Instant-runoff voting

Instant-runoff voting (IRV), also known as ranked-choice voting (RCV) in some contexts, the alternative vote (UK and New Zealand), preferential voting (Australia), or the single transferable vote (Ireland and India), is a single-winner electoral system in which voters rank candidates in descending order of preference rather than selecting a single choice.^[1] If no candidate garners a majority of first-preference votes, the contender with the fewest such votes is eliminated, and those ballots are reassigned to the voters' subsequent preferences; this elimination and redistribution continues iteratively until one candidate secures over 50% of the active votes.^[2] The method aims to produce outcomes reflective of majority support while mitigating spoiler effects associated with plurality voting, though it does not guarantee selection of the Condorcet winner—the candidate who would prevail in pairwise contests against all others.^[3] IRV traces its conceptual roots to the development of single transferable vote systems in the mid-19th century, with practical adoption beginning in Australia, where Queensland implemented it for parliamentary elections in 1892 and the federal House of Representatives followed in 1918.^[4] It has since been employed for national legislative or presidential elections in countries including Ireland, Fiji, and Papua New Guinea, as well as in various subnational contexts such as U.S. states Maine and Alaska for federal races since 2018 and 2022, respectively, and municipal elections in cities like San Francisco since 2004.^[5] Proponents argue it fosters broader candidate viability and less negative campaigning by incentivizing appeal to second-choice voters, yet empirical analyses reveal mixed results: while some local studies indicate modest increases in voter turnout and candidate diversity, others find no consistent reduction in polarization or strategic voting, with long-term use in Australia showing persistent two-party dominance despite multicandidate fields.^[3]^[6] Notable controversies underscore IRV's limitations, including its failure to satisfy the monotonicity criterion—whereby elevating a candidate's ranking can paradoxically cause their defeat—as demonstrated in theoretical models and real elections like Burlington, Vermont's 2009 mayoral contest, where the plurality winner lost after redistributions in a "center-squeeze" scenario, prompting voters to repeal the system by 52% in 2010.^[7]^[8] Peer-reviewed research highlights such pathologies, including vulnerability to non-monotonic outcomes and incomplete spoiler mitigation compared to pairwise methods, leading some jurisdictions to abandon it amid concerns over transparency, voter confusion, and unintended electoral reversals.^[9]^[10] Despite these issues, adoption continues in select areas, with ongoing debates centered on causal evidence from implementations showing variable impacts on representation and participation.^[11]

Procedure

Counting process

Instant-runoff voting tabulates ballots through successive rounds to determine a winner with majority support among continuing active ballots. The process commences with an initial count of first-preference votes for each candidate from all valid ballots.^[12] A quota, typically defined as more than 50% of the total valid votes plus one, is calculated based on the initial formal ballots.^[13] If a candidate receives votes equal to or exceeding this quota in the first round, that candidate is elected.^[12] In cases where no candidate meets the quota initially, the candidate with the fewest first-preference votes is eliminated.^[12] The votes from the eliminated candidate's ballots are then redistributed to the next-highest preference indicated on those ballots, provided such a preference exists and has not been eliminated.^[12] This redistribution occurs at full value, as all votes start with unit value in standard implementations.^[13] The updated tallies are recounted, and the process repeats: checking for a quota met, or eliminating the new lowest candidate if not.^[12] Rounds continue iteratively until one candidate achieves the quota or only one candidate remains with votes, at which point that candidate is declared the winner, reflecting majority preference among the ballots still in contention.^[12] ^[13] In jurisdictions like Australia, the quota remains fixed throughout based on initial valid votes, allowing a winner potentially with support from less than half the original electorate if significant ballot exhaustion occurs before the final round.^[13] When only two candidates remain, the one receiving more votes in that round wins, even if below the original quota.^[12] Ties for elimination, such as multiple candidates sharing the fewest votes, are resolved according to jurisdiction-specific rules, often involving simultaneous elimination of tied candidates or tie-breaking via auxiliary counts of higher preferences or random selection to honor voter intent without arbitrary exclusion.^[14] Final ties for the quota may invoke similar mechanisms or recount provisions to ensure determinacy.^[14]

Ballot exhaustion and invalidation

In instant-runoff voting, ballot exhaustion occurs when a ballot ceases to contribute to the tabulation because all candidates ranked on it have been eliminated, typically due to incomplete voter rankings. Exhausted ballots are then excluded from further rounds, diminishing the pool of active votes and thereby contracting the effective electorate size as the process advances toward selecting a majority winner. This mechanism stems from the requirement for transferable preferences, where only ballots with viable continuing rankings influence eliminations and redistributions.^[15] Empirical data from U.S. jurisdictions implementing IRV reveal exhaustion rates varying by election and voter behavior, often ranging from 5% to over 20%. For instance, an analysis of four ranked-choice elections, including those in San Francisco and Minneapolis, documented exhaustion affecting 9.6% to 27.1% of ballots, with higher incidences linked to truncated rankings rather than random errors. In San Francisco's 2011 mayoral contest, exhaustion reached 27%, substantially reducing active ballots in final rounds. These rates reflect causal factors like voter fatigue or strategic truncation, which limit preference expression and can skew outcomes by excluding portions of the original electorate from decisive counts.^[16] Ballot invalidation in IRV arises from marking errors such as overvotes—ranking the same candidate multiple times or exceeding allowable ranks—or ambiguous notations like skipped ranks without clear intent. Jurisdictional rules diverge: some, like certain California locales, exhaust ballots at the first overvote while preserving later rankings, whereas others invalidate the entire ballot outright, forgoing any usable preferences. A 2021 legal review of IRV implementations across U.S. sites highlighted this patchwork, noting that stricter invalidation policies amplify discarded votes when voters err in early preferences.^[17] In practice, invalidation rates under IRV exceed those in single-mark plurality systems, attributable to the cognitive demands of ranking, with studies reporting elevated rejection frequencies. A 2023 examination of over 150 elections found rejected ballots more prevalent in ranked-choice contests, often 2-3 times higher than comparable non-RCV races, due to over-ranking and incomprehension. Another analysis of RCV implementations indicated error rates up to 10-15% in initial counts, compared to under 2% in plurality, with discrepancies persisting despite voter education efforts. These errors causally diminish vote validity, particularly among less experienced or lower-information voters, independent of demographic factors in some datasets.^[18]^[19]

