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"Cardinal voting" redirects here. For the voting system used by Cardinals to elect the pope, see Papal conclave § voting.
A joint Politics and Economics series
Social choice and electoral systems
Single-winner methods
Single vote – plurality methods

Condorcet methods

Positional voting

Cardinal voting

Proportional representation
Party-list

Quota-remainder methods

Approval-based committees

Fractional social choice

Semi-proportional representation

Mixed systems
By results of combination
By mechanism of combination

By ballot type

Paradoxes and pathologies
Spoiler effects

Pathological response

Strategic voting

Paradoxes of majority rule

Social and collective choice
Impossibility theorems

Positive results

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A theoretical ballot with the instructions "Rate each between negative ten and ten." There are five options, each one with a number corresponding to it. The numbers, from top to bottom, are seven, ten, negative three, zero, and ten.
On a rated ballot, the voter may rate each choice independently.
A theoretical ballot with the instructions "Vote for any number of options." Two choices are marked, three are not. There is no difference between the markings.
An approval voting ballot does not require ranking or exclusivity.

Rated, evaluative,[1][2] graded,[1] or cardinal voting rules are a class of voting methods that allow voters to state how strongly they support a candidate,[3] by giving each one a grade on a separate scale.[1]

The distribution of ratings for each candidate—i.e. the percentage of voters who assign them a particular score—is called their merit profile.[4] For example, if candidates are graded on a 4-point scale, one candidate's merit profile may be 25% on every possible rating (1, 2, 3, and 4), while a perfect candidate would have a merit profile where 100% of voters assign them a score of 4.

Since rated methods allow the voters to express how strongly they support a candidate, these methods are not covered by Arrow's impossibility theorem,[5] and their resistance to the spoiler effect becomes a more complex matter. Some rated methods are immune to the spoiler effect when every voter rates the candidates on an absolute scale, but they are not when the voters' rating scales change based on the candidates who are running.[6]

Variants

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A scan of a real ballot that was already marked, with instructions to mark each candidate from A to F, where A is best. Spaces left blank are considered as F. The options from top to bottom are Eleanor Roosevelt, graded C, Cesar Chavez, graded B, Walter Lum, graded C, John Hancock, graded F, Martin Luther King Jr, graded B, and Nancy Reagan, graded A.
A majority judgment ballot is based on grades like those used in schools.

There are several voting systems that allow independent ratings of each candidate, which allow them to be immune to the spoiler effect given certain types of voter behavior. For example:

  • Score voting systems, where the candidate with the highest average (or total[7]) rating wins.
  • Highest median rules, where the candidate with the highest median grade wins. The various highest median rules differ in their tie-breaking methods.

However, other rated voting methods have a spoiler effect no matter what scales the voters use:

  • Quadratic voting is unusual in that it is a cardinal voting system that does not allow independent scoring of candidates.
  • Cumulative voting could be classified as a cardinal rule with unconditional spoiler effects.
  • STAR (score then automatic runoff) is a hybrid of ranked and rated voting systems. It chooses the top 2 candidates by score voting, who then advance to a runoff round (where the candidate is elected by a simple plurality).
  • Any hybrid of a ranked and rated voting system that reduces to majority rule when only two candidates are running fails independence of irrelevant alternatives due to the Condorcet paradox.

In addition, there are many different proportional cardinal rules, often called approval-based committee rules.

Relationship to rankings

[edit]

Ratings ballots can be converted to ranked/preferential ballots, assuming equal ranks are allowed. For example:

Rating (0 to 99) Preference order
Candidate A 99 First
Candidate B 55 Second
Candidate C 20 Third
Candidate D 20 Third

Analysis

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Arrow's impossibility theorem does not apply to cardinal rules.

Psychological research has shown that cardinal ratings (on a numerical or Likert scale, for instance) convey more information than ordinal rankings in measuring human opinion.[11][12][13][14]

Cardinal methods can satisfy the Condorcet winner criterion, usually by combining cardinal voting with a first stage (as in Smith//Score).

Strategic voting

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Like all (deterministic, non-dictatorial, multicandidate) voting methods, rated methods are vulnerable to strategic voting, due to Gibbard's theorem.

Cardinal methods where voters give each candidate a number of points and the points are summed are called additive. Both range voting and cumulative voting are of this type. With a large number of voters, the strategic Myerson-Weber equilibria for such methods are the same as for methods where only extreme ballots are allowed.[15] In this setting, the optimal strategy for Range voting is the same as for approval voting, and the optimal strategy for cumulative voting is the same as for first-past-the-post. For approval voting (and thus Range voting), the optimal strategy involves approving (or rating maximum) everybody above a certain utility threshold, and not approving (or rating minimum) everybody below it.[16]

See also

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References

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Revisions and contributorsEdit on WikipediaRead on Wikipedia

Rated voting

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Rated voting encompasses a class of cardinal electoral systems in which voters assign explicit ratings—such as numerical scores or ordinal grades—to multiple candidates, with the winner determined by aggregating these ratings, typically via the highest average score or median evaluation. These methods contrast with ordinal systems like ranked-choice voting by capturing the intensity of preferences rather than mere orderings, aiming to produce outcomes more aligned with collective utilities under assumptions of voter rationality. Prominent variants include , where candidates receive scores within a fixed range (e.g., 0 to 10) and the one with the greatest mean prevails, and majority judgment, which selects the candidate whose median grade is highest, resolving ties by comparing the proportion of superior grades. Proponents contend that rated voting mitigates issues in traditional plurality and runoff systems, such as vote-splitting and the spoiler effect, by enabling sincere expression of support levels without fear of wasting votes on non-viable options. Theoretical analyses suggest it satisfies criteria like more robustly in certain models, though it remains susceptible to strategic inflation of scores to manipulate aggregates. Despite these attributes, rated voting has seen limited adoption in public elections, confined largely to experimental trials, organizational decisions, and niche applications like certain or municipal pilots in places such as , . Empirical studies, including parallel simulations with actual elections, indicate higher voter satisfaction and reduced no-show paradoxes compared to pairwise , but reveal challenges in scale interpretation and potential for polarized outcomes when ratings cluster at extremes. Ongoing debates center on its practical superiority over established systems, with causal evidence hampered by the scarcity of large-scale implementations and confounding factors in small trials.

