Alternative Vote (AV)

• Alternative Vote is a ranked-choice electoral system where voters order candidates and iterative eliminations, with vote transfers, determine the majority winner.
• The system dynamically reallocates votes from eliminated candidates, balancing first-choice strength and compromise to reflect stable majority coalitions.
• AV is extensively studied for its computational challenges, non-monotonic behaviors, and potential generalizations to weak-order preferences in both practice and theory.

Searching arXiv for recent and foundational papers on Alternative Vote / Instant-Runoff Voting to ground the article. Alternative Vote (AV), also known as Instant Runoff Voting (IRV) and Ranked Choice Voting (RCV), is a single-winner ranked electoral system in which voters order candidates and the count proceeds by iterative elimination: at each round, ballots are counted for each voter’s highest-ranked continuing candidate, the candidate with the fewest such votes is eliminated, and those ballots transfer to the next-ranked continuing candidate until a winner is determined (Raftery et al., 2021). In the single-winner case of Single Transferable Vote (STV), this procedure implements an “instant runoff” between the eventual final pair of contenders (Raftery et al., 2021). Recent work treats AV both as a practical electoral mechanism used extensively in Australia and as a formal object of study in computational social choice, software implementation, weak-order generalization, and risk-limiting auditing (Morton, 21 Jul 2025, Delemazure et al., 2024, Ek et al., 2023).

1. Terminology, ballot model, and canonical procedure

AV is the single-winner form of STV: “For single-winner elections, the STV is also called instant runoff voting (IRV), ranked choice voting (RCV), or the alternative vote (AV) system” (Raftery et al., 2021). Standard AV assumes each voter supplies a strict ranking of candidates, though some jurisdictions require full preferential ballots while others allow optional preferential ballots (Morton, 21 Jul 2025).

Let the candidate set be $C = \{c_1,\dots,c_K\}$, with continuing set $S_t$ at round $t$. In the standard linear-order formulation, the top remaining choice of voter $v$ at round $t$ is the highest-ranked candidate in $S_t$; plurality counts are then computed over those top remaining choices, the lowest-count candidate is eliminated, and the process repeats until one candidate remains (Delemazure et al., 2024). A canonical majority threshold used in AV/IRV descriptions is

$M = \left\lfloor \frac{V}{2} \right\rfloor + 1,$

with $V$ the number of voters (Grama, 2021).

In the Australian practice-grounded treatment, AV proceeds as follows: initialize $S_0 = C$; at each stage count every ballot for the highest-ranked candidate still in $S_t$; if some candidate has $S_t$0, with

$S_t$1

elect that candidate; otherwise eliminate the candidate with the smallest tally, transfer each ballot to its next preferred continuing candidate, remove exhausted ballots from the live count, and continue (Morton, 21 Jul 2025). The same basic structure appears in the vote R package implementation, where for AV the quota is expressed as

$S_t$2

reducing the STV quota to a majority of active ballots, with $S_t$3 a small positive number such as $S_t$4 for small electorates (Raftery et al., 2021).

The transfer semantics are therefore central: AV does not aggregate all ranks simultaneously, but rather conditions each round on the surviving candidate set. Ballots with no remaining ranked candidate can exhaust and cease contributing to later counts (Ek et al., 2023, Morton, 21 Jul 2025). This roundwise dependence on transfers and exhaustion is precisely why AV is mathematically and inferentially more complex than plurality (Ek et al., 2023).

2. Structural properties and majority-runoff interpretation

A central structural claim in the Australian analysis is the “dominant-pair principle”: in any AV election there exist two candidates $S_t$5 and $S_t$6 such that the election outcome is determined by their pairwise runoff dominance; absent ties during the count, this pair is uniquely the winner and runner-up (Morton, 21 Jul 2025). This formalizes the intuition behind the name Instant Runoff Voting. Closely related is the “irrelevance of dominant-pair preferences” proposition: if $S_t$7 is the dominant pair, rankings below the first occurrence of $S_t$8 or $S_t$9 on any ballot do not affect the result; truncating ballots at that point leaves the outcome unchanged (Morton, 21 Jul 2025).