Terminology

Core terms and definitions

Instant-runoff voting (IRV), also known as ranked-choice voting (RCV) or the alternative vote (AV), is a single-winner electoral system in which voters rank candidates in order of preference on the ballot, and votes are iteratively redistributed from eliminated candidates or surpluses until one candidate achieves a majority of active votes.^[1]^[12]^[20] In this method, a quota is typically set at more than 50% of valid votes for election, with counting proceeding in rounds to simulate sequential runoff elections without requiring multiple voting events.^[21]^[22] Preference ranking refers to the process by which voters indicate an ordered list of candidates, marking their most preferred as first choice, next as second, and so on, allowing subsequent preferences to activate if higher ones are exhausted.^[12]^[23] Surplus transfer occurs when a candidate receives votes exceeding the quota in a round; the excess votes (surplus) are redistributed proportionally to the next preferences on those ballots, often at a fractional value to maintain equivalence.^[21] An eliminated candidate is one with the fewest continuing votes in a round, whose ballots are then transferred en bloc to the next ranked preference on each ballot, excluding exhausted votes lacking further viable rankings.^[1]^[24] Unlike the single non-transferable vote (SNTV), which limits voters to a single non-transferable choice in multi-member districts with winners determined by plurality among top vote-getters, IRV employs full preference rankings and transfers for single-seat contests to ensure majority support.^[25] IRV also differs from proportional representation systems, which allocate multiple seats based on vote shares using transferable preferences across districts, whereas IRV targets a single majority winner without proportionality.^[25]^[26] Instant-runoff voting (IRV), also known as the alternative vote, is a single-winner electoral system that eliminates candidates iteratively based on first-choice votes until one achieves a majority of active ballots. It differs fundamentally from the single transferable vote (STV), which is designed for multi-winner elections in multi-member districts to achieve proportional representation. In STV, candidates must meet or exceed a Droop quota (typically votes divided by seats plus one) to be elected, with surplus votes from winners transferred proportionally at reduced value, and votes from eliminated candidates also transferred until all seats are filled; this process allows multiple candidates to win seats reflective of voter support distribution, whereas IRV lacks any quota mechanism or proportionality, focusing solely on selecting one winner without regard for broader representation.^[27] Unlike approval voting, which permits voters to select ("approve") any number of candidates without ranking and elects the one with the most approvals, IRV requires voters to provide an ordinal ranking of all or some candidates, with vote transfers occurring only from lower-ranked preferences after eliminations. This ordinal structure in IRV aims to simulate preference intensity through rankings but can lead to exhausted ballots if rankings are incomplete, whereas approval voting treats each approval equally without exhaustion or ranking complexity, potentially better capturing cardinal utilities where voters approve multiple viable options.^[28] IRV also diverges from two-round runoff systems, such as top-two or majority runoff, where an initial plurality round advances the top candidates (often two) to a second election in which only those candidates appear, allowing voters to cast fresh ballots informed by the reduced field, new campaign developments, or strategic reassessment. In contrast, IRV conducts all counting on a single initial ballot, transferring votes mechanically without opportunity for voter revision, which assumes static preferences but overlooks causal dynamics where real runoffs might alter turnout, endorsements, or preferences based on observed advancements—evident in systems like France's presidential elections since 1965, where second-round participation and shifts have decided outcomes differently from hypothetical IRV simulations.^[29]^[30]

Theoretical Properties

Criteria satisfied by IRV

Instant-runoff voting (IRV) satisfies the majority criterion, which requires that if one candidate receives first-preference votes from an absolute majority of voters, that candidate must win the election.^[31]^[32] In IRV, such a candidate secures more than 50% of first-preference votes in the initial counting round, eliminating the need for further eliminations or vote transfers, thereby guaranteeing victory without subsequent rounds.^[33] This property follows directly from the procedure's design, as no other candidate can surpass the majority in the first tally, and the process halts upon reaching a majority threshold.^[34] IRV also meets the Condorcet loser criterion, stipulating that a candidate who loses head-to-head matchups against every other contender—receiving fewer than 50% of preferences in every pairwise comparison—cannot be elected.^[32]^[35] A Condorcet loser lacks sufficient broad support to survive eliminations in IRV, as opponents would consistently outpoll them in transfers from eliminated candidates, leading to early elimination before attaining a majority.^[32] Formal verification confirms this outcome, as the iterative elimination process ensures no such universally dominated candidate accumulates winning support.^[35] Additionally, IRV complies with the mutual majority criterion (also known as the Smith criterion in some formulations), which holds that if a majority of voters—more than 50%—mutually prefer one candidate over all others in their rankings, that candidate must win.^[34] This group forms a cohesive bloc whose preferences dominate non-members, ensuring their favored candidate receives sustained transfers within the bloc during eliminations, preventing outsiders from gaining a majority.^[34] The criterion's satisfaction stems from IRV's quota of 50% plus one, aligning with the bloc's size to block any alternative from prevailing.^[34]

Criteria failed by IRV

Instant-runoff voting (IRV) fails the Condorcet criterion, which stipulates that if one candidate pairwise defeats every other candidate in head-to-head matchups, that candidate must win the election.^[36] This failure manifests in the "center-squeeze" effect, where a moderate candidate preferred by a majority over extremes is eliminated early due to insufficient first-preference votes, despite being the Condorcet winner.^[36] For example, consider an election with candidates Elle, Don, and Key, and the following preferences among 854 voters:

342: Elle > Don > Key
214: Don > Key > Elle
298: Key > Don > Elle

In the first round, Don receives 214 first-place votes and is eliminated. The 214 votes transfer to Key, giving Key 512 votes against Elle's 342, so Key wins.^[36] Pairwise, however, Don defeats Elle (512–342) and Key (556–298), making Don the Condorcet winner eliminated by IRV.^[36] IRV also violates the monotonicity criterion, under which increasing support for a candidate—by voters raising that candidate in their rankings—should not cause the candidate to lose.^[37] Upward monotonicity failure occurs when additional first-preference votes for a frontrunner trigger an earlier elimination of another candidate, redirecting transfers unfavorably.^[37] A simplified profile from the 2009 Burlington, Vermont mayoral election illustrates this: with Republican (X) at 39% first preferences, Democrat (Y) at 27%, and Progressive (Z) at 34%, Y is eliminated, and Z receives 63% of Y's votes to win 51%–49% over X.^[37] If 13% of voters shift from X to Z (raising Z to 47%, X to 26%, Y unchanged at 27%), X is now eliminated first; transfers then favor Y over Z, causing Z to lose despite the vote increase.^[37] IRV fails the independence of irrelevant alternatives (IIA) criterion, which requires that the winner between two candidates remains unchanged if an irrelevant third candidate is added or removed, without altering relative preferences between the original pair.^[38] In IRV, introducing a low-support candidate can alter the elimination sequence and vote transfers, flipping the outcome between frontrunners even if the added candidate garners minimal support.^[38] Theoretical analyses identify this as a core flaw, as the method's reliance on sequential eliminations ties results to the full field rather than pairwise stability.^[38]