Fundamentals

Definition and Core Principles

Rated voting, also referred to as or range voting, is an electoral method where voters evaluate each candidate by assigning a numerical score from a predefined discrete scale, such as 0 to 5 or 0 to 10, to indicate the level of support independent of other candidates. This approach enables the expression of preferences in cardinal terms, quantifying the intensity of approval or disapproval for multiple options simultaneously, rather than merely selecting one or ordering them. The system aggregates these scores to determine outcomes, typically selecting the candidate or candidates with the highest total or average score in single-winner contests. At its core, rated voting rests on the principle of aggregation, drawing from utilitarian frameworks where voter are treated as measurable utilities that can be summed or averaged to reflect collective welfare. Unlike ordinal voting systems, which capture only relative rankings and assume equal preference strength between adjacent ranks, rated voting allows differentiation in preference intensity—for instance, distinguishing a strong endorsement from a lukewarm one through score magnitude. This facilitates a more granular representation of voter sentiments, theoretically aligning outcomes closer to the mean voter utility by incorporating variance in support levels across the electorate. Tallying in rated voting commonly employs the mean score, calculated as the sum of individual scores divided by the number of voters, to identify winners, though variants may normalize for non-participation or use medians in specific contexts. In multi-winner scenarios, such as , scores can inform seat allocation by selecting the highest-scoring candidates until quotas are met, emphasizing aggregate support over pairwise comparisons. This method's foundation in direct score summation promotes outcomes that maximize expressed satisfaction, predicated on the assumption that higher scores reliably signal greater perceived value to voters.

Distinction from Other Voting Methods

Rated voting, or cardinal voting, fundamentally differs from —commonly known as first-past-the-post—where voters must select a single , with victory awarded to the one receiving the most votes regardless of support. This single-choice constraint compels voters into zero-sum decisions, often amplifying spoiler effects as votes for ideologically similar candidates fragment support for frontrunners. In contrast, rated voting enables assignment of independent scores (e.g., 0 to 10) to each , allowing voters to express varying degrees of support or opposition without allocating a fixed "one vote" quota, thus reducing the strategic pressure to abandon preferred but weaker options. Approval voting serves as a binary variant of cardinal evaluation, permitting voters to approve or reject multiple candidates without ordinal , with winners determined by total approvals received. Rated voting extends this by incorporating graduated scales rather than dichotomous judgments, capturing nuanced intensities—such as strong enthusiasm versus mild acceptability—that approval's yes/no framework cannot convey. This generalization aligns rated methods more closely with underlying voter utilities, facilitating aggregation of strengths over simplistic counts. Unlike ordinal systems such as ranked-choice voting, which elicit only relative orderings of candidates, rated voting quantifies support levels, circumventing limitations inherent to ranking-based aggregation. demonstrates that no ordinal voting procedure can simultaneously satisfy non-dictatorship, universal domain, , and without producing inconsistent or manipulative outcomes. By treating votes as measurable utilities rather than mere sequences, rated voting avoids these ordinal paradoxes, enabling potential utilitarian optimizations that ordinal methods inherently forego.

Historical Development

Pre-20th Century Precursors

In ancient Sparta, the assembly employed acclamation voting for selecting members of the Gerousia, the council of elders, where candidates were evaluated based on the intensity of shouts from the citizen body, effectively introducing a proto-scalar measure of support rather than mere binary approval. Plutarch recounts in his Life of Lycurgus (c. 75–100 AD) that after candidates were presented, the assembly's volume of acclamation determined ranking, with the loudest response indicating strongest collective preference; this process repeated until the required number were chosen, lasting from around the 8th century BC until Sparta's decline in the 4th century BC. While not a numerical rating system, this method captured graduated enthusiasm, distinguishing it from uniform voice votes and foreshadowing later quantified evaluations, though its reliance on auditory perception limited precision and susceptibility to manipulation. The provides another early analog in its Doge elections from 1268 to 1797, where a multi-stage process involving the Great Council incorporated elements of multi-candidate endorsement after nominations, allowing electors to express support across options in ways that some scholars interpret as limited-range scoring to mitigate factionalism. Historical analyses, drawing on Venetian , describe final rounds among 41 electors using qualified majorities that effectively weighted preferences on a coarse scale (e.g., approval thresholds akin to a 3-point range), preventing dominance by narrow pluralities over five centuries of continuous use. However, primary accounts emphasize selection and single-vote eliminations more than explicit grading, with scoring claims resting on interpretive reconstructions rather than unambiguous archival evidence of numerical ballots. Such practices remained exceptional, as pre-20th-century societies predominantly favored , , or consensus mechanisms over systematic rating, reflecting smaller-scale polities where personal knowledge obviated complex aggregation and quantified scales risked alienating communal unity. No widespread 19th-century electoral proposals for scoring emerged amid reforms focused on franchise expansion and secret ballots, underscoring rated voting's novelty in formal theory.

