offs.
2. Check which axioms you care about (definitions):
– Majority criterion: if a majority ranks X first, X must win.
– Condorcet criterion: if someone beats all others pairwise, they must win.
– Monotonicity: ranking X higher should not cause X to lose.
– Independence of Irrelevant Alternatives (IIA): relative ranking of X and Y should not be affected by a third candidate Z (this is the IIA Arrow shows cannot be satisfied with other reasonable axioms).
3. Simulate with realistic preference profiles:
– Build a small pairwise matrix from sample ballots.
– Look for Condorcet winner: if none, run your chosen completion method and see outcomes across plausible scenarios.
4. Consider operational constraints:
– Ballot complexity, voter education, tabulation resources, auditability.
Step-by-step: detect a Condorcet paradox in practice
1. Collect ranked ballots.
2. For each ordered pair (X, Y), count voters who prefer X over Y. Fill the pairwise matrix.
3. If some candidate has wins against every other candidate → Condorcet winner; stop.
4. If not, apply your agreed completion method (e.g., Ranked Pairs or Schulze).
5. Document tie-breaking rules and audit trails before the election.
Worked mini-example of Ranked Pairs on a small cycle
– Using the A/B/C 33/33/33 groups above, pairwise margins are all +33 in the direction of the wins.
– Ranked Pairs orders those three wins arbitrarily (all equal); the first locked-in win
the first locked-in win becomes the initial locked edge; the algorithm then considers the next strongest pair and locks it if doing so does not create a directed cycle with edges already locked. In the A/B/C 33/33/33 mini-example the three pairwise wins all have equal strength, so the order in which those equal-strength pairs are considered (a tie-break) determines the final ranking.
Worked numeric demonstration — Ranked Pairs (Tideman)
1. Setup (from the 99-voter example):
– Voter blocks (each 33): A>B>C, B>C>A, C>A>B.
– Pairwise tallies:
– A vs B: A 66, B 33 (A beats B by 33).
– B vs C: B 66, C 33 (B beats C by 33).
– C vs A: C 66, A 33 (C beats A by 33).
– All three wins have identical margin/strength.
2. Ranked Pairs algorithm (tie-breaker choice matters):
– Case 1 — order pairs A>B, then B>C, then C>A:
– Lock A→B (no cycle).
– Lock B→C (
– Lock B→C (no cycle).
– Consider C→A: locking C→A would create a directed cycle A→B→C→A, so skip C→A.
– Final locked graph: A→B→C (A beats B, B beats C). Final ranking: A > B > C.
– Case 2 — order pairs B>C, then C>A, then A>B:
– Lock B→C (no cycle).
– Lock C→A (no cycle).
– Consider A→B: locking A→B would create cycle B→C→A→B, so skip A→B.
– Final locked graph: B→C→A. Final ranking: B > C > A.
– Case 3 — order pairs C>A, then A>B, then B>C:
– Lock C→A (no cycle).
– Lock A→B (no cycle).
– Consider B→C: locking B→C would create cycle C→A→B→C, so skip B→C.
– Final locked graph: C→A→B. Final ranking: C > A > B.
Key point: with these symmetric tallies (each pair won by the same margin), the Ranked Pairs procedure produces every possible cyclic order depending only on the tie-break order used to process equal-strength pairs. In other words, the algorithm’s outcome is not uniquely determined by voter preferences alone when pair strengths tie; the tie-breaking rule (explicit or implicit) decides the result.
Checklist to analyze ties in Ranked Pairs (practical steps)
1. Compute all pairwise tallies and margins (winner votes minus loser votes).
2. Sort pairs by strength (margin). Identify groups of pairs with identical strength.
3. For each tied group, decide a tie-break policy before locking edges:
– random lottery among tied pairs; or
– fixed external priority (candidate ID, ballot order, etc.); or
– deterministic secondary metric (e.g., total first-place votes or an auxiliary voting rule).
4. Run the lock-in process in that tie-resolved order, skipping any pair that would create a cycle.
5. Report final locked graph and derive ranking by topological order.
Implications and practical notes
– Condorcet-respecting methods like Ranked Pairs still require tie-breaking when pair strengths equal; the tie-break can be arbitrary and materially affect the outcome.
– When designing or evaluating an election method, specify the tie-break rule in advance to avoid contention and to make outcomes reproducible.
– If you want to minimize dependence on arbitrary tie-breaks, consider (a) using a secondary deterministic metric as the tie-break, or (b) using another Condorcet completion method (e.g., Schulze) and compare sensitivity across methods.
Worked numeric summary (recap)
– Pairwise tallies in the 99-voter example: A vs B = 66–33, B vs C = 66–33, C vs A = 66–33.
– All margins = 33 → all three pair strengths tie.
– Different processing orders produce A>B>C, B>C>A, or C>A>B respectively.
References (explainers and method sources)
– Tideman, “Ranked Pairs” (original method discussion) — see William F. Tideman’s work and summaries.
– Wikipedia — Ranked Pairs: https://en.wikipedia.org/wiki/Ranked_pairs
– Condorcet criterion and methods — Wikipedia: https://en.wikipedia.org/wiki/Condorcet_method
– Arrow’s impossibility theorem (background on social choice limits) — Investopedia: https://www.investopedia.com/terms/a/arrows-impossibility-theorem.asp
– FairVote (electoral reform perspective): https://www.fairvote.org
Educational disclaimer
This explanation is educational and conceptual. It is not personalized advice about elections or institutional design.