RCV: Voting Methods, Fairness, and Computation

• Ranked Choice Voting (RCV) is an electoral system where voters rank candidates and winners are determined through iterative elimination based on vote transfers.
• RCV encompasses multiple methods like IRV for single-winner races and STV for proportional committee selection, each using specific quotas and transfer rules.
• Research on RCV highlights its robustness and fairness, with empirical studies showing outcome stability (e.g., over 96% winner agreement) when ballots rank at least three candidates.

Searching arXiv for recent and foundational papers on Ranked Choice Voting to ground the article in the cited literature. Ranked Choice Voting (RCV) denotes electoral systems in which voters rank candidates in order of preference rather than selecting only one candidate. In the literature, the term is used in two closely related ways. In many public-election settings it refers specifically to the single-winner elimination rule also called Instant Runoff Voting (IRV), the Alternative Vote, or the Hare method. In other work it denotes a broader family of ranked-ballot systems, including pairwise, positional, committee, and transfer-based rules such as Borda Count, Tideman, Chamberlin–Courant, Expanding Approvals Rule, and Single Transferable Vote (STV) (Graham-Squire et al., 2024, Onur et al., 2022).

1. Definitions, scope, and rule families

In the single-winner setting, RCV/IRV proceeds in rounds. Voters submit a ranked ballot; first-place votes are counted; if no candidate has a majority of active ballots, the candidate with the fewest first-place votes is eliminated; ballots for that candidate transfer to the next-ranked remaining candidate; and the process repeats until one candidate has a majority of the remaining ballots. One formalization defines, for remaining candidate set $R \subseteq C$,

$$\text{First}(c,R,P)=\#\{ i\in N : c \text{ is the highest-ranked candidate in } R \text{ on ballot } i \},$$

with the majority threshold in that round given by $\frac12\sum_{c\in R}\text{First}(c,R,P)$ (Graham-Squire et al., 2022).

In the multi-winner setting, ranked ballots are used to elect a committee of size $k$. The output is a committee $W(P,k)\subseteq C$ with $|W(P,k)|=k$. STV implements this through quotas, election of candidates who exceed quota, surplus transfers, and elimination of low candidates; Chamberlin–Courant instead maximizes aggregate voter satisfaction by assigning each voter to their most-preferred committee member; and the Expanding Approvals Rule (EAR) gradually expands an approval threshold until candidates pass quota (Graham-Squire et al., 2024). The vote package in R treats STV as the general multi-winner rule and notes that, when $m=1$, STV reduces exactly to IRV/RCV/AV (Raftery et al., 2021).

A second terminological distinction concerns whether RCV denotes a tally rule or a ballot format. ElectAnon, for example, treats RCV as the ballot format—complete rankings over candidates—while making the tallying layer algorithm-agnostic, with Borda Count and Tideman as pluggable libraries (Onur et al., 2022). This suggests that the contemporary literature contains both an IRV-centered usage tied to public election administration and a broader social-choice usage in which ranked ballots support multiple tally rules.

2. Ballot structure, truncation, and preference representation

A central empirical fact about ranked elections is that voters frequently cast partial rather than complete rankings. A truncated ballot ranks only a subset of candidates, such as “$A>B$” or “$A>C>D$,” leaving the rest unranked. One formal model treats a preference profile $P$ as a multiset of ballots that are linear orders over a subset of candidates, with unranked candidates tied at the bottom (Graham-Squire et al., 2024). In the Scottish dataset analyzed for multi-winner RCV, about 14% of ballots rank only one candidate, about 58% rank fewer than the number of seats $$\text{First}(c,R,P)=\#\{ i\in N : c \text{ is the highest-ranked candidate in } R \text{ on ballot } i \},$$0, and only 13.2% rank $$\text{First}(c,R,P)=\#\{ i\in N : c \text{ is the highest-ranked candidate in } R \text{ on ballot } i \},$$1 or $$\text{First}(c,R,P)=\#\{ i\in N : c \text{ is the highest-ranked candidate in } R \text{ on ballot } i \},$$2 candidates (Graham-Squire et al., 2024).