20th Century Formalization

Approval voting, a form of binary rated voting, was formally proposed by political scientists Steven J. Brams and Peter C. Fishburn in their 1978 paper published in the American Political Science Review. In this system, voters select all candidates they approve of, with the candidate receiving the most approvals declared the winner; this allows expression of support for multiple options without ranking, distinguishing it from ordinal methods by incorporating a simple cardinal (yes/no) evaluation. Brams and Fishburn argued that approval voting satisfies key criteria like monotonicity and more robustly than , while reducing incentives for strategic abstention from weaker preferences, based on axiomatic analysis of voter utilities. Parallel developments in during the mid-20th century explored broader cardinal aggregation, drawing from utilitarian . Following Kenneth Arrow's 1951 impossibility theorem, which focused on ordinal preferences, scholars like John C. Harsanyi advocated incorporating cardinal utilities for welfare judgments, suggesting aggregation via summation of individual utilities to reflect intensity of preferences rather than mere rankings. This utilitarian framework, rooted in Harsanyi's 1955 analysis of interpersonal utility comparisons, laid groundwork for score-like voting by implying that numerical ratings could yield more informative collective decisions than ordinal methods, though direct application to electoral scoring awaited later refinements. Rated voting concepts also emerged in non-electoral contexts, such as and , where multi-attribute scoring evaluated alternatives on scales. For instance, in the 1960s and 1970s, methods like those in multi-criteria decision analysis—pioneered by figures such as with (1977)—employed numerical ratings to aggregate judgments, influencing processes in academia where referees assigned scores to papers or grants. These applications demonstrated practical aggregation of ratings to rank options, prefiguring electoral uses by highlighting reduced strategic manipulation compared to pairwise comparisons.

21st Century Advocacy and Refinements

In the early 2000s, mathematician Warren D. Smith founded the Center for Range Voting in 2005 alongside Jan Kok, launching RangeVoting.org as a platform for advocating range voting, a cardinal rating system where voters score candidates on a scale such as 0-9. Smith's work emphasized simulations and theoretical analyses showing range voting's superiority in criteria like monotonicity, , and utilitarian efficiency compared to plurality or instant-runoff systems, with detailed tables contrasting outcomes across thousands of simulated elections. These efforts proliferated through online mathematical and election reform communities, including forums discussing voting theory, where Smith's datasets and proofs highlighted range voting's resistance to spoilers and vote-splitting absent in ordinal methods. The Center for Election Science, established in 2012, further advanced —a variant of range voting—in the 2010s by prioritizing computational simulations over sparse real-world data to model voter behavior and strategic incentives. Their analyses demonstrated 's potential to maximize social utility by allowing nuanced expressions of preference intensity, though they noted challenges like tactical exaggeration, leading to a partial pivot toward as a simpler proxy. This period saw increased advocacy via podcasts, AMAs, and policy briefs, framing as a reform to reduce polarization in first-past-the-post systems. Post-2015 refinements addressed empirical observations of scale effects, where voters' inconsistent use of rating ranges—such as compression toward extremes or middles—distorts aggregates. Proposals for normalized scoring, involving post-hoc adjustments like z-score or affine transformations to align individual ballots to a common scale, aimed to mitigate these biases while preserving ordinal information. Such tweaks, explored in academic papers and studies, enhanced robustness against heterogeneous voter , though critics argued they introduce arbitrary assumptions risking loss of raw preference data. These developments, disseminated via specialized websites and peer-reviewed outlets, underscored a shift toward hybrid cardinal systems blending ratings with safeguards against manipulation.

Variants and Implementations

Approval Voting

functions as the simplest form of rated voting, where each voter assigns a binary score—either approval (1) or disapproval (0)—to each , effectively selecting any number of acceptable options on the . The tallying process sums the approvals received by each , with the winner determined as the one garnering the highest total. This method contrasts with single-mark plurality by allowing expression of support for multiple candidates, thereby reducing vote-splitting among similar options. As a special case of range voting restricted to a 0-1 scale, generalizes the binary choice of traditional voting into a rudimentary rated system but limits nuance by omitting gradations of strength. Critics contend this binary structure fails to capture the intensity of voter preferences, potentially undervaluing candidates with broad but lukewarm support compared to those with fervent backing from fewer voters, a limitation addressed in expanded score variants. Municipal adoptions illustrate practical implementation: in , voters approved the system via ballot initiative on June 12, 2018, for city commission elections, utilizing it in cycles through 2022 before a state law enacted April 16, 2025, mandated reversion to plurality. Similarly, St. Louis, Missouri, enacted through Proposition D, passed November 3, 2020, applying it to primary elections including the March 2, 2021, mayoral contest where top candidates exceeded 40% approval rates. Professional organizations, such as certain mathematical societies, have employed for internal elections since the late , predating widespread civic use.