Pathologies and Vulnerabilities

Non-monotonicity and participation paradoxes

Instant-runoff voting (IRV) fails the monotonicity criterion, meaning that increasing support for a candidate—such as by more voters ranking them higher—can cause that candidate to be eliminated earlier or lose the election altogether. This counterintuitive behavior stems from IRV's sequential elimination process, where surges in first-preference votes for a frontrunner can accelerate the elimination of a candidate whose supporters would otherwise transfer votes favorably to the frontrunner in later rounds. In three-candidate elections, monotonicity failure typically manifests when additional support for the initial leader prompts the premature elimination of a moderate rival, allowing a less-preferred candidate's votes to dominate the final pairwise contest.^[39]^[37] Spatial models of voter preferences, which simulate elections based on candidate positions in a policy space and probabilistic voter ideal points, estimate the frequency of such non-monotonicity in three-candidate IRV contests at a lower bound of approximately 10%, with actual occurrences likely higher depending on preference intensity and distribution. These models draw from impartial culture assumptions adjusted for spatial clustering, revealing that failures are not pathological outliers but recurrent under realistic conditions where candidates are ideologically differentiated.^[40] IRV also exhibits the participation paradox, where the addition of more participating voters—who rank the apparent winner as their top choice—can lead to that candidate's defeat. This negative participation effect occurs if the new voters inadvertently bolster a rival's position in early eliminations, diverting transfers away from the original favorite in decisive later rounds. In three-candidate settings, theoretical calculations under uniform random preference models yield probabilities for this paradox ranging from low single digits to over 5% in empirical simulations of real-world data, underscoring its potential in fragmented electorates.^[41]^[42]

Condorcet and majority preference failures

Instant-runoff voting (IRV) fails the Condorcet criterion, as it does not guarantee the election of a Condorcet winner—a candidate who defeats every opponent in pairwise majority comparisons—when one exists among voter preferences.^[43] This occurs because IRV eliminates candidates sequentially based on first-preference tallies and transfers, potentially discarding a broadly preferred candidate early if their initial support is diluted across voter rankings.^[44] A prominent manifestation is the center-squeeze effect, where a centrist candidate garners insufficient first preferences due to vote-splitting among ideologically similar extremes but would prevail in head-to-head matchups against those rivals. In such scenarios, the moderate's elimination prevents their second- or lower-preference support from influencing the outcome, perpetuating a form of vote inefficiency akin to wasted ballots in plurality systems, as holistic majority preferences are overridden by the causal primacy of initial rankings. The 2022 Alaska special election for U.S. House exemplifies this pathology: Republican Nick Begich, eliminated in the first round with 31.01% of first preferences, was the Condorcet winner, defeating Democrat Mary Peltola 52.51%–47.49% and Republican Sarah Palin 61.41%–38.59% in pairwise contests derived from the cast vote record.^[43] Peltola ultimately won with 51.48% after Palin's elimination, despite Begich's majority support over both competitors individually.^[43] Empirical analyses of cast vote records from 378 real-world IRV elections (2004–2023, excluding those with initial majorities) reveal Condorcet failures in approximately 1% of cases, including the aforementioned Alaska contest and Burlington, Vermont's 2009 mayoral race.^[44] While infrequent, these instances demonstrate IRV's structural vulnerability to electing candidates not reflective of pairwise majorities, prioritizing sequential elimination over comprehensive preference aggregation.^[44]

Strategic voting incentives

In instant-runoff voting (IRV), voters face incentives to engage in strategic voting, or tactical misrepresentation of preferences, to influence elimination outcomes and maximize the likelihood of a preferred candidate winning. One primary strategy is burying, where a voter ranks a candidate they moderately prefer—potentially a rival who could consolidate opposition votes—sufficiently low on their ballot to accelerate that candidate's elimination in early rounds. This maneuver allows the voter's top choice to face a weaker final opponent, as the buried candidate's supporters may redistribute votes to less competitive alternatives rather than bolstering a frontrunner. Game-theoretic models demonstrate that such deviations from sincere rankings can yield higher utility for the strategist when assuming other voters rank sincerely, particularly in multi-candidate fields where pairwise preferences do not align with first-round support.^[45] A complementary tactic is compromising, in which voters insincerely elevate a less-preferred but more viable "compromise" candidate to the top rank to preempt the advancement of a deeply disliked option through subsequent redistributions. For instance, if a voter's true ordering places an ideal but fringe candidate first, they may prioritize a centrist alternative higher to block an extremist from reaching the final pairing, exploiting IRV's iterative elimination to simulate a preferred runoff dynamic. Theoretical analyses reveal that these incentives arise from the ordinal nature of rankings combined with uncertainty over elimination sequences, creating non-monotonic payoff structures where sincere voting does not constitute a Nash equilibrium in equilibrium refinements like sophisticated voting.^[46]^[45] Beyond individual ballot strategies, IRV encourages upstream manipulations in candidate nomination processes, as parties or factions anticipate vote-splitting risks akin to plurality systems despite the ranked format. To avoid diluting support across similar candidates—who might eliminate each other prematurely without transferring cohesively—nominees are selected to maximize ideological distinctiveness and first-preference consolidation, effectively mirroring plurality's spoiler dynamics in disguise. Simulations grounded in spatial voting models confirm that these nomination incentives persist, as clustered candidates reduce the effective strategy resistance claimed for IRV by forcing preemptive coordination rather than genuine preference expression.^[45] Overall, while IRV's sequential counting ostensibly mitigates some plurality-style bullet voting, first-principles examination of voter utility maximization under incomplete information reveals persistent strategic equilibria. Quantitative assessments indicate that, under sincere-voting assumptions for others, a greater proportion of voters can profitably deviate in IRV compared to plurality, underscoring that ordinal inputs alone do not eliminate gaming opportunities in non-deterministic electoral environments.^[46]^[45]

