Single Transferable Vote (STV) Overview

• Single Transferable Vote (STV) is a preferential electoral system where voters rank candidates and winners are determined using quotas, surplus transfers, and eliminations.
• STV procedures vary by jurisdiction, with differences in surplus distribution, transfer values, rounding, and exhaustion rules that affect electoral outcomes.
• STV’s path-dependent nature, potential monotonicity failures, and auditing complexities pose practical challenges for computation, verification, and strategic behavior.

Single Transferable Vote (STV) is a preferential electoral system for choosing multiple winners from ranked ballots. In STV, voters rank candidates, candidates are elected when they attain a quota, surpluses are transferred, and otherwise the lowest candidate is excluded and their ballots move onward. In the single-winner case, the same transfer logic is also called instant runoff voting (IRV), ranked choice voting (RCV), or the alternative vote (AV) (Conway et al., 16 Sep 2025, Raftery et al., 2021). Contemporary research treats STV less as one canonical rule than as a family of related procedures whose concrete behavior depends on jurisdiction-specific definitions of quota, surplus transfer, ballot exhaustion, rounding, and tie-handling (Conway et al., 2024).

1. Core counting model

A standard formalization takes a multiwinner election with candidate set $C$, ballots $B$, and committee size $k$, with each voter submitting a strict ranking of some subset of candidates (McCune et al., 2024). Another auditing-oriented formalization writes an election as $E=(C,B,Q,N)$, where $Q$ is the quota and $N$ the number of seats (Blom et al., 19 Mar 2025). In both presentations, each ballot begins by counting for its highest-ranked continuing candidate.

The quota is usually the Droop quota. In the Scottish formulation this is

$q=\left\lfloor \frac{n}{k+1}\right\rfloor +1,$

and in audit work the equivalent notation is

$Q=\left\lfloor \frac{|B|}{N+1} \right\rfloor +1.$

The purpose of the quota is to ensure that no more candidates than seats can simultaneously attain it (McCune et al., 2024, Blom et al., 19 Mar 2025).

The counting process proceeds in rounds. If a candidate has at least quota, that candidate is elected. If enough candidates are elected, the process stops; if the number of continuing candidates equals the number of remaining vacancies, all continuing candidates are declared elected. Otherwise, surpluses are distributed. If no candidate has quota, the lowest-tally candidate is eliminated and their ballots transfer to the next eligible preference (Conway et al., 16 Sep 2025, McCune et al., 2024).

In the Scottish fractional-transfer formulation, if candidate $C_i$ has $x_i \ge q$ current votes and candidate $B$0 is next available on $B$1 of those ballots, then $B$2 receives

$B$3

votes from $B$4’s surplus (McCune et al., 2024). Under exclusion, by contrast, ballots transfer at current value rather than at a reduced surplus value (Conway et al., 16 Sep 2025). This creates the characteristic STV distinction between elimination transfers, which move whatever value remains, and surplus transfers, which preserve only the portion not retained to hold an elected candidate at quota.

Research papers repeatedly emphasize that STV is path-dependent. Later tallies depend not only on first preferences, but on the full sequence of elections, exclusions, transfer values, and exhausted ballots. That dependence is central to both its representational claims and its computational difficulty in auditing, margin computation, and counterfactual analysis (Blom et al., 2017, Blom et al., 24 Jan 2025).

2. STV as a family of implemented rules

Recent work on Australia makes explicit that “STV” is not a single legislated procedure. Australian variants differ especially in surplus distribution, transfer-value calculation, parcel definition, rounding, and the treatment of exhausted ballots (Conway et al., 2024). The ACT uses a last-parcel rule; NSW local government may use “the last few sets of most recently received votes”; NSW may also choose surplus ballots randomly; Federal and Victoria assign transferred ballots the same outgoing transfer value; and almost everyone rounds down to integers, while ACT from 2020 onward rounds down to six decimal places (Conway et al., 2024).