Score or Range Voting

Score voting, also known as range voting, requires voters to assign each candidate a numerical score within a predefined range, such as integers from 0 (least preferred) to 5 or 10 (most preferred), allowing independent evaluation without ranking constraints. The scores reflect the intensity of support, enabling voters to express approval for multiple candidates by assigning them high values while penalizing disliked ones with low or zero scores. To determine the winner, the total score for each is computed as the sum of ratings received from all voters, divided by the number of voters to yield the score; the with the highest score is elected. This averaging process equates to summing scores when no are blank for a , as the remains constant across candidates. Blank or unrated candidates typically receive a default score of zero, though some implementations treat abstentions differently to avoid penalizing lesser-known entrants. A variation known as normalized range voting scales each voter's scores proportionally to utilize the full range (e.g., if a voter assigns scores from 0 to 3 on a 0-5 scale, they are multiplied by 5/3), aiming to counteract strategic compression of ratings but introducing and potential bias toward extreme expressions. Equal-sum normalization, less common, constrains each voter's total scores across candidates to a fixed sum (e.g., 10 points to distribute), resembling point allocation systems and altering incentives by forcing trade-offs between candidates. In practice, score voting has seen limited adoption in formal elections but appears in student governance contexts, such as the 2007 elections in France, where voters rated candidates on a 0-20 scale for representative selection. It is also employed in online platforms for aggregating preferences, including some non-binding polls and rating systems for decisions like conference speakers or product reviews. Simulations conducted by Warren D. Smith in the 2000s, using probabilistic models of voter utilities drawn from multivariate normal distributions, demonstrate that maximizes expected social utility—defined as the sum of voters' true preference intensities for the winner—outperforming methods like plurality and instant-runoff by minimizing Bayesian regret, a measure of outcome inefficiency relative to the optimal candidate. These models, averaging over millions of simulated elections with 3 to 25 candidates and honest or assumptions, consistently show yielding 5-10% higher utility efficiency than alternatives.

Specialized and Hybrid Forms

Cumulative voting serves as a specialized application of rated principles in multi-winner contexts, where participants distribute a fixed pool of votes—equivalent to shares held multiplied by seats available—across candidates to emphasize support intensity. This allocation mechanism, akin to assigning differential points, enables minority interests to concentrate votes on fewer candidates, potentially securing representation disproportionate to raw shareholding. In U.S. corporate boards, gained prominence after the 1980s as a statutory option in states like , fostering board diversity by countering majority dominance in director elections. Hybrid forms integrate rated scoring with sequential or comparative stages to address limitations in standalone systems. exemplifies this by requiring voters to rate candidates on a 0-5 scale, advancing the top two scorers to an automatic runoff resolved by pairwise preference between them. Developed in the mid-2010s by advocates seeking to combine score expressiveness with runoff consensus, STAR reduces incentives for score inflation compared to pure range voting while avoiding exhaustive rankings. Graded multi-winner extensions adapt rated ballots for proportional outcomes, such as through reweighted mechanisms where initial scores inform iterative seat allocation, adjusting voter weights post-election to simulate proportionality. These systems, explored in theoretical electoral design since the early , prioritize cardinal utilities over ordinal preferences to better capture voter intensities in committee selections. Niche analogs appear in non-electoral domains, like fantasy sports leagues where participants rate player performances on scales to aggregate team rankings, mirroring rated aggregation without formal vote constraints.

Real-World Adoptions

Adoptions of rated voting systems remain scarce as of 2025, confined largely to the variant in select U.S. municipalities, with no implementations at the state or national level in sovereign elections. , where voters select all acceptable candidates and the one with the most approvals wins, has seen limited uptake due to legislative resistance and preference for traditional plurality or ranked systems. Other rated forms, such as score or range voting—which allow numerical scores across a scale—and majority judgment—which uses ordinal grades to find the highest —lack verified governmental use, appearing instead in experimental, organizational, or software-based contexts like online polls or private decision-making tools. In , voters approved a citizen-initiated measure for in city commission elections on June 12, 2018, with 67% support, leading to its first implementation in the June 9, 2020, municipal election. The system was used again in 2022 and 2024 elections, where candidates received varying approval percentages, such as leading contenders garnering over 50% in some races, though detailed satisfaction surveys post-implementation are limited. However, on April 16, 2025, Governor signed legislation banning in all statewide elections, effectively ending its use in Fargo beyond prior cycles. St. Louis, Missouri, adopted approval voting through Proposition D, approved by 65% of voters on November 5, 2019, instituting nonpartisan primaries where voters approve multiple candidates, advancing the top two to a general election. The method debuted in the March 2, 2021, primary, with mayoral frontrunners Tishaura Jones and Cara Spencer receiving 57% and 46% approval rates, respectively, reducing vote-splitting in multi-candidate fields. Subsequent elections, including 2025 analyses, show about 33% of voters approving more than one candidate, indicating partial multi-approval usage amid ongoing nonpartisan reforms. Unlike North Dakota, Missouri's 2024 ban targeted only ranked-choice voting, preserving approval's local application. Organizational and non-binding uses of occur in platforms like ElectionBuddy for internal decisions, where participants assign scores to options, but these do not extend to public elections. Similarly, majority judgment has been tested in contexts like the 2012 French presidential experiment and Eurovision analyses but has no routine electoral adoption. data from approval implementations, such as Fargo's, show no consistent dramatic increases over prior plurality systems, with municipal participation remaining context-dependent rather than system-driven. Overall, these cases highlight rated voting's niche role in addressing local vote-splitting without broader systemic shift.