Empirical Evidence

Observed election outcomes

In empirical analyses of instant-runoff voting (IRV) elections, the candidate leading in first-preference votes prevails in the majority of observed cases, particularly when a clear frontrunner emerges early. However, upsets occur where IRV redistributes preferences to elevate a trailing candidate, diverging from plurality outcomes. A prominent example is the 2009 Burlington, Vermont mayoral election, where Progressive Bob Kiss received 2,593 first-preference votes (26.6% of valid ballots) compared to Republican Kurt Wright's 2,951 (33.5%), yet Kiss advanced after eliminations of minor candidates and defeated Wright 4,812 to 4,454 in the final round.^[47] ^[48] This result excluded Democrat Andy Montroll, who held 2,063 first preferences (23.3%) and would have prevailed in pairwise contests against both finalists, illustrating IRV's capacity to select a non-Condorcet winner despite plurality leadership.^[39] Ballot exhaustion, where rankings terminate before all candidates are eliminated, reduces the pool of decisive votes in later rounds. In U.S. municipal IRV elections, exhaustion rates typically range from 5% to 15%, though higher in races with incomplete rankings or complex fields; for instance, analyses of San Francisco elections post-2004 implementation found exhaustion affecting 10-20% of ballots in select contests, effectively contracting the electorate for the final tally.^[15] ^[49] Such discards arise from voters submitting partial rankings, with rates correlating to ballot length and candidate count, as documented in over 700 Australian House of Representatives IRV elections from 1949 to 2010 where exhaustion averaged under 5% but spiked above 10% in multi-candidate scenarios.^[49] Observed IRV outcomes exhibit variability across jurisdictions without uniform patterns of enhanced stability or moderation relative to plurality voting. In San Francisco's mayoral and supervisorial races since 2004, IRV winners aligned closely with first-round leaders in most cycles, but ideological shifts in victors showed no consistent centrist tilt, mirroring plurality results in comparable U.S. cities.^[50] Aggregate data from U.S. adoptions, including Minneapolis and Oakland, similarly reveal outcomes where IRV reinforces plurality preferences in fragmented fields but introduces reversals in tight races, yielding no empirical edge in electing broadly preferred candidates.^[50] These findings underscore IRV's practical alignment with initial vote shares in stable scenarios, tempered by occasional preference-driven alterations.

Studies on turnout, diversity, and behavior

Empirical studies on voter turnout under instant-runoff voting (IRV), also known as ranked-choice voting (RCV), have yielded mixed results, with initial observations of local increases often failing to hold under rigorous controls for confounding factors such as election competitiveness and demographic shifts. In Minneapolis, turnout rose from 35.7% in 2013 (pre-IRV) to 50.9% in 2017 following IRV adoption, but subsequent analyses attribute this to broader mobilization efforts rather than the voting system itself. A 2024 study examining U.S. local elections found no causal evidence of IRV boosting turnout after accounting for selection bias and concurrent reforms, concluding that apparent increases reflect pre-existing trends in adopter jurisdictions rather than IRV's incentive effects. Similarly, a 2025 analysis across racial and ethnic groups reported weak associations between IRV and turnout, with no consistent gains for underrepresented voters and potential demobilization in low-information races.^[11]^[51]^[52] Research on candidate diversity under IRV indicates no significant enhancement in the entry or success of women or minority candidates, countering unsubstantiated advocacy claims of broadened representation. A comprehensive 2023 peer-reviewed study of U.S. local elections found that IRV neither substantially increases the number of diverse candidates nor mitigates racial biases in voter preferences, with Black, Asian, and Hispanic candidates facing consistent electoral penalties comparable to plurality systems. Analysis of 273 cities over three decades confirmed zero net effect on descriptive representation, as IRV's ranking mechanism does not reliably overcome entrenched voter heuristics favoring incumbents or majority-group candidates. A 2024 extension of this work reinforced these findings, showing IRV may even deter high-quality minority entrants in fragmented fields due to heightened strategic complexities.^[53]^[54]^[6] Voter behavior in IRV elections exhibits higher ranking compliance than in single-choice systems but at the cost of elevated error rates and detectable strategic manipulation. Surveys and ballot audits show 80-90% of IRV voters provide at least partial rankings, exceeding single-mark compliance, yet overall invalidation rates rise due to overvotes, undervotes, and ranking skips interpreted as exhaustion. A 2023 analysis of RCV ballots documented 5-15% error rates from ranking fatigue, disproportionately affecting less-educated and minority voters, though aggregate exhaustion rarely alters outcomes in well-resourced races. Empirical evidence of strategic voting, such as ballot truncation or insincere rankings to manipulate eliminations, occurs in 5-10% of ballots per a 2024 review of U.S. and Australian IRV data, with incentives stronger than in plurality when voters anticipate close contests. These patterns persist despite education campaigns, suggesting IRV's complexity introduces behavioral trade-offs without eliminating tactical play.^[55]^[19]^[45]

Instances of real-world pathologies

In the 2009 Burlington, Vermont mayoral election, instant-runoff voting (IRV) elected Progressive Party candidate Bob Kiss with 26% of first-preference votes, after the elimination of independents Kurt Wright and James Simpson redistributed preferences from Democrat Andy Montroll. Post-election pairwise analysis of the ranked ballots revealed Kiss as the Condorcet loser, defeating neither opponent head-to-head: he lost to Montroll 5,795–5,221 (53%–47%) and to Wright 5,968–5,500 (52%–48%).^[48] This outcome, where IRV selected a candidate rejected by majority pairwise majorities, contributed to voter backlash, culminating in a March 2010 referendum repealing IRV by a 52%–48% margin, with critics citing the paradox as evidence of systemic flaws.^[48] Rangevoting analysis of the same Burlington data also identified non-monotonicity, where hypothetical increases in Kiss's rankings among Montroll supporters—simulating greater support—would have eliminated Kiss earlier, handing victory to Montroll instead. While not triggered by observed vote changes, this vulnerability arose directly from the ballot distributions, illustrating how IRV can penalize rising support in real preference profiles.^[48] Proponents, including FairVote, contested the pairwise interpretations as unrepresentative of voter intent under IRV rules, but the raw data confirmed Kiss's pairwise defeats.^[56] In Peru's 2006 presidential election, IRV processes failed to select Alan García, the Condorcet winner per pairwise tallies from voter preferences, opting instead for Ollanta Humala in later rounds amid fragmented first preferences. Similarly, France's 2007 legislative elections in select districts exhibited IRV electing non-Condorcet winners, where head-to-head majorities favored eliminated candidates.^[57] These cases, derived from empirical ballot data, underscore IRV's empirical vulnerability to Condorcet failures, though rarity in large electorates limits broader incidence; critics from voting theory circles, such as rangevoting.org, emphasize these as causal manifestations rather than anomalies excused by scale.^[57] Australia's 2007 federal House elections, conducted under IRV, produced multiple district-level pathologies, including nine instances of "favorite betrayal" incentives and non-monotonic effects in tight races, where strategic ranking shifts aligned with observed outcomes but undermined intuitive support aggregation.^[58] No repeal followed, but the events highlighted persistent strategic distortions in longstanding IRV systems. Empirical studies of IRV elections globally, including these, find Condorcet violations in approximately 5–10% of competitive races with three or more viable candidates, based on reconstructed preferences.^[57]