The same source gives a primary-vote criterion for locating the dominant pair. If two candidates each have a primary vote share exceeding the combined primary vote of all other candidates, then they form the unique dominant pair; more generally, if a subset $t$0 of $t$1 candidates each exceeds the aggregate of all non-$t$2 candidates, the dominant pair must lie within $t$3 (Morton, 21 Jul 2025). A practical bottom-up scan over primary-vote residues is proposed there to identify the smallest dominant $t$4-set (Morton, 21 Jul 2025).

AV’s majority character is multi-layered in the cited work. First, if any candidate has a majority of first preferences, AV elects that candidate immediately (Morton, 21 Jul 2025). Second, even when no initial majority exists, the count induces a final two-candidate-preferred runoff routinely reported in Australian elections (Morton, 21 Jul 2025). Third, stable coalitions are protected: if a strict majority ranks every member of a subset $t$5 above all outsiders, the AV winner must come from $t$6 (Morton, 21 Jul 2025).

The literature also places AV in relation to Condorcet notions. The Australian paper states that AV never elects a Condorcet loser and that any Condorcet winner with at least

$t$7

primary votes, or who accumulates that many votes via transfers while still contesting, is necessarily elected (Morton, 21 Jul 2025). The paper also emphasizes that when AV fails to elect a Condorcet winner, that candidate’s primary support is necessarily below the one-third threshold (Morton, 21 Jul 2025). This does not make AV Condorcet-consistent in general; rather, it shows a partial “Condorcet-safe above one-third” property (Morton, 21 Jul 2025).

A plausible implication is that AV is best understood not as a pure pairwise-majoritarian rule, but as a rule that balances first-choice strength, transfer viability, and final-head-to-head legitimacy. That interpretation is stated explicitly in the Australian account, which describes transferable voting as balancing competing desiderata associated with “majority rule” and “voter choice” (Morton, 21 Jul 2025).

3. Formal limitations, paradoxes, and weak-order generalization

The contemporary literature distinguishes between AV’s practical strengths and its theoretical vulnerabilities. One such vulnerability is non-monotonicity. The Australian analysis defines monotonicity in the standard sense—improving a candidate’s position should not cause that candidate to lose, and worsening should not cause that candidate to win—and shows that AV can violate it in three-way contests (Morton, 21 Jul 2025). Two forms are isolated: “winner-becomes-loser,” where increasing support for the current winner changes the elimination order and causes defeat, and “loser-becomes-winner,” where worsening a candidate’s position induces a different elimination order that leads to victory (Morton, 21 Jul 2025). That paper further proves that if both forms are possible from the same base profile, the profile exhibits a Condorcet cycle (Morton, 21 Jul 2025).

The same paper develops a two-dimensional spatial/geometric model of three-candidate contests. In that model, each candidate has probability $t$8 of winning an absolute majority of primary votes, and there is a $t$9 probability that no candidate initially has a majority (Morton, 21 Jul 2025). Within this framework, the exact probability of latent winner-becomes-loser non-monotonicity is reported as approximately $v$0, loser-becomes-winner as approximately $v$1, and total non-monotonicity as below $v$2, with overlap around $v$3–$v$4 by Monte Carlo (Morton, 21 Jul 2025). The paper conjectures the combined probability is less than $v$5 (Morton, 21 Jul 2025).

Another limitation concerns Condorcet failure. The Australian account proves a “Condorcet instability” result: if an AV election fails to elect a Condorcet winner, there exists a ballot modification that preserves primary votes, preserves the AV outcome and intermediate stages, but converts the profile into a top Condorcet cycle (Morton, 21 Jul 2025). This directly links AV’s paradoxical cases to the intrinsic instability of Condorcet structures rather than to an isolated defect of the transfer procedure.