Theoretical Analysis

Satisfaction of Standard Criteria

Rated voting methods satisfy the monotonicity criterion, wherein increasing support for a —through higher scores on ballots without reductions elsewhere—cannot cause that to lose an election they would otherwise win, as aggregate scores non-decreasingly reflect voter preferences. This holds logically for score and range variants, where scores are summed or averaged, ensuring consistent responsiveness to intensified preference. In contrast, rated voting fails the Gibbard-Satterthwaite theorem, which demonstrates that no non-dictatorial voting system for elections with three or more candidates and ordinal-like unrestricted preferences is fully strategy-proof, allowing scenarios where voters benefit by misreporting utilities to manipulate outcomes. Although cardinal inputs in rated voting provide expressive nuance beyond rankings, strategic exaggeration or compression of scores remains possible, undermining full . Rated voting does not satisfy independence of irrelevant alternatives (IIA) in strict theoretical terms, as adding a non-winning candidate can redistribute voter scores, altering relative standings among originals and potentially inverting winners. However, logical and simulation-based tests reveal superior performance over plurality, where spoiler effects dramatically violate IIA; rated systems exhibit muted disruptions in modeled electorates, preserving outcome stability more effectively. For multi-winner contexts, plain rated voting fails full proportionality, as score summation may overrepresent clustered preferences without mechanisms like reweighting, leading to disproportional seat allocation. It excels, nonetheless, in utilitarian efficiency, directly optimizing aggregate voter satisfaction by maximizing summed utilities, outperforming ordinal methods in expected welfare under assumptions. Analyses by Warren D. Smith in the 2000s, including criteria tables comparing systems across metrics like majority criterion adherence and Condorcet efficiency, position range voting—a core rated form—as superior to on more than ten benchmarks in simulated scenarios with realistic distributions. These evaluations underscore rated voting's strengths in empirical criterion satisfaction despite theoretical shortfalls.

Mathematical Foundations

Rated voting systems derive their aggregation rule from cardinal utility theory, where voter scores represent intensities of preference akin to von Neumann-Morgenstern utilities derived from preferences over lotteries. Under the von Neumann-Morgenstern axioms—totality, continuity, dominance, and —preferences can be represented by a utility function uu such that the utility of a lottery is the : u(p⋅a+(1−p)⋅b)=p⋅u(a)+(1−p)⋅u(b)u(p \cdot a + (1-p) \cdot b) = p \cdot u(a) + (1-p) \cdot u(b), unique up to positive affine transformations. In rated voting, candidates are treated as deterministic outcomes, with scores si(c)∈[0,1]s_i(c) \in [0,1] (or a bounded scale) serving as normalized utilities ui(c)u_i(c) for voter ii and candidate cc. Aggregation proceeds via the utilitarian , summing or averaging individual utilities to select the candidate maximizing collective welfare. The total score for candidate cc is S(c)=∑i=1nsi(c)S(c) = \sum_{i=1}^n s_i(c), and the winner is c∗=arg⁡max⁡cS(c)c^* = \arg\max_c S(c); since nn is fixed, this equates to maximizing the Sˉ(c)=1n∑i=1nsi(c)\bar{S}(c) = \frac{1}{n} \sum_{i=1}^n s_i(c). This represents the expected utility for an impartial observer behind a veil of ignorance, equally likely to be any voter, as formalized by Harsanyi's utilitarian theorem: social utility is the unweighted average of individual utilities, maximizing expected personal utility under identity uncertainty. Normalization to a common scale (e.g., [0,1]) ensures interpersonal comparability and invariance to affine rescaling, preventing strategic inflation of all scores without altering relative utilities. Probabilistic models extend this by interpreting aggregation as approximating social choice over lotteries. If scores reflect utilities over uncertain outcomes, the mean score aligns with linear expected utility maximization, avoiding ordinal restrictions. Unlike ordinal methods, which collapse preference strengths and thus fall under —no non-dictatorial rule satisfying unanimity, non-dictatorship, and —rated voting elicits cardinal intensities, enabling and utilitarian aggregation without dictatorial outcomes. This direct measurement circumvents ordinal paradoxes, such as cycles from incomparable intensities, by quantifying causal preference differences rather than mere rankings.

Strategic Vulnerabilities

In rated voting systems, strategic manipulation can manifest through insincere score assignments that deviate from a voter's true cardinal preferences to alter candidate averages. One tactic is bullet voting, where voters assign the maximum score (e.g., 10) to their most preferred candidate and the minimum (e.g., 0) to all others, effectively concentrating support and mimicking plurality outcomes while discarding information on relative intensities among non-favorites. Another approach involves equal-rating, where voters assign identical scores to multiple candidates to inflate or deflate group averages, potentially to block a frontrunner by diluting scores across rivals or to bolster a coalition. Exaggeration occurs when voters inflate scores beyond their felt intensities to amplify a favorite's relative standing, while compression—restricting scores to a narrow band—avoids overemphasizing differences but risks underutilizing the ballot's expressive range. Despite these possibilities, theoretical analyses indicate weaker incentives for such tactics in rated voting compared to ordinal systems like (IRV), where discrete ranking manipulations (e.g., burying a strong contender below a weaker preferred alternative) can pivot eliminations more decisively due to the sequential recount process. In rated systems, the continuous scoring allows nuanced expression of preferences, reducing the payoff for deviation since truthful ratings better capture voter utilities and aggregate more stably under varied preference distributions. Simulations from the , modeling multi-candidate elections with probabilistic voter utilities drawn from normal distributions, demonstrate that rated voting exhibits lower strategic vulnerability than IRV; in these models, coordinated insincere blocs achieve smaller utility gains (often under 5% relative to sincere equilibria) due to the averaging mechanism's resistance to outlier manipulations. Truthful scoring frequently emerges as a robust , particularly when voters face uncertainty about others' exact scores or turnout, as over-exaggeration risks backfiring if opponents reciprocate similarly, converging outcomes toward sincere aggregates. Claims of pervasive gaming in rated systems are thus overstated, as empirical game-theoretic models show dominating in iterated play under realistic asymmetries, unlike the sharper tactical pivots incentivized in IRV's elimination stages.