The Australian Senate variant analyzed by Blom defines the transfer value for a candidate $B$5 elected in round $B$6 as

$B$7

where $B$8 is the candidate’s tally and $B$9 the number of ballots in the pile (Blom, 2024). Weighted Inclusive Gregory audit work instead describes surplus transfer by multiplying each ballot’s current value by surplus divided by tally (Blom et al., 19 Mar 2025). These are not merely notational differences; they encode materially different ballot-value dynamics.

Scottish local government elections use a fractional-transfer STV that is also the benchmark in several empirical studies, including work on spoiler effects, monotonicity, exhaustion, and audits (McCune et al., 2024, McCune et al., 12 May 2026). In proportionality studies, Scottish STV is often compared with Meek-STV, the Expanding Approvals Rule, Sequential Ranked-Choice Voting, and SNTV (Bardal et al., 1 May 2025).

Software implementations also reveal the plurality of STV variants. The R package vote implements standard STV, STV with equal preferences, and a reserved-seats STV variant in which a minimum number of winners from a marked group is required (Raftery et al., 2021). In the equal-preferences version, a voter’s support is split equally among tied top-ranked remaining candidates; in the reserved-seats version, elimination can be altered so that marked candidates are retained to satisfy the group constraint (Raftery et al., 2021). This suggests that, even within software, STV is treated as a configurable family rather than a single invariant algorithm.

3. Ballot flow, exhaustion, and explainability

One of the central interpretive problems in STV is that it is often unclear to whom an individual vote “went.” Conway and Egan address this directly with a public demonstration system for recent Australian Federal Senate elections that lets a voter enter an example ranked ballot and see both “how that vote would have been transferred between candidates” and “how much that vote would have contributed to the tallies of relevant candidates, across rounds of tabulation” (Conway et al., 16 Sep 2025). Their motivating point is that in Australian STV a ballot can begin with one candidate, later move at reduced value to another, skip over candidates who have already been elected or excluded, and ultimately help elect someone far down the ranking.

The system’s Victorian 2025 example illustrates the underlying accounting. A Greens-first ballot remains with Steph HODGINS-MAY until count 259, then transfers to Fiona Patten at value $k$0 because $k$1 of the ballot has already been used to elect the Greens candidate; when Fiona Patten is excluded at count 284, the same residual $k$2 transfers to Michelle ANANDA-RAJAH and helps elect her (Conway et al., 16 Sep 2025). The paper’s point is that the relevant object is not merely an ordinal path through preferences but a sequence of candidate assignments annotated by counts and fractional ballot values.

Work on exhaustion in Scottish local elections refines the same theme. McCune and Naber distinguish exhausted ballots, non-first-choice exhausted ballots, unrepresented exhausted ballots, and weight exhaustion (McCune et al., 12 May 2026). In their 1,070-election dataset, $k$3 of ballots are exhausted by the final round, but the weight exhaustion rate is only $k$4; $k$5 of ballots are non-first-choice exhausted and $k$6 are unrepresented exhausted (McCune et al., 12 May 2026). They therefore argue that raw exhaustion rates alone “substantially overstate the extent to which voters lose influence or fail to obtain representation under STV” (McCune et al., 12 May 2026).

The same study shows that most exhaustion occurs late in the count and often after a ballot has already contributed to electing someone. It also reports that $k$7 of seats are won by candidates who have not achieved quota when the authors’ STV algorithm terminates, and that even if the count is extended until only $k$8 candidates remain for $k$9 seats, $E=(C,B,Q,N)$0 of seats—$E=(C,B,Q,N)$1 of $E=(C,B,Q,N)$2—are still filled by candidates who do not reach quota (McCune et al., 12 May 2026). This sharpens the distinction between ceasing to transfer, losing remaining weight, and failing to obtain representation.