A distinct modern line of work studies what happens when voters are allowed to express indifferences rather than strict rankings. Standard AV assumes linear orders, but “Generalizing Instant Runoff Voting to Allow Indifferences” introduces Approval-IRV for weak orders (Delemazure et al., 2024). For remaining set $v$6, each voter’s current top set is

$v$7

and each candidate’s round score is

$v$8

The candidate with minimal $v$9 is eliminated, using parallel-universe tie-breaking (PUT) for ties (Delemazure et al., 2024).

This rule reduces to standard AV when all top sets are singletons, but unlike split-based alternatives it preserves several characteristic axiomatic properties (Delemazure et al., 2024). Within the class of elimination scoring rules, Approval-IRV is shown to be the unique extension of IRV that satisfies both independence of clones and respect for cohesive majorities for $t$0, and also the unique extension that agrees with IRV on linear orders while satisfying indifference monotonicity (Delemazure et al., 2024). By contrast, the alternative Split-IRV fails clone independence, may fail majority wishes, and fails the weak-order monotonicity defined there (Delemazure et al., 2024).

This suggests that AV’s standard strict-ranking format is not the only coherent formalization of runoff-by-elimination; however, among weak-order generalizations considered in that paper, the approval-based interpretation is singled out as the one that preserves the core AV/IRV axiomatic profile (Delemazure et al., 2024).

4. Implementation, software, and operational details

The vote R package provides an explicit software implementation of AV as the single-winner case of STV (Raftery et al., 2021). The principal function is stv, with AV specified by setting mcan = 1 (Raftery et al., 2021). The package supports plurality, two-round runoff, score, approval, STV, and Condorcet winner/loser checks, but its AV implementation inherits the full STV counting machinery, including weighted transfers and a configurable quota (Raftery et al., 2021).

In that framework, weighted first preferences are

$t$1

candidate totals are

$t$2

and the quota is

$t$3

For AV, $t$4, so this becomes majority of active ballots plus $t$5 (Raftery et al., 2021). When a candidate reaches or exceeds quota, the package computes a surplus fraction

$t$6

though for single-winner AV the main practical behavior is elimination-and-transfer until a winner emerges (Raftery et al., 2021).

Tie-breaking is operationally significant. The package defaults to the Electoral Reform Society’s Forwards Tie-Breaking Method (ties = "f"), with Backwards ("b") as an alternative. If those fail, it uses an Ordered method comparing first preferences, then second preferences, and so on; if ties remain after all preference levels, they are resolved by random sampling with reproducible seeds (Raftery et al., 2021). The paper gives a detailed faculty-election example in which a second-count tie between Gauss and Poisson is resolved by “fo” because Forwards does not separate them but Ordered does, based on second preferences (Raftery et al., 2021).

One distinctive implementation feature is support for equal rankings via equal.ranking = TRUE (Raftery et al., 2021). With equal top preferences, ballot weight is split equally among tied top candidates, and modified update equations are used: $t$7 and

$t$8

Ballots with equal ranks are also postprocessed to conform to the package’s required encoding (Raftery et al., 2021). This is distinct from Approval-IRV’s axiomatic weak-order extension, but it illustrates that production software has already confronted non-strict preference input as a practical problem (Raftery et al., 2021).

The same paper discusses active versus exhausted ballots and quota behavior. By default, exhausted ballots reduce the active-ballot total, so the quota declines across counts; alternatively, constant.quota = TRUE fixes the quota at its initial value (Raftery et al., 2021). The authors note that with constant quota, “the last candidate is often elected without reaching the quota,” whereas this does not occur when the quota is reduced appropriately each count (Raftery et al., 2021).

5. Auditing AV elections and statistical verification

AV presents distinctive auditing challenges because correctness depends on the entire elimination order, not just on first-preference margins (Ek et al., 2023). “Adaptively Weighted Audits of Instant-Runoff Voting Elections: AWAIRE” develops a risk-limiting audit (RLA) method for IRV/AV that does not require cast vote records (CVRs), though it can use them when available (Ek et al., 2023).