Empirical Evidence

Simulations and Modeling Results

Warren D. Smith's computational simulations, conducted from the early 2000s onward, demonstrate that range voting— a form of rated voting—achieves higher social utility than plurality or (IRV) in electorates with diverse voter preferences. Using the Voter Satisfaction Index (VSI), a metric approximating the expected utility of elected outcomes relative to the optimal candidate, Smith's models simulated millions of elections across varied scenarios. In one set with 5 candidates and 20 voters drawn from random utility distributions, range voting yielded a VSI of 96.71%, compared to 67.63% for plurality and 78.49% for IRV. A second set with 5 candidates and 50 voters, where utilities derived from positions on two ideological issues, produced VSIs of 94.66% for range voting, 62.29% for plurality, and 76.32% for IRV. These results stem from Monte Carlo-style enumerations testing 720 parameter combinations, including voter honesty levels and ignorance, favoring range voting's ability to aggregate cardinal preferences effectively in multidimensional preference spaces. Further modeling by Smith employs Bayesian regret (BR), the expected loss in social utility from electing a suboptimal candidate, to compare systems across honest and strategic voter behaviors. Range voting consistently exhibits the lowest BR among tested methods, including approval voting, plurality, and IRV, particularly in simulations with larger electorates and multiple candidates where preference diversity amplifies the pitfalls of ordinal systems. For instance, in scenarios with 200 voters and 5 candidates aligned on two issues, range voting minimizes regret by allowing full expression of intensities, reducing the incidence of spoilers—candidates who alter the winner without winning themselves—more effectively than IRV, which can exacerbate non-monotonicity in ranked ballots. Complementary Monte Carlo analyses confirm rated voting's robustness, with approval variants showing lower spoiler vulnerability under tactical conditions than ranked methods, though range extends this by quantifying gradations. These simulations assume underlying utility models and often sincere cardinal inputs as baselines, with extensions testing strategic deviations; however, outcomes remain sensitive to in voter utilities or incomplete , potentially inflating range voting's advantages if real-world cardinal data deviates from simulated distributions. Smith's models prioritize empirical proxies over axiomatic criteria, but critics note reliance on specific utility generators may not capture all causal dynamics in polarized or low-information settings.

Field Studies and Voter Behavior Data

In , approval voting—a binary form of rated voting where voters approve or disapprove multiple candidates—was implemented following a 2018 ballot initiative, with its first use in the 2020 municipal elections and subsequent application in 2022. Voter turnout in the 2020 election stood at approximately 22%, comparable to prior plurality elections, while approval percentages for winners ranged from 27% to 35%, reflecting broader participation without reported widespread voter confusion in initial post-election analyses. Administrative challenges emerged, including errors in tabulating exhausted ballots in 2020 and 2022, but these did not indicate systemic voter misunderstanding. The system was repealed by state legislation in April 2025, amid debates over its consistency with traditional methods, though pre-repeal data showed stable outcomes relative to plurality, with reduced vote splitting evident in higher approval shares for non-top candidates.) Field experiments on finer-scale rated voting, such as evaluative voting with grade assignments, provide limited insights into voter behavior. In a 2012 in-situ experiment parallel to the first round of the French presidential election, 2,340 participants assigned grades to candidates using scales from binary (0-1) to 21 levels (0-20); inconsistency rates—defined as ballots incompatible with a total ordering—ranged from 3.41% to 8.9% across sites, suggesting voters could apply grades coherently despite scale variations. Grade distributions shifted with scale granularity, with negative options (e.g., -1,0,1) lowering averages by up to 46% compared to positive-only scales, and small candidates receiving inflated scores on broader scales (80-300% uplift), indicating expressive but potentially strategic use. Participants expressed preference for longer scales enabling nuance, though the 21-point scale was deemed overly complex, with no direct measures of overall satisfaction or turnout impact. Data from organizational and small-scale implementations of score or range voting remain anecdotal and sparse, often from self-selected groups like professional societies or online communities favoring reform. Surveys in such contexts report subjective increases in voter utility or satisfaction, with some claiming 10-20% higher perceived expressiveness over plurality, but these lack peer-reviewed validation and suffer from selection bias toward reform-adopting entities. No large-scale public elections have employed continuous score voting, limiting generalizable evidence on turnout or long-term behavior; existing adopters tend to be in ideologically progressive or experimental settings, raising concerns about external validity.

Advantages

Expressive Power and Voter Satisfaction

Rated voting enables voters to express intensities by assigning numerical scores to candidates, capturing not only orderings but also the strength of support or opposition, which ordinal systems like plurality or ranked-choice omit. This cardinal input aligns with utility theory, where aggregating intensities better approximates social welfare by weighting according to their perceived value to individuals. Simulations modeling diverse voter preferences demonstrate that score voting variants achieve Voter Satisfaction Efficiency (VSE) scores approximately 15-20% higher than ; for instance, in clustered spatial models with five candidates, score methods yield 99.2% VSE compared to plurality's 83.6%. These results stem from reduced instances of suboptimal winners, as voters avoid vote-splitting by assigning partial scores to compromise candidates, thereby diminishing "wasted" votes that plague single-choice systems. Empirical preference diversity, evidenced by multi-candidate races where ordinal methods enforce strategic truncation or , finds better accommodation in rated systems, potentially alleviating two-party dominance by allowing weak support for alternatives without risking preferred outcomes. Small-scale mock ballot experiments and polls, while limited by familiarity biases favoring methods, report participants appreciating the ability to signal nuanced views, with post-trial satisfaction linked to fuller expression reducing regret over uncast preferences.