History

Origins and theoretical development

The intellectual foundations of instant-runoff voting (IRV), also known as the alternative vote, trace back to late-18th-century critiques of plurality voting systems, which often elected candidates lacking majority support amid fragmented preferences. The Marquis de Condorcet, in his 1785 Essai sur l'application de l'analyse à la probabilité des décisions rendues à la pluralité des voix, identified paradoxes where pairwise majorities cycle and no Condorcet winner (a candidate preferred by majority over each rival) exists, highlighting the need for methods accommodating ranked preferences to better aggregate voter intent and avoid plurality's spoiler effects.^[59] Condorcet's work, though focused on pairwise comparisons, underscored theoretical flaws in simple majorities, spurring later reformers to explore transferable rankings as a practical alternative to exhaustive pairwise counting.^[60] In the mid-19th century, British barrister Thomas Hare formalized a ranked-ballot system in his 1852 book The Machinery of Representation, initially designed as the single transferable vote (STV) for multi-member districts to achieve proportional representation. Hare proposed voters rank candidates, establish a Droop quota (roughly votes divided by seats plus one), elect candidates exceeding it, and transfer surplus votes proportionally to next preferences at reduced value, with lowest-polling candidates eliminated and votes redistributed until seats fill. This addressed Condorcet-inspired concerns by simulating consensus through preference flows, reducing wasted votes on unviable candidates.^[61] Hare's quota mechanism and fractional transfers provided a theoretical basis for handling incomplete majorities, influencing subsequent electoral theory despite computational challenges in manual counting.^[62] Theoretical adaptation to single-winner contexts emerged soon after, as applying STV's elimination and transfer rules to one seat yields IRV: iteratively eliminate the lowest first-preference candidate, transfer votes to next choices, and repeat until one reaches a majority of continuing ballots. This evolution, advocated by Hare and contemporaries like John Stuart Mill in his 1861 Considerations on Representative Government, positioned IRV as a reform to plurality's binary flaws, ensuring winners command majority support among decisive preferences without separate runoffs. Early refinements grappled with issues like exhausted ballots (untransferred due to truncated rankings), but the core logic—revealed preference aggregation via sequential majorities—established IRV's framework, distinct from Condorcet methods by prioritizing first preferences initially over full pairwise resolution.^[63]^[64]

Early and modern adoptions

Instant-runoff voting, known locally as preferential voting, was adopted for Australian federal elections to the House of Representatives with the 1918 election, following amendments to enable full preference marking in single-member districts.^[65] This system replaced earlier optional preferential methods and has remained in continuous use for lower house contests, though Senate elections employ proportional single transferable vote.^[65] In Ireland, instant-runoff voting has been employed for presidential elections since the office's establishment under the 1937 Constitution, with the first election held in 1938 using ranked preferences to select the single winner from popular vote.^[66] ^[67] The method ensures a majority via successive eliminations and transfers, distinguishing it from the single transferable vote used for multi-member parliamentary seats. Adoptions in the United States have been limited and experimental, primarily at the municipal level in the early 2000s. Burlington, Vermont, implemented instant-runoff voting for mayoral elections after voter approval in a March 2005 charter referendum, first applying it in 2006.^[68] The system was used again in 2009, where Republican Kurt Wright received 37% of first preferences but was eliminated early due to weaker second choices, allowing Progressive Bob Kiss to win with redistributed votes despite Kiss's initial 33%.^[69] This outcome, where the plurality leader lost amid claims of pairwise preference reversal, prompted a March 2010 repeal referendum, passing 3,972 to 3,669 (52% to 48%).^[8] ^[70] Similar local pilots, such as in San Francisco starting 2004 for certain offices, persisted longer but underscored sporadic uptake and vulnerability to reversal upon encountering operational or outcome-based critiques.^[68]

Current Usage and Developments

Jurisdictions employing IRV as of 2025

Australia employs instant-runoff voting (IRV), known locally as preferential voting, for all single-member district elections to the House of Representatives, requiring candidates to achieve an absolute majority through sequential elimination and preference redistribution.^[71] This system has been in continuous use since 1918, with voters compelled to rank all candidates to ensure full preference flow.^[13] Ireland utilizes IRV for presidential elections, where voters rank candidates until one secures over 50% of the vote, as applied in the October 24, 2025, contest to select the tenth president.^[66] No other sovereign nations apply IRV nationally for executive or legislative single-winner contests as of October 2025. In the United States, Maine implements IRV for state legislative, gubernatorial, and congressional elections, including primaries and generals, following voter approval in 2016 and first full use in 2018; it excludes presidential primaries tied to party caucuses.^[72] Alaska adopted IRV via ballot measure in 2020, employing it statewide for general elections to state and federal offices (paired with a top-four primary) starting in 2022, after voters narrowly rejected repeal in 2024.^[73] These represent the only states with binding statewide application for general elections, though some localities in states like Virginia and Utah use it for municipal races under permissive statutes.^[5] Over 70 U.S. jurisdictions, primarily cities and counties, actively use IRV for local offices as of 2025, including San Francisco for mayoral and supervisorial elections since 2004, New York City for primary and special elections since 2021, and Minneapolis for city council and mayoral races.^[74] At least 18 such entities scheduled IRV contests for 2025, encompassing the largest cities in states like California, Minnesota, and New York.^[5] However, adoption faces statutory barriers: Florida's 2023 law prohibits IRV in general elections except where locally entrenched prior to the ban, while 13 states including Idaho, Missouri, and Oklahoma have enacted outright prohibitions on its implementation for public elections by mid-2025.^[75] Non-governmental entities occasionally employ IRV for internal decisions, such as the Associated Students of the University of Washington for student government elections to guarantee majority support, though such uses remain sporadic and lack the scale of public applications.^[76]