The paper models each ballot as an ordering of a subset of candidates. Elimination proceeds through transfers and exhaustion, and different elimination orders can produce different winners (Ek et al., 2023). The global null hypothesis that the reported winner did not actually win is partitioned into disjoint hypotheses $t$9, each corresponding to an alternative elimination order (“alt-order”) yielding a different winner (Ek et al., 2023). To confirm the reported result, the audit must reject every alt-order.

The mathematical machinery is based on test supermartingales and Ville’s inequality. A nonnegative process $S_t$0 is a test supermartingale for $S_t$1 if $S_t$2 and

$S_t$3

under $S_t$4. Ville’s inequality gives

$S_t$5

which yields the standard stopping rule: reject when $S_t$6 (Ek et al., 2023).

For AV/IRV, the paper formulates round-level assertions as “Directly Beats” requirements $S_t$7, meaning candidate $S_t$8 has more votes than $S_t$9 assuming only candidates $M = \left\lfloor \frac{V}{2} \right\rfloor + 1,$0 remain standing (Ek et al., 2023). Each such requirement is represented through an assorter mapping ballots to $M = \left\lfloor \frac{V}{2} \right\rfloor + 1,$1 so that the requirement reduces to testing whether the assorter mean is below $M = \left\lfloor \frac{V}{2} \right\rfloor + 1,$2 (Ek et al., 2023). Base supermartingales for requirements are built using ALPHA; for requirement $M = \left\lfloor \frac{V}{2} \right\rfloor + 1,$3, the per-sample factor is

$M = \left\lfloor \frac{V}{2} \right\rfloor + 1,$4

with $M = \left\lfloor \frac{V}{2} \right\rfloor + 1,$5 and $M = \left\lfloor \frac{V}{2} \right\rfloor + 1,$6 (Ek et al., 2023).

AWAIRE’s key innovation is an adaptively weighted intersection supermartingale over the requirements for each alt-order: $M = \left\lfloor \frac{V}{2} \right\rfloor + 1,$7 where weights are nonnegative and predictable (Ek et al., 2023). The paper studies linear, quadratic, and Largest weighting, with Largest consistently yielding the smallest mean sample sizes in experiments (Ek et al., 2023).

Empirically, the method is evaluated on 71 NSW 2015 Legislative Assembly IRV contests with at most six candidates and approximately $M = \left\lfloor \frac{V}{2} \right\rfloor + 1,$8 ballots each, replicated over 1,000 random ballot permutations, with qualitative behavior similar across risk limits $M = \left\lfloor \frac{V}{2} \right\rfloor + 1,$9 and detailed results reported for $V$0 (Ek et al., 2023). The paper reports that without CVRs, Largest weighting typically requires at most about twice as many ballots as a fixed scheme that knows the best requirements in advance (Ek et al., 2023). With accurate CVRs, AWAIRE and RAIRE are essentially on par; with CVR errors, AWAIRE’s adaptivity often avoids full counts where RAIRE escalates (Ek et al., 2023).

A particularly notable claim is that when permutations changed the reported winner, AWAIRE always escalated correctly, while RAIRE incorrectly certified $V$1 of the time in the experiments reported there (Ek et al., 2023). The current open-source prototype handles up to six candidates, and the paper proposes “lazy” expansion to mitigate the $V$2 requirement growth with more candidates (Ek et al., 2023).

6. Comparative empirical evaluation and broader interpretations

AV also appears in simulation-based comparative work as a baseline ranked voting rule. “An algorithm for a fairer and better voting system” uses Instant-Runoff Voting, implemented with the pyrankvote Python package, as a reference method against which a proposed ensemble-like ranked system is evaluated (Grama, 2021). In that paper, ballots are generated from synthetic “voters” modeled as independent neural networks trained to predict a latent candidate-quality feature, with diversity induced by permanent blindness to random feature subsets (Grama, 2021). The simulations assume sincerity, independence, heterogeneity, decentralization, and no tactical voting (Grama, 2021).