Reduction of Pathological Outcomes

Rated voting systems mitigate the spoiler effect prevalent in , where a minor candidate similar to a frontrunner draws sufficient votes to cause the frontrunner's defeat by a less preferred opponent. In plurality, voters must select only one option, forcing trade-offs that benefit dissimilar candidates; rated methods like score and allow positive ratings for multiple candidates, preserving support for preferred frontrunners even if a spoiler appears. Voters can assign high scores or approvals to both a major candidate and a similar third option without penalty, as totals reflect cumulative support rather than exclusive choices. A historical analog is the 2000 U.S. presidential election in , where garnered 97,488 votes—many from supporters who preferred over —contributing to Gore's 537-vote loss. Under , Nader voters could have approved both candidates, bolstering Gore's total against Bush without aiding the spoiler dynamic, as empirical models of voter preferences indicate such dual approvals align with revealed rankings. similarly avoids this by enabling granular scoring; adding a low-scoring spoiler does not redistribute votes from similars, maintaining relative strengths unless voters explicitly downgrade, which sincere cardinal preferences discourage. This reduction stems from rated voting's cardinal aggregation of utilities, which sums independent evaluations across candidates rather than pitting them in pairwise or exclusive contests that amplify splitting. Unlike plurality's for vote concentration on frontrunners, rated systems permit expression of full preference profiles, diminishing pathological inversions where irrelevant similars overturn majority will. Compliance with criteria like weak independence of clones—where similar candidates do not harm each other—further insulates outcomes from such distortions, as verified in theoretical analyses of score rules.

Criticisms and Limitations

Potential for Strategic Manipulation

In rated voting, voters face incentives to exaggerate scores by assigning maximum values to preferred candidates and minimum values to rivals, rather than providing nuanced ratings that reflect relative utilities. This strategy, known as range compression or polarization, can amplify the impact of a single , as extreme scores disproportionately influence totals compared to sincere intermediate ratings. For instance, a small of strategic voters can elevate a low-rated candidate's score sufficiently to overtake frontrunners, as demonstrated in theoretical examples where two manipulators shift outcomes against preferences. Critiques from organizations like FairVote, which advocate for ranked-choice alternatives, highlight this vulnerability, arguing that inconsistent rating scales across voters—such as one using a narrow 1-7 range and another a wide 0-10—further distorts aggregates and encourages tactical extremes over honest expression. Such analyses, often rooted in advocacy for competing systems, emphasize how strategic play may converge toward binary (max/min) scoring, effectively degrading rated voting to a less expressive form akin to . However, these concerns warrant given the source's institutional preference for rank-based methods, which themselves permit strategic or tactics. Simulation models incorporating mixtures of honest and strategic voters reveal that detectable strategic deviations arise in 10-20% of scenarios, particularly under assumptions of partial information and , though the overall loss from such behavior remains limited compared to baseline sincere voting. Game-theoretic analyses further indicate that truthful rating equilibria are viable in large electorates, where pivotal vote probabilities are low and uncertainty about others' ballots discourages aggressive manipulation, as deviations risk backfiring without guaranteed gains. Empirical proxies from experimental voting studies corroborate this resilience, showing rated systems elicit lower rates of insincere ballots than plurality under similar strategic pressures.

Challenges in Implementation and Comprehension

Implementing rated voting necessitates ballot designs that accommodate multiple ratings per candidate, imposing greater cognitive demands on voters than single-choice plurality systems and risking . demonstrates that extended sets on ballots elevate undervoting rates, as voters confronted with additional evaluations prior to a contest exhibit heightened or reliance on heuristics like straight-ticketing. In simulated and small-scale trials of score variants, partial on candidate ratings has approached 5% beyond plurality baselines, attributed to the effort of discerning nuanced scores across fields of 5-10 contenders. Transitioning to rated systems incurs infrastructural expenses, including modifications to voting machines for multi-entry capture and aggregation algorithms to compute scores, distinct from plurality's simple counts. While precise figures for remain sparse, analogous reforms have demanded $25-40 million in jurisdictions for hardware and programming updates, with rated tabulation potentially less iterative than ranked alternatives but still requiring validation against errors in score normalization. Incumbent political actors often oppose adoption, citing unproven disruptions to entrenched dynamics, as evidenced by voter rejections of propositions in in November 2022 despite advocacy for its simplicity. Selection of rating granularity sparks ongoing contention, balancing expressiveness against usability; finer scales (e.g., 0-99) enable precise preference differentiation, as tested in a 2004 U.S. presidential where granular inputs revealed subtleties obscured by coarser metrics, yet they amplify marking complexity and risk inconsistent voter application. Coarser options (e.g., 0-5) mitigate overload but compress information, prompting debates in evaluative voting studies where scale endpoints influence grade distributions, with negative scores carrying outsized symbolic weight that unevenly penalizes lesser-known candidates. , a binary-rated subset, exhibits minimal comprehension hurdles in adoptions like (2018 onward), where education emphasizes additive approvals over exclusionary choices, though full-score variants demand targeted instruction to prevent defaulting to extremes or uniformity.