This matters both normatively and mathematically. The order-theoretic analysis in “Preferences on Ranked-Choice Ballots” formalizes top-truncated ballots as weak orders in which only the minimal elements are tied, and proves that a ranked-choice ballot is a finite join semilattice. In that framework, top-truncated orders are modular, ranked-choice ballots have a submodular representation, and the voting record can be rationalized by a strictly concave or strictly quasiconcave utility function (Duricy, 2023). The paper’s interpretation is that the tied unranked block is not missing data but part of the preference structure itself.

By contrast, some computational and cryptographic systems assume complete rankings. ElectAnon requires each ballot to be a permutation of all candidate IDs, encodes that permutation as a single integer $$\text{First}(c,R,P)=\#\{ i\in N : c \text{ is the highest-ranked candidate in } R \text{ on ballot } i \},$$3, and then tallies under Borda Count or Tideman after a commit–reveal process backed by zero-knowledge proofs (Onur et al., 2022). The contrast between such complete-ranking protocols and the empirical prevalence of truncation in public elections is one of the recurrent fault lines in current RCV research.

Truncation also interacts directly with ballot exhaustion. In single-winner IRV, if all candidates ranked on a ballot have been eliminated, the ballot becomes exhausted and drops out of later rounds. In a dataset of 1171 real ranked-choice elections, artificial truncation levels were imposed to test outcome sensitivity. When the truncation level $$\text{First}(c,R,P)=\#\{ i\in N : c \text{ is the highest-ranked candidate in } R \text{ on ballot } i \},$$4 was 1, the winner agreed with the untruncated RCV winner in 76.1% of useable elections; at $$\text{First}(c,R,P)=\#\{ i\in N : c \text{ is the highest-ranked candidate in } R \text{ on ballot } i \},$$5, in 89.7%; at $$\text{First}(c,R,P)=\#\{ i\in N : c \text{ is the highest-ranked candidate in } R \text{ on ballot } i \},$$6, in 96.8%; at $$\text{First}(c,R,P)=\#\{ i\in N : c \text{ is the highest-ranked candidate in } R \text{ on ballot } i \},$$7, in 98.9%; and at $$\text{First}(c,R,P)=\#\{ i\in N : c \text{ is the highest-ranked candidate in } R \text{ on ballot } i \},$$8, in 99.7% (Dickerson et al., 2023). The general empirical finding is that if the truncation level is at least three, restricting the number of candidates that can be ranked rarely affects the election winner (Dickerson et al., 2023).

3. Normative properties, paradoxes, and failure modes

The standard evaluative vocabulary for RCV includes Condorcet consistency, monotonicity, participation, spoiler effects, and related criteria. A Condorcet winner is a candidate who defeats every rival in pairwise majority comparisons; a Condorcet loser loses to every rival. A monotone rule should not harm a candidate when that candidate is moved up in some voters’ rankings, and should not help a candidate when moved down (Weaver, 2023). IRV is known not to satisfy monotonicity, and the 3-candidate case admits an exact characterization. With candidates $$\text{First}(c,R,P)=\#\{ i\in N : c \text{ is the highest-ranked candidate in } R \text{ on ballot } i \},$$9, winner $\frac12\sum_{c\in R}\text{First}(c,R,P)$0, and $\frac12\sum_{c\in R}\text{First}(c,R,P)$1 last in the first round, upward monotonicity failure is possible if and only if

$\frac12\sum_{c\in R}\text{First}(c,R,P)$2

while downward monotonicity failure for $\frac12\sum_{c\in R}\text{First}(c,R,P)$3 is possible if and only if

$\frac12\sum_{c\in R}\text{First}(c,R,P)$4

with $\frac12\sum_{c\in R}\text{First}(c,R,P)$5 if $\frac12\sum_{c\in R}\text{First}(c,R,P)$6 is odd and $\frac12\sum_{c\in R}\text{First}(c,R,P)$7 if $\frac12\sum_{c\in R}\text{First}(c,R,P)$8 is even (Weaver, 2023).

A concrete illustration is the 2022 Alaska special election for the U.S. House. Under IRV, Mary Peltola won after Nick Begich was eliminated first, yet Begich was the Condorcet winner and Sarah Palin was the Condorcet loser. The same election also exhibited a monotonicity paradox and a no-show paradox: some Palin-only voters changing to $\frac12\sum_{c\in R}\text{First}(c,R,P)$9 would have caused Peltola to lose, and some $k$0 voters would have been better off abstaining because their participation caused Begich to lose to Peltola (Graham-Squire et al., 2022). Related analysis of the same election concluded that IRV selected a non-Condorcet winner, that Palin acted as a spoiler, and that Approval Voting and STAR might have produced different outcomes under explicit behavioral assumptions (Clelland, 2023).