4. Proportionality, coalition guarantees, and committee behavior

STV is widely described as a proportional rule, but recent work distinguishes several senses in which that claim may hold. A central axiomatic account is proportionality for solid coalitions (PSC). In the formulation used by Lackner, Nardi, and Skowron, a committee $E=(C,B,Q,N)$3 satisfies PSC if for any $E=(C,B,Q,N)$4 and any $E=(C,B,Q,N)$5-large group $E=(C,B,Q,N)$6 forming a solid coalition over $E=(C,B,Q,N)$7, it holds that

$E=(C,B,Q,N)$8

On this basis, STV is “generally considered proportional because it satisfies PSC” (Bardal et al., 1 May 2025).

Morton gives a closely related coalition theorem: if, in a STV election with $E=(C,B,Q,N)$9 winners, $Q$0 ballots, and quota $Q$1, there is a subset $Q$2 of candidates such that $Q$3 voters rank all candidates in $Q$4 ahead of all candidates outside $Q$5, then STV elects at least $Q$6 from $Q$7, provided $Q$8 contains enough candidates; if $Q$9 contains fewer than $N$0 candidates, STV elects all candidates in $N$1 (Morton, 21 Jul 2025). The same paper argues that under strict party allegiance STV behaves “in essentially the same manner” as a largest-remainder list system, differing mainly in how final residual seats are decided (Morton, 21 Jul 2025).

Empirical work complicates the axiomatic picture. Using Scottish local-election data, Lackner, Nardi, and Skowron show that solid coalitions of the size required by PSC are often absent: in $N$2 elections ($N$3), every committee of size $N$4 satisfies PSC; in $N$5 elections ($N$6), there is only one solid coalition that earns a single seat; and only $N$7 elections ($N$8) contain multiple solid coalitions deserving seats under PSC (Bardal et al., 1 May 2025). They therefore argue that binary axioms alone can be empirically toothless, and propose quantitative versions of PSC, LS, EJR$N$9, and priceability to assess proportionality in practice (Bardal et al., 1 May 2025).

Comparative work also distinguishes STV from other multiwinner ranked rules. McCune and Graham-Squire argue that Sequential Ranked-Choice Voting is “best understood as an excellence-based method” rather than a proportional one, whereas STV is “a proportional representation method by design” (McCune et al., 2023). On Scottish data, when the two rules disagree, STV more often yields broader party representation: in $q=\left\lfloor \frac{n}{k+1}\right\rfloor +1,$0 elections STV results in more parties represented than sequential RCV, in $q=\left\lfloor \frac{n}{k+1}\right\rfloor +1,$1 elections STV yields two more represented parties, and there are no elections in which sequential RCV yields more parties than STV (McCune et al., 2023).

At the same time, STV is not immunity-based in the manner sometimes claimed. In Scottish empirical data its spoiler-effect rate on actual ballots is $q=\left\lfloor \frac{n}{k+1}\right\rfloor +1,$2, lower than Bloc and SNTV but not zero (McCune et al., 2024). The same paper concludes that STV is “not spoiler-proof in any meaningful sense in the multiwinner context,” even though it performs well relative to many alternatives (McCune et al., 2024).

5. Monotonicity failures, no-show paradoxes, and idiosyncratic pathologies

A large recent literature documents that STV can fail multiple monotonicity-related criteria. In 1,079 Scottish local government STV/IRV elections, McCune and Graham-Squire found some kind of anomaly in 62 elections, including 61 multiwinner contests (McCune et al., 2023). Their breakdown reports $q=\left\lfloor \frac{n}{k+1}\right\rfloor +1,$3 committee-size anomalies, $q=\left\lfloor \frac{n}{k+1}\right\rfloor +1,$4 upward monotonicity anomalies, $q=\left\lfloor \frac{n}{k+1}\right\rfloor +1,$5 downward monotonicity anomalies, and $q=\left\lfloor \frac{n}{k+1}\right\rfloor +1,$6 no-show anomalies (McCune et al., 2023). They characterize these as the first documented monotonicity anomalies in real-world multiwinner elections for which complete preference data are available (McCune et al., 2023).

