# Cardinal voting
**Cardinal voting** is a family of electoral systems in which each voter assigns a numerical score to every candidate on a shared scale, rather than ranking candidates against one another as ordinal (ranked) systems require. Because scores can be ordered to produce a ranking, possibly with ties, but a ranking does not specify score magnitudes, so a rated ballot can in principle carry more information than a ranked one, and the family includes approval voting, score voting, majority judgment, cumulative voting, and [STAR voting](https://www.edgepedia.org/star-voting).[1](https://electowiki.org/wiki/Cardinal_voting_systems)
| Key fact | Detail |
|---|---|
| Defining feature | Each voter assigns a score to each candidate on the same scale; scores can be ordered to produce a ranking, possibly with ties, while a ranking does not specify score magnitudes, so rated ballots can pass more information than ranked ones[1](https://electowiki.org/wiki/Cardinal_voting_systems) |
| Approval voting | Voters approve any subset of candidates; the candidate with the most approvals wins; invented in the 1970s[2](https://doi.org/10.1257/jep.9.1.3) |
| Score voting | Ballots assign grades from a finite set of numbers; the candidate with the largest average grade wins[3](https://plato.stanford.edu/entries/voting-methods/) |
| Majority judgment | Proposed by Michel Balinski and Rida Laraki (2007, 2010); the candidate with the largest median grade wins[3](https://plato.stanford.edu/entries/voting-methods/) |
| STAR voting | Voters score candidates 0 to 5, and the two highest-scoring candidates face an automatic runoff[4](https://www.eac.gov/sites/default/files/electionofficials/Final_Alternative_Voting_Methods_in_the_United_States_508.pdf) |
| Current U.S. use | Approval voting is used in Fargo, North Dakota, and St. Louis, Missouri, where 68% of voters approved Proposition D in 2020[4](https://www.eac.gov/sites/default/files/electionofficials/Final_Alternative_Voting_Methods_in_the_United_States_508.pdf) |
| Historical reach | Forty-one popes were elected from 1294 to 1621 under an approval-style system in which cardinals could approve candidates without limit[5](https://electionscience.org/education/early-history) |
## Definition and core idea
The dividing line between cardinal and ordinal voting is what the ballot records. An ordinal ballot records only the voter's relative ordering of candidates; a cardinal ballot records an independent judgment of each candidate on a common scale. Two consequences follow. First, cardinal ballots can express ties among candidates.[6](https://link.springer.com/chapter/10.1007/978-3-319-09810-4_7) Second, the electorate as a whole can express overall approval or overall disapproval of a candidate, which makes cardinal systems useful where the number of candidates to be elected is not fixed in advance, and sometimes where it is possible for no one to be elected.[6](https://link.springer.com/chapter/10.1007/978-3-319-09810-4_7)
**Approval voting sits at the simplest end of the spectrum.** Each citizen approves as many candidates as he or she wants, the winner is the candidate with the most approvals, and the rule fails ordinality because a citizen's decision problem is not captured by a single ranking.[7](https://scholar.harvard.edu/files/maskin/files/elections_and_strategic_voting_condorcet_and_borda_dasgupta_11.04.2019.pdf) It was invented in the 1970s and is the only method considered in one survey of vote-counting schemes that does not require voters to rank the candidates.[2](https://doi.org/10.1257/jep.9.1.3) In the taxonomy used by election-methods writers, approval voting and score voting are the "pure" cardinal methods, because the outcome depends only on the scores themselves; STAR voting and similar methods are "semi-cardinal," because a runoff stage is added on top of the scores.[1](https://electowiki.org/wiki/Cardinal_voting_systems)
## Major cardinal methods
**Approval voting** gives each voter a binary choice per candidate: approve or not. With three candidates A, B, and C, a voter can vote for A or against A (by approving both B and C); approving all three is equivalent to not voting, since it has no differential impact on the count.[2](https://doi.org/10.1257/jep.9.1.3) The winner is the candidate with the largest number of approvals.[7](https://scholar.harvard.edu/files/maskin/files/elections_and_strategic_voting_condorcet_and_borda_dasgupta_11.04.2019.pdf)
**Score voting**, also called range voting, uses ballots that assign grades drawn from a finite set of numbers to the candidates; a citizen might grade each candidate on a ten-point scale.[3](https://plato.stanford.edu/entries/voting-methods/) • [7](https://scholar.harvard.edu/files/maskin/files/elections_and_strategic_voting_condorcet_and_borda_dasgupta_11.04.2019.pdf) The candidate with the largest average grade is declared the winner.[3](https://plato.stanford.edu/entries/voting-methods/) Tallying is by sum or average.[1](https://electowiki.org/wiki/Cardinal_voting_systems)
**Majority judgment**, proposed by Balinski and Laraki in 2007 and 2010, uses the same kind of graded ballot but aggregates differently: the candidate with the largest median grade is declared the winner, with tie-breaking refinements.[3](https://plato.stanford.edu/entries/voting-methods/)
**Cumulative voting** looks similar to score voting but imposes a budget. Each voter must distribute a fixed number of points, for example ten, among the candidates, so the sum of grades assigned by each voter is the same; in score voting the sum is unconstrained.[3](https://plato.stanford.edu/entries/voting-methods/)
**STAR voting**, short for "Score Then Automatic Runoff," allows voters to score each candidate on a scale of 0 to 5, with 0 the least favorite and 5 the most favorite; the two highest-scoring candidates then face an automatic runoff.[4](https://www.eac.gov/sites/default/files/electionofficials/Final_Alternative_Voting_Methods_in_the_United_States_508.pdf) Per the U.S. Election Assistance Commission, STAR was not then used in any jurisdiction, though bills had been introduced in Oregon and Utah.[4](https://www.eac.gov/sites/default/files/electionofficials/Final_Alternative_Voting_Methods_in_the_United_States_508.pdf)
## Strategic voting and formal criteria
**Monotonicity and independence.** In pure cardinal voting, if any set of voters raises a candidate's score, that can help the candidate but cannot hurt him; a voter's score for candidate C does not affect the contest between A and B; and there is never an incentive for favorite betrayal, meaning a voter need never abandon a top choice to avoid a worse outcome. Many cardinal methods nevertheless fail later-no-harm, the criterion that adding lower-ranked preferences should not harm a higher one, because they use all ballot information at once.[1](https://electowiki.org/wiki/Cardinal_voting_systems)
**Normalization.** The characteristic tactical behavior under score voting is normalization: a voter rescales the rated ballot so that the highest-scored candidate receives