Research has also asked whether IRV can elect a “least popular” candidate. In three-candidate elections, the answer depends on which weakness notion is used. One 2026 study considered four definitions: Borda loser, Bucklin loser, candidate with the most last-place votes, and candidate with minimum social utility. It found that IRV can select the weakest candidate under each definition, but that such outcomes are generally rare; across most models, the probability that IRV selects a given type of weakest candidate is at most 5%, with larger probabilities arising only when the electorate is extremely polarized (McCune et al., 25 Feb 2026).

This paradox literature has motivated comparison with other ranked methods. An evaluation of five Borda variations on 421 U.S. RCV elections concluded that these methods avoid monotonicity paradoxes, and that majority failures are rare or nonexistent depending on the variation, but that truncation and compromise failures occur more frequently than under IRV as a trade-off for avoiding monotonicity paradoxes (Fox et al., 2024). A plausible implication is that the choice among ranked methods is not about eliminating pathology altogether, but about which failure modes are being minimized.

4. Multi-winner RCV, proportionality, and fairness under truncated ballots

Multi-winner RCV introduces an additional objective absent from single-winner IRV: proportional representation. STV and related committee rules are often justified through proportionality for solid coalitions (PSC), under which sufficiently large cohesive groups should obtain a proportional number of seats (Graham-Squire et al., 2024). The complication studied in “New fairness criteria for truncated ballots in multi-winner ranked-choice elections” is that PSC-type analyses usually assume complete rankings even though real ballots are often heavily truncated.

That paper introduces two voter-bloc independence criteria. Let $k$1 be the winning committee and $k$2 the losers. Independence of Losing Voter Blocs (ILVB) requires that removing ballots that rank only losing candidates should leave the committee unchanged: $k$3 Independence of Winning Voter Blocs (IWVB) requires that if ballots ranking only some winners $k$4 are removed, then each candidate in that subset who previously won should continue to win. A stricter version, $k$5, requires that if all candidates in the subset $k$6 still win after those ballots are removed, then the entire committee must remain unchanged (Graham-Squire et al., 2024).

The comparative results are sharp. Chamberlin–Courant under both optimistic and pessimistic partial-ballot models satisfies ILVB and $k$7. EAR admits worst-case ILVB, IWVB, and $k$8 violations as severe as possible. Scottish STV and Meek STV also admit worst-case violations, including cases where removing ballots that rank only losers or only winners changes the rest of the committee completely. In a heuristic search over 1070 Scottish multi-winner wards with parameters $k$9 and $W(P,k)\subseteq C$0, ILVB violations were found in 40 Scottish STV elections, 19 Meek STV elections, 54 EAR elections, and 0 CC OM/PM elections; $W(P,k)\subseteq C$1 violations were found in 104 Scottish STV elections, 103 Meek STV elections, 181 EAR elections, and 0 CC OM/PM elections (Graham-Squire et al., 2024).

The paper therefore places the main multi-winner rules on a spectrum: Chamberlin–Courant performs best with respect to the new truncated-ballot fairness criteria, EAR performs worst, and STV lies in between (Graham-Squire et al., 2024). At the same time, Chamberlin–Courant is not generally PSC-proportional, whereas STV and EAR are designed to satisfy PSC-type axioms. Proposition 5.1 of the same paper shows that if $W(P,k)\subseteq C$2, then any $W(P,k)\subseteq C$3-PSC scoring rule violates ILVB (Graham-Squire et al., 2024). This suggests a structural tension between strong proportionality guarantees and robustness to truncated-ballot voter-bloc independence.

5. Computation, software, cryptography, and probabilistic inference

A substantial part of RCV research now concerns computation rather than only axioms. The vote package in R implements plurality, two-round runoff, score, approval, STV, and methods for selecting the Condorcet winner and loser, while emphasizing STV as a practical rule for small-electorate multi-winner elections such as committees, councils, and multiple-job-candidate selection (Raftery et al., 2021). It also implements STV with equal preferences and a variant with reserved seats for specified groups. This makes it possible to study single-winner IRV as the $W(P,k)\subseteq C$4 special case of a more general transfer system (Raftery et al., 2021).