The no-show paradox also persists under structured preference domains. Imbriovis, Mandal, and Roy show that under one-dimensional preference models such as 1D-Euclidean, single-peaked, or single-crossing preferences, STV is “highly vulnerable” to the group no-show paradox, in sharp contrast to Condorcet rules, for which the paradox cannot occur on these domains (Mohsin, 11 Jun 2026). Their experiments show that voters at the extremes of the one-dimensional spectrum are particularly likely to cause the paradox by abstaining, and that the likelihood increases substantially as the number of alternatives grows (Mohsin, 11 Jun 2026).

Australian implementation studies reveal even more unusual phenomena. Conway and Egan show that, under some Australian rules, a candidate can in principle acquire a negative tally, ballots can increase in transfer value as they move through the count, and a candidate can be made to win by taking votes away from them at an earlier stage (Conway et al., 2024). For NSW local government, the crucial surplus fraction is

$q=\left\lfloor \frac{n}{k+1}\right\rfloor +1,$7

where $q=\left\lfloor \frac{n}{k+1}\right\rfloor +1,$8 is tally, $q=\left\lfloor \frac{n}{k+1}\right\rfloor +1,$9 quota, and $Q=\left\lfloor \frac{|B|}{N+1} \right\rfloor +1.$0 exhausted-value total; because $Q=\left\lfloor \frac{|B|}{N+1} \right\rfloor +1.$1 may be rounded while $Q=\left\lfloor \frac{|B|}{N+1} \right\rfloor +1.$2 is effectively exact, the denominator can become negative, yielding a negative transfer value (Conway et al., 2024). For the Australian Senate, they note a real example from 2016 Tasmania in which ballots received by J. Dunian at value $Q=\left\lfloor \frac{|B|}{N+1} \right\rfloor +1.$3 later transferred onward at value $Q=\left\lfloor \frac{|B|}{N+1} \right\rfloor +1.$4, so ballot value literally increased during tabulation (Conway et al., 2024).

Blom’s Australian Senate analysis supplies a related synthetic example: in a 5-seat contest with parties $Q=\left\lfloor \frac{|B|}{N+1} \right\rfloor +1.$5, $Q=\left\lfloor \frac{|B|}{N+1} \right\rfloor +1.$6, and $Q=\left\lfloor \frac{|B|}{N+1} \right\rfloor +1.$7, party $Q=\left\lfloor \frac{|B|}{N+1} \right\rfloor +1.$8 has 410 first-preference votes, more than four quotas when $Q=\left\lfloor \frac{|B|}{N+1} \right\rfloor +1.$9, yet $C_i$0 fails to win four seats because a set of 101 ballots that left candidate $C_i$1 at value $C_i$2 later leave $C_i$3 at value $C_i$4, pushing candidate $C_i$5 to quota instead of electing $C_i$6 (Blom, 2024). Such examples do not show that every STV implementation behaves this way, but they demonstrate that statutory details can generate pathologies beyond the familiar monotonicity critiques.

6. Auditing, verification, and victory margins

STV has become a substantial subject in election auditing and computational verification because its outcome depends on a long sequence of elections, exclusions, transfer values, and changing ballot weights. Blom, Stuckey, and Teague define the margin of victory (MOV) for an STV election as “the smallest number of vote manipulations required to ensure that a set of candidates $C_i$7 are elected to a seat” (Blom et al., 2017). They introduce the MINLP formulation DistanceTo$C_i$8 and the branch-and-bound algorithm margin-stv, obtaining exact margins in some small elections and lower or upper bounds in larger ones (Blom et al., 2017). Teague, Blom, and Stuckey later improve lower-bounding efficiency for STV margins by using transfer-path reasoning, improved elimination-quota lower bounds, a displacement lower bound, stronger MINLP integration, and structural dominance pruning (Blom et al., 24 Jan 2025).