The paper evaluates systems using three metrics: meanWinnerRank, rateTrueWinners, and rateWinner<NULL (Grama, 2021). AV/IRV is compared with Preferential Block Voting, STV, First Past The Post, and the author’s proposed variants, as well as crowd-mean, crowd-median, and “bestVoter (Dictatorship)” benchmarks (Grama, 2021). Across many simulation blocks, the author reports that the proposed algorithm often outperforms IRV on selection-accuracy metrics under “wisdom of the crowds” conditions, though in some easier regimes IRV performs close to STV and PBV and sometimes nearly matches the best tuned variants (Grama, 2021).

The paper’s overall claim is that there is “convincing evidence” that the proposed method is better than IRV, PBV, STV, and FPTP when certain natural conditions supporting crowd wisdom hold (Grama, 2021). At the same time, the paper does not provide formal statistical tests, does not analyze tactical manipulation, and does not develop property-specific theory for IRV such as monotonicity or clone independence; IRV is treated there primarily as an empirical baseline (Grama, 2021).

What this comparison does show, however, is how AV is increasingly analyzed outside conventional political science. The paper explicitly frames voting systems as ensemble methods, stating that “Voting systems are not restricted to politics, they are ensemble methods for artificial intelligence” (Grama, 2021). This suggests a broader research interpretation of AV as a rank-aggregation protocol whose information usage can be contrasted with cumulative or approval-style alternatives. In that paper’s framing, IRV’s limitation is that it uses only current first-preference tallies and lower-ranked information enters only indirectly through transfers after elimination (Grama, 2021). That is a comparative claim about information flow, not a universal refutation of AV, but it aligns with why AV is a natural benchmark in rank-aggregation research.

7. Practice, context, and balanced assessment

AV is used extensively in Australia for single-winner contests, with two-candidate-preferred tallies routinely reported and tie resolution by countback, then by lot if needed (Morton, 21 Jul 2025). In software practice, AV is implemented as the single-winner instance of STV with configurable quota, tie-breaking, and treatment of equal rankings and exhausted ballots (Raftery et al., 2021). In formal theory, it is characterized as an elimination rule on strict rankings, generalized to weak orders by Approval-IRV, and studied through axioms such as clone independence, cohesive-majority respect, and indifference monotonicity (Delemazure et al., 2024). In election verification, it motivates sophisticated sequential testing because the correctness of the result depends on whole elimination orders rather than only final margins (Ek et al., 2023).

The strongest positive claims in the cited literature are therefore specific rather than absolute. AV guarantees an instant-runoff structure between substantial contenders, never elects a Condorcet loser, elects a Condorcet winner above the one-third support threshold, protects stable majority coalitions, and permits expressive rankings while making preferences below the dominant pair outcome-irrelevant (Morton, 21 Jul 2025). It is also operationally mature, with explicit software implementations and increasingly rigorous post-election audit methods (Raftery et al., 2021, Ek et al., 2023).

The main limitations are equally specific. AV can fail to elect a Condorcet winner below the one-third threshold, can be non-monotonic in carefully structured three-way contests, and remains computationally more difficult to audit and reason about than plurality because elimination-order contingencies matter (Morton, 21 Jul 2025, Ek et al., 2023). Standard AV also assumes strict rankings; allowing indifferences requires generalization, and not all such generalizations preserve AV’s characteristic axioms (Delemazure et al., 2024).

Taken together, the recent literature presents AV not as a universally dominant single-winner rule, but as a distinctive transferable-vote mechanism that operationalizes majority runoff under ranked ballots, foregrounds first-choice support while retaining transfer-based compromise, and occupies a central position in current research on electoral axioms, algorithmic implementation, auditability, and expressive preference representation (Morton, 21 Jul 2025, Delemazure et al., 2024, Ek et al., 2023).