Secure and scalable implementations form a second strand. ElectAnon encodes each complete ranked ballot as a permutation rank, commits to it through a hash, and uses Semaphore-based zero-knowledge proofs to guarantee eligibility, uniqueness, and anonymity before reveal-time tallying (Onur et al., 2022). The protocol is algorithm-agnostic at the tally stage, currently supporting Borda Count and Tideman, and reports gas-consumption reductions of up to 89% compared with prior works (Onur et al., 2022). Here RCV is treated as a ranked-ballot substrate rather than as IRV specifically.

A third strand concerns probabilistic forecasting. “Ahead of the Count” introduces an algorithm for probabilistic prediction of IRV elections that takes discrete probability distributions over ranking counts, enumerates all possible elimination sequences, and computes the probability that each candidate wins (Kapoor et al., 2024). The method is effective for elections with five or fewer candidates and is intended for real-time election-night modeling and recount prediction. A complementary Bayesian approach for RCV polling argues that the usual plurality-style margin of error is not straightforward to define for RCV because winning is not a function of a single population parameter. It instead models the electorate’s ballot-type distribution with a Dirichlet–multinomial framework and reports posterior win probabilities for each candidate, illustrated with the 2021 New York City Democratic mayoral primary and the 2022 Alaska special election (Rosenman et al., 30 Jun 2026).

These computational developments extend the meaning of “RCV analysis.” The rule is no longer studied only as a mapping from ballots to winners, but also as a forecasting target, a software object, and a cryptographic protocol.

6. Empirical dynamics, robustness, and current research directions

Large-scale empirical work increasingly emphasizes that worst-case complexity need not imply opaque practical dynamics. “Simpler Than You Think: The Practical Dynamics of Ranked Choice Voting” studies 54 New York City 2021 Democratic primaries, 52 Alaska 2024 statewide elections, and 4 Portland 2024 multi-winner city council elections. Using a candidate-reduction and ballot-addition framework, it finds that after RCV adoption competitiveness increased substantially compared to prior plurality elections, with average margins of victory declining by 9.2 percentage points in NYC and 11.4 points in Alaska; that complex ballot-addition strategies are not more efficient than simple ones; and that ballot exhaustion altered outcomes in only 3 of 110 elections (Deshpande et al., 15 Feb 2026). The paper’s conclusion is that RCV’s practical dynamics are usually simple and transparent despite its theoretical intricacy.

At the level of electoral theory, one active frontier concerns robustness to random error. “Noise Stability of Ranked Choice Voting” conjectures that among fair voting methods with small influences satisfying the Condorcet Loser Criterion, Borda count is the ranked-choice method that best preserves election outcomes under randomly corrupted votes. The paper proves this for three candidates when the corrupted votes are nearly uncorrelated with the original votes, and develops a low-dimensionality result for optimal rules under the associated Gaussian program (Heilman, 2022). Another frontier concerns ideological moderation. Using CES-based one-dimensional spatial models, “Candidate Moderation under Instant Runoff and Condorcet Voting” finds that under more realistic models incorporating partial ballots and related behaviors, the differences between Condorcet methods and IRV largely disappear, implying that the moderating effect of Condorcet methods may not be nearly as strong as suggested by more theoretical models (McCune et al., 4 Mar 2026).

Across these empirical and theoretical literatures, a consistent pattern emerges. Ranked ballots create additional expressive capacity, but they also create dependence on transfer paths, truncation conventions, and counting architecture. Some analyses stress robustness: truncation levels of at least three rarely change single-winner outcomes (Dickerson et al., 2023), practical ballot-addition manipulations are usually simple (Deshpande et al., 15 Feb 2026), and Borda may be especially stable under noise (Heilman, 2022). Others stress trade-offs: IRV can violate monotonicity and participation (Weaver, 2023), STV-style proportional rules can be brittle with respect to truncated-ballot fairness (Graham-Squire et al., 2024), and even highly moderate methods can lose their theoretical advantage once realistic ballot behavior is introduced (McCune et al., 4 Mar 2026). A plausible synthesis is that RCV is best understood not as a single rule with a single normative profile, but as a family of ranked-ballot institutions whose practical behavior depends critically on how rankings are elicited, truncated, transferred, scored, and interpreted.