Risk-limiting audits (RLAs) pose a distinct problem. An RLA, in the formulation used by Blom, Sanderson, and Teague, is a statistical procedure that examines paper ballots and guarantees that if the reported outcome is wrong, there is at least a prescribed minimum chance that the audit will escalate to a full hand count rather than incorrectly accept the outcome (Blom et al., 19 Mar 2025). Their 2025 work extends STV RLAs from the 2-seat case to 3+-seat contests under the first winner criterion and for the Weighted Inclusive Gregory Method, using an assertion-based SHANGRLA-style framework with assertions $C_i$9, $x_i \ge q$0, $x_i \ge q$1, $x_i \ge q$2, $x_i \ge q$3, and the new $x_i \ge q$4 (Blom et al., 19 Mar 2025).

Earlier 2-seat work introduced the basic assertion types. The first approach to RLAs for 2-seat STV elections defined $x_i \ge q$5 assertions verifying first-round quota, $x_i \ge q$6 assertions bounding transfer values from above, and $x_i \ge q$7 and $x_i \ge q$8 assertions proving that some candidates are always ahead of others (Blom et al., 2021). The later revisiting paper adds $x_i \ge q$9, a lower-transfer-value assertion, and argues that lower as well as upper transfer-value bounds can materially reduce audit cost by certifying how much transferred support must reach the second winner (Blom et al., 2024).

The empirical results show both progress and restriction. In 513 Scottish local-council contests satisfying the first winner criterion, the 3+-seat audit verified all winners in $B$00 of $B$01 three-seat contests ($B$02) and $B$03 of $B$04 four-seat contests ($B$05); it more often produced partial audits that verified most, but not all, winners (Blom et al., 19 Mar 2025). The 2024 revisit also explicitly reframes the older “general” 2-seat approach as a partial-audit mechanism when a full audit is unavailable (Blom et al., 2024). This suggests that for STV, unlike simpler methods, auditing is likely to remain a combination of exact verification, partial certification, and lower-bound reasoning.

7. Strategic behavior, implementation error, and non-electoral applications

The complexity of STV creates vulnerabilities not only in auditing but also in strategic behavior and implementation error. Conway, Bohan, and Teague show that “random errors are not necessarily politically neutral” in Australian Senate STV (Blom et al., 2020). The dominant mechanism is ballot invalidation: BTL ballots are much more vulnerable than ATL ballots because they require six consecutive correctly interpreted preferences, while ATL ballots require only one valid preference (Blom et al., 2020). In simulations over the 2016 and 2019 Senate elections, the only contest whose elected set changed for error rates between $B$06 and $B$07 was Tasmania 2016; under the pairwise digit-confusion model with average per-digit error rate $B$08, Kate McCulloch was elected in $B$09 of simulations and Nick McKim in only $B$10 (Blom et al., 2020). The paper’s conclusion is not that STV is generically unstable, but that close STV elections can be directionally biased by random error because ballot styles are not equally fragile.

Manipulation results point in a different direction. Walsh’s empirical study of single-winner STV finds that although STV was among the earliest rules shown NP-hard to manipulate, “in almost every election in our experiments, it was easy to compute how a single agent could manipulate the election or to prove that manipulation by a single agent was impossible” (Walsh, 2010). The paper studies the single-winner case only, but its broader implication is that worst-case complexity is a weak practical defense unless hard instances are common. In coalition settings for multi-carrier network path selection, by contrast, Mellia and colleagues find STV “largely more resistant to manipulability than the natural system which tries to get the economical optimum,” while still selecting paths close to the economical optimum (Durand et al., 2012). In their reference scenario, STV attains $B$11 sincere efficiency, $B$12 manipulability with $B$13, and $B$14 manipulated average efficiency, compared with range voting at $B$15, $B$16, and $B$17 respectively (Durand et al., 2012).

These results jointly suggest that STV’s practical profile is domain-dependent. In public elections, its complexity can create opacity, auditing difficulty, non-monotonicity, and sensitivity to formality rules or implementation details. In committee selection, network path choice, and other structured settings, the same transfer mechanism can be attractive precisely because it preserves minority representation, uses rich ordinal information, and avoids simple plurality-style exclusion.