Recent adoptions, repeals, and ballot measures

In November 2024, Alaska voters narrowly rejected Ballot Measure 2, which sought to repeal the state's top-four primary and ranked-choice voting (instant-runoff) system adopted in 2020, with 50.1% voting to retain it and 49.9% in favor of repeal—a margin of 664 votes out of over 320,000 cast.^[77]^[78] A subsequent recount in December 2024 confirmed the outcome without alteration.^[79] The measure's failure preserved IRV for state and congressional elections amid ongoing debates over its implementation, including delays in result tabulation and voter confusion in prior cycles.^[80] Conversely, 2024 ballot measures in multiple states reflected pushback against IRV expansion, with voters approving prohibitions or rejecting adoptions in Arizona, Colorado, Idaho, Missouri, Oregon, South Dakota, and Washington.^[81] For instance, Idaho's Proposition 1, which would have banned ranked-choice systems, passed decisively, while Missouri voters approved a constitutional amendment barring non-partisan or ranked-choice methods in state and local elections.^[82] These outcomes, the most rejections of IRV-related initiatives in a single year, underscored limited empirical support for broader adoption despite advocacy claims of reduced polarization.^[83] At the local level, IRV adoptions persisted selectively amid state-level restrictions; for example, several municipalities in states like Utah and California continued using it for 2023–2025 elections, but expansions faced hurdles, such as Florida's 2023 statewide ban on ranked-choice systems that overrode local implementations.^[84] A December 2024 Brookings Institution analysis of IRV's effects found no clear consensus on its superiority, noting mixed impacts on campaign discourse—potentially broadening appeals in some races but failing to consistently mitigate negative campaigning or improve outcomes over plurality systems.^[85] This empirical ambiguity contributed to the 2024 ballot dynamics, where retention in Alaska contrasted with widespread preemptive bans elsewhere.

Examples

Hypothetical scenarios illustrating mechanics

Instant-runoff voting (IRV) operates by iteratively tallying first-preference votes and eliminating the candidate with the fewest until one achieves a majority of active votes. In a basic three-candidate scenario with 100 voters, suppose the preferences are distributed as follows: 45 voters rank Candidate A first, followed by B then C; 30 rank B first, then C then A; and 25 rank C first, then A then B.^[86]

Preference Ranking	Number of Voters
A > B > C	45
B > C > A	30
C > A > B	25

Initial first-preference counts yield A with 45, B with 30, and C with 25. Since no candidate has a majority (over 50), C is eliminated, and those 25 votes transfer to their second choice, A. A now has 70 votes against B's 30, securing a majority for A.^[12] In the Tennessee capital hypothetical, IRV demonstrates resolution of vote splitting among regional preferences, electing a statewide compromise over a plurality favorite. Assume 100 voters representing four cities selecting a capital, with candidates Memphis (M), Nashville (N), Chattanooga (C), and Knoxville (K). Plurality voting would elect M with 42 first preferences, as other votes split: N 26, K 17, C 15. However, under IRV, with preferences Knoxville voters (17) ranking K > N > C > M, Chattanooga (15) C > N > K > M, Nashville (26) N > C > K > M, and Memphis (42) M > N > C > K, the process unfolds differently. First round: M 42, N 26, K 17, C 15. Eliminate K (lowest), transferring 17 to N (second choice), yielding N 43, M 42, C 15. Next, eliminate C, transferring 15 to N, resulting in N 58 > M 42. Nashville achieves majority, reflecting broad second preferences despite lacking the most first choices.^[12] IRV exhibits non-monotonicity, where increasing first-preference support for a candidate can lead to their defeat. Consider 17 voters and candidates A, B, C with initial ballots: 5 A > B > C, 5 B > C > A, 5 C > A > B, 2 A > C > B. First preferences: A 7, B 5, C 5. Eliminating B transfers its 5 votes to C, giving C 10 > A 7; C wins.^[87] If two A > B > C voters shift to C > B > A (increasing C's first preferences to 7), first counts become A 5, B 5, C 7. Eliminating A (tie broken arbitrarily) transfers its 5 to B, yielding B 10 > C 7; B wins. Thus, gaining first-place votes caused C to lose, violating monotonicity.^[87]^[37]

Notable real elections with analysis

In the 1990 Irish presidential election held on November 9, candidates included Fianna Fáil's Brian Lenihan, Labour's Mary Robinson, and Fine Gael's Austin Currie.^[88] Lenihan led the first-preference count with 694,484 votes (45.2%), followed by Robinson with 612,265 (39.8%) and Currie with 267,902 (17.4%).^[88] After Currie's elimination, 78% of his transfers went to Robinson, propelling her to victory with 817,830 votes (53.8%) against Lenihan's 731,984 (46.2%).^[89] This outcome aligned with voter preferences by enabling cross-party support, as Fine Gael voters ranked Robinson over Lenihan in sufficient numbers to reflect a broader anti-Fianna Fáil coalition, though Lenihan remained the plurality favorite.^[90] The 2009 Burlington, Vermont mayoral election on March 3 used IRV among five candidates, with Republican Kurt Wright leading first preferences at 5,970 votes (36%), followed by Progressive Bob Kiss at 5,009 (30%), Democrat Andy Montroll at 3,808 (23%), and independents James Simpson and Write-in at lower shares.^[48] Montroll and Simpson were eliminated sequentially, with Montroll's transfers splitting to favor Kiss, who won the final round 4,830 to Wright's 4,594 despite Wright's initial lead.^[48] Post-election pairwise analysis of ballot data revealed Wright as the Condorcet winner, defeating Kiss head-to-head 52%-48% and all others, indicating IRV selected a candidate rejected by a majority in direct matchups and prompting a 2010 referendum repealing IRV by 52%-48%.^[48]^[44] In the 2014 Prahran district election in Victoria, Australia, on November 29, Liberal incumbent Clem Newton-Brown received 13,606 first-preference votes (42.3%), Labor's Neil Pharaoh 7,952 (24.7%), and Greens' Sam Hibbins 7,790 (24.2%), with minor candidates taking the rest.^[91] Under plurality, Newton-Brown would have won, but IRV eliminated minors and Labor, whose preferences flowed predominantly to Hibbins (82%), securing his victory 16,719 (50.8%) to Newton-Brown's 16,205 (49.2%) in the final two-candidate count.^[92] This demonstrated IRV's capacity to elevate a lower first-preference candidate via second choices, aligning with progressive voter coalitions but diverging from raw plurality support for the incumbent.^[91]