Comparisons and Debates

Versus Plurality Voting

, or first-past-the-post, incentivizes strategic abandonment of third-party or independent due to the spoiler effect, where a preferred by a minority can inadvertently cause the election of the least-preferred option among major contenders by splitting votes from the more similar rival. This dynamic, formalized in , empirically correlates with the emergence and persistence of two-party systems in single-member districts, as observed in national elections across the and , where effective party numbers remain below 2.5 despite occasional third-party candidacies. Rated voting counters this by aggregating scores across all , enabling voters to express partial support without binary trade-offs; a third receives additive contributions to their total only from genuine preference intensity, rather than subtracting from a rival via vote normalization to one. In the 2000 U.S. presidential election in , Ralph Nader's 97,488 votes under plurality split primarily from Al Gore's base, contributing to George W. Bush's 537-vote margin, though ballot-level analysis indicates at least 40% of Nader voters would have shifted to Bush absent Nader, underscoring plurality's vulnerability to entry effects regardless. Under rated systems, simulations of similar preference distributions—drawing from spatial voter models—demonstrate reduced spoiler incidence, as voters assign high scores to both Gore and Nader while low to Bush, yielding Gore's victory via higher mean scores reflecting broader acceptability. Empirical tests of confirm plurality amplifies such distortions, with voters deviating from sincere preferences up to 30% more often than in cardinal methods like . Plurality's equal weighting of votes normalizes preferences, disregarding intensity differences—treating a mild equivalent to intense opposition—contrasting rated voting's summation, which approximates utilitarian welfare by maximizing total expressed utility across the electorate. Simulations under probabilistic spatial models reveal plurality elects candidates farther from the voter 15-25% more frequently than range voting, as the former rewards peak mobilization over broad appeal, fostering polarization; range, by averaging scores, selects moderates who minimize variance in voter utilities. Field data from experimental elections, such as those comparing score and plurality in controlled settings, show rated methods yield winners with 10-20% higher average satisfaction scores, as they capture nuanced trade-offs absent in plurality's ordinal .

Versus Ranked-Choice Voting

Rated voting, also known as score or range voting, differs fundamentally from ranked-choice voting (RCV), or (IRV), in that it elicits cardinal utilities—voters assign numerical scores to candidates reflecting intensities—whereas RCV relies on ordinal rankings that capture only relative order without magnitude. This cardinal approach enables rated voting to aggregate voter utilities more accurately, potentially electing candidates closer to the social optimum in spatial models, as demonstrated by simulations where range voting achieves a voter satisfaction index of 96.71% under honest voting, outperforming IRV's lower scores in comparable tests. Ordinal methods like RCV, by contrast, discard intensity information, leading to suboptimal outcomes when preferences vary in strength, such as in elections with broad consensus favorites versus narrow ones. Theoretical criteria further highlight rated voting's advantages. IRV violates the monotonicity criterion, where increasing support for a leading candidate can paradoxically cause its defeat through vote redistribution, as shown in constructed examples and real-world analyses. Rated voting satisfies monotonicity, as higher scores unambiguously improve a candidate's total, preserving intuitive fairness. Simulations by Warren D. Smith in the 2000s and 2010s, using probabilistic models of voter preferences, consistently favor range voting over IRV for criteria like from irrelevant alternatives and overall winner quality, with range electing the optimal candidate more frequently across thousands of trials. RCV's reliance on complete rankings introduces ballot exhaustion, where incomplete ballots are discarded in later rounds, reducing effective turnout; in Alaska's August 2022 U.S. special , approximately 6.2% of ballots exhausted by the final round, disenfranchising voters who did not rank all candidates. Rated voting mitigates this by allowing partial scoring without invalidation, ensuring all expressed preferences contribute fully. While RCV proponents argue exhaustion rates remain low (often under 10% in U.S. implementations), models indicate rated systems elicit more expressive and sincere ballots, avoiding such losses. On strategic manipulation, game-theoretic analyses reveal higher susceptibility in IRV, where voters may benefit from insincere rankings to manipulate eliminations—studies show more voters gain from in IRV than in plurality under sincere assumptions, with risks amplified by in others' rankings. Rated voting, while not immune, exhibits lower strategic incentives in simulations, as truthful scoring often dominates due to the aggregation of intensities, reducing the payoff for or tactics compared to RCV's ordinal distortions. These differences underscore rated voting's edge in capturing true voter utilities, though empirical field data remains limited for both systems.

Broader Electoral Impacts

Rated voting systems incentivize candidates to cultivate broad appeal by maximizing average scores across the electorate, which theoretical analyses suggest promotes ideological moderation and counters the extremism associated with plurality voting's enforcement of . Under plurality, voters concentrate support on frontrunners to avoid wasting votes, fostering two-party dominance and polarized platforms; rated voting mitigates this by allowing nuanced scoring without penalty for expressing full preferences, potentially enabling third-party viability through a "nursery effect" where smaller parties accumulate support without spoiling major contests. Debates persist regarding whether this broad-appeal dynamic risks excessive , suppressing niche ideologies in favor of consensus candidates, or instead empowers diverse parties by eliminating vote-splitting incentives inherent in ranked systems like RCV, which can exhibit center-squeeze effects. Simulations and historical analogs, such as score-like systems in ancient and that sustained multi-party competition without two-party decay, indicate rated voting supports multi-party equilibria while maintaining stability, though empirical data remains limited due to sparse implementations. Proponents argue this reduces systemic polarization by rewarding compromise over base mobilization, enhancing legitimacy in divided electorates as evidenced by higher perceived fairness in complex, polarized scenarios. As of October 2025, rated voting reforms have stalled amid dominance of ranked-choice voting initiatives in U.S. jurisdictions, with adoption confined to niche applications like Fargo's school board elections using related approval variants, yet advocacy grounded in simulations continues to highlight its potential for viable multi-party systems without RCV's drawbacks.

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