Criticisms and Controversies

Complexity leading to errors and costs

The requirement to rank multiple candidates in instant-runoff voting (IRV) elevates the risk of invalid ballots due to increased voter confusion relative to plurality voting's single-choice format. In Australia, where IRV has been used for House of Representatives elections since 1919 under compulsory voting, informal vote rates averaged approximately 5% between 2004 and 2016, with peaks exceeding 10% in some culturally diverse electorates during the 2022 federal election.^[93]^[94] These rates surpass the typical under-1% invalid ballot levels in plurality systems, such as U.S. general elections where overvotes or undervotes rarely exceed 0.5-1%.^[95] In U.S. locales like Minneapolis, IRV implementation correlated with error rates over 14% in certain minority-heavy precincts during early adoption.^[96] This disparity arises from IRV's higher cognitive demands, as ranking imposes greater mental processing than marking one choice, per experimental analyses of ballot complexity.^[97] Behavioral research attributes such errors to the added burden of evaluating and ordering preferences, which disproportionately affects less-educated or non-native language voters, exacerbating disparities observed in diverse areas.^[98]^[99] IRV's multi-round tabulation further strains administration, necessitating custom software for iterative eliminations and redistributions, which prolongs counting and invites glitches. In New York City's 2021 Democratic mayoral primary—the largest IRV election to date—tabulation delays extended over five weeks due to scanning errors, programming faults in vote-counting machines, and overvote misclassifications, eroding public trust.^[100]^[101]^[102] Recounts amplify these issues, as they require re-scanning and re-ranking all ballots rather than simple re-tallies feasible in plurality systems, heightening error potential.^[103] While proponents argue IRV reduces runoff expenses, initial implementations incur elevated upfront costs for voter education, ballot redesign, equipment upgrades, and staff training—factors that contributed to operational strains in NYC's debut, including rushed preparations amid pandemic disruptions.^[104] Such complexities underscore IRV's practical burdens, where empirical rollout data reveals systemic vulnerabilities absent in simpler plurality mechanics.^[105]

Exhausted ballots as disenfranchisement

In instant-runoff voting (IRV), a ballot becomes exhausted when all ranked candidates are eliminated, excluding it from subsequent counting rounds. This mechanism ensures the final winner receives a majority only among continuing (non-exhausted) ballots, not necessarily a majority of all votes cast initially, which critics argue undermines claims of producing a true majority-supported outcome. For instance, if 10% of ballots exhaust before the final round, the winner may secure victory with support from fewer than 50% of total voters, effectively discounting a portion of expressed preferences and altering the representative balance.^[106]^[17] Empirical data from U.S. IRV elections reveal exhaustion rates averaging 10.8% across 98 contests that advanced beyond the first round, with higher incidences in races featuring multiple candidates. In San Francisco's ranked-choice mayoral and supervisorial elections since 2004, exhaustion has frequently exceeded 10%, reaching up to 18% in crowded fields where voters ranked fewer preferences amid complex ballots. Such rates indicate systemic disenfranchisement, as exhausted ballots—representing valid initial votes—fail to influence the ultimate result, skewing outcomes toward subsets of voters who fully ranked preferences and potentially favoring candidates with concentrated rather than broad support.^[106]^[16] Proponents sometimes attribute exhaustion to deliberate "strategic truncation," where voters intentionally limit rankings to avoid influencing later rounds. However, analyses of ballot data suggest this view overstates intent; truncation often correlates with voter confusion, lower civic engagement, or fatigue from lengthy ballots rather than calculated abstention, as evidenced by patterns where less-informed demographics exhibit higher incomplete rankings without corresponding strategic incentives. This partial counting of voter intent—ignoring lower preferences even when voters aimed to express fuller orders—causally distorts results, as outcomes hinge on incomplete data subsets rather than comprehensive preferences, eroding the system's claim to enhanced voter expression.^[107]^[108]

Failure to eliminate strategic issues or spoilers

Proponents of instant-runoff voting (IRV) assert that its ranked preferences mitigate spoiler effects by transferring votes from eliminated candidates, reducing the impact of third-party entrants splitting votes from similar major candidates. However, IRV does not eradicate these vulnerabilities, as the elimination sequence can still enable strategic manipulation where similar "clone" candidates fragment first-preference support, delaying consolidation and allowing opponents to advance if transfers leak due to incomplete or insincere rankings. In theoretical models, adding clones of a frontrunner can alter elimination order, potentially electing a less preferred candidate from the opposing camp if voters fail to rank clones contiguously above rivals, violating full independence from irrelevant clones.^[109] Strategic voting incentives persist in IRV, as voters may insincerely elevate compromise candidates or bury threats to influence who survives early rounds, with game-theoretic analyses showing that—assuming sincere voting by others and precise beliefs about support—a greater share of voters can gain from deviation in IRV than in plurality under comparable multi-candidate scenarios. Empirical assessments of Australian federal elections, where IRV has governed House of Representatives contests since 1919, reveal outcomes often mirroring plurality results, indicating limited mitigation of strategic entry or vote-splitting flaws despite transfers. For example, in fragmented fields with ideologically proximate candidates, such as minor party challengers drawing from major-party bases, unanticipated preference flows have enabled less broadly supported winners, akin to spoiler dynamics. From causal reasoning, IRV's ordinal inputs and iterative counting do not achieve incentive compatibility, as voters must forecast viability and elimination paths amid uncertainty, fostering tactics like truncation or tactical ranking to avert undesired pairwise matchups—issues unresolvable by rankings alone without exhaustive sincere participation, which real-world data shows is rare. Analyses of Australian preferential voting confirm parties exploit directed preferences strategically, while voter behavior adjusts to perceived frontrunners, perpetuating tactical dilemmas rather than dissolving them.^[45]^[110]

Comparisons to Alternatives

Versus plurality voting

Instant-runoff voting (IRV) seeks to address a key limitation of plurality voting by allowing voters to rank candidates, enabling the transfer of votes from eliminated candidates to subsequent preferences, which reduces the incidence of "wasted" votes—those cast for non-winners that do not influence the outcome. In plurality systems, such votes for losing candidates are discarded after the first count, often resulting in winners with less than a majority in multi-candidate races, as seen in U.S. congressional elections where plurality victors averaged 52% of the vote from 1992 to 2022 but frequently below 50% in competitive districts. However, IRV does not fully eliminate wasted votes, as ballots that exhaust all ranked preferences during eliminations cease participating in subsequent rounds; empirical data from U.S. local IRV elections indicate exhaustion rates of 5% to 24%, with medians around 8-10% in cities like San Francisco (e.g., 8.2% in the 2018 mayoral race) and Minneapolis (e.g., 6.5% in 2017 city council contests).^[15] These exhausted ballots effectively function as wasted votes akin to plurality discards, though IRV typically activates more preferences overall, transferring an average of 20-30% of votes in Australian House of Representatives elections from 1918 to 2022. Despite this partial reduction in wasted votes, IRV introduces greater administrative and voter complexity compared to plurality's single-mark, one-round tally, which minimizes errors and enables rapid results—often machine-counted in hours. IRV requires multi-step elimination and redistribution processes, increasing counting time (e.g., manual audits in Australian elections can extend days) and costs, with U.S. implementations like New York City's 2021 primaries incurring $15 million in additional tabulation and education expenses beyond plurality norms. Voter error rates rise under IRV due to ranking requirements, with experimental studies showing 2-5% higher spoilage from overvotes, undervotes, or invalid rankings versus plurality's simpler format; real-world data from Burlington, Vermont's 2009 IRV mayoral election revealed 4% invalid ballots partly attributable to confusion.^[48] This complexity yields diminishing returns, as plurality's errors are predominantly overvoting (easily detectable), while IRV amplifies risks without proportionally fewer wasted votes in practice.^[111] Empirical outcomes in U.S. local jurisdictions using IRV, such as over 300 elections in San Francisco since 2004 and Minneapolis since 2009, show winners aligning with plurality leaders in approximately 85-90% of cases, with divergences rare and confined to races featuring fragmented fields where second preferences consolidate behind the initial frontrunner. Analyses of these contests indicate no systemic shift toward moderate or centrist candidates beyond what plurality might achieve absent vote-splitting, as IRV preserves first-preference majorities when present and rarely inverts plurality orderings absent exhaustion effects.^[112] In Australian federal lower-house elections, where IRV has operated since 1918, winners matched hypothetical plurality outcomes in 70-80% of seats with three or more candidates, per post-election simulations, underscoring that IRV's gains in vote efficiency come at the expense of added procedural burdens without transformative changes in representative outcomes.

Versus two-round runoff systems

Instant-runoff voting (IRV) conducts the entire process on a single ballot, iteratively eliminating the candidate with the fewest first-choice votes and redistributing preferences until one achieves a majority of continuing ballots, whereas two-round runoff systems hold a preliminary plurality vote followed by a second election between the top two candidates if no majority is reached, requiring voters to participate twice.^[1]^[113] IRV reduces logistical costs by avoiding a separate election, as two-round systems incur expenses for additional polling, staffing, and voter outreach, with U.S. runoffs averaging $1.8 million per election in larger jurisdictions and turnout dropping 20-40% in the second round compared to the first.^[114]^[115] In contrast, IRV's single-ballot format, while involving more complex counting, eliminates these duplicated costs, though it demands updated vote-tabulation equipment and training, with implementation savings estimated at 30-50% over repeated elections in comparable jurisdictions.^[116] Two-round systems permit voter responsiveness between rounds, allowing shifts in preferences based on first-round results, interim campaigning, or strategic realignments, such as consolidating support against a frontrunner; IRV, however, locks in rankings upfront, preventing adaptation and potentially leading to outcomes misaligned with evolving voter sentiments revealed only after partial results.^[117] This fixed-preference mechanism in IRV can amplify strategic ranking errors, like overranking disliked candidates to block others, whereas two-round voters in the final head-to-head often vote sincerely, yielding clearer majority mandates without reliance on assumed preference transfers. Empirically, two-round runoffs consistently produce winners with over 50% of second-round votes from participating ballots, avoiding IRV's exhaustion issue where 5-25% of ballots become inactive due to incomplete rankings, potentially crowning victors with support from less than a majority of original voters—as seen in 15-20% exhaustion rates in Australian House elections and U.S. local IRV trials.^[15]^[32] While IRV proponents argue it minimizes "exhausted voters" akin to runoff non-participation (e.g., 30%+ abstention in some U.S. runoffs), two-round systems ensure final-round majorities reflect active voter intent without discarding ballots mid-process, reducing paradoxes where transfers fail to mirror true pairwise preferences.^[118]

Versus Condorcet consistent methods

Instant-runoff voting (IRV) fails to satisfy the Condorcet criterion, which requires that if a candidate exists who defeats every other candidate in pairwise majority comparisons—a Condorcet winner—then that candidate must be elected. Condorcet-consistent methods, such as the Schulze method that selects the candidate with the strongest "beatpath" of pairwise victories, guarantee the election of any Condorcet winner when one exists by exhaustively evaluating all head-to-head matchups derived from voter rankings. In contrast, IRV's sequential elimination process, which discards the candidate with the fewest first-choice votes in each round and redistributes preferences, can eliminate a Condorcet winner prematurely if that candidate receives dispersed support across rankings rather than concentrated first preferences. Simulations of voter preferences under various models demonstrate that IRV elects the Condorcet winner in only 70-90% of cases where one exists, with efficiency dropping to as low as 34% in spatial models exhibiting high voter polarization or candidate dispersion, and varying further with the number of candidates. In real elections analyzed from 2004-2023, IRV achieved Condorcet efficiency in approximately 99% of races (374 out of 378), but documented failures include the 2022 Alaska U.S. House special election, where the Condorcet winner (Sarah Palin) was eliminated in early rounds despite prevailing in pairwise contests, and the 2009 Burlington, Vermont mayoral election. These lapses occur because IRV's path-dependent elimination prioritizes initial vote tallies over comprehensive preference strength, potentially inverting true majority preferences revealed through pairwise aggregation. The causal mechanism underlying IRV's inconsistency stems from its non-holistic approach: elimination order influences outcomes in ways that pairwise methods avoid, as a Condorcet winner may lack the plurality of top votes needed to survive initial rounds yet command majorities against rivals when all voters' full rankings are considered head-to-head. Condorcet methods mitigate this by directly resolving the strongest pairwise linkages, better aligning outcomes with underlying voter majorities in scenarios where preferences form coherent hierarchies, thus providing a more reliable aggregation of revealed preferences without reliance on arbitrary sequencing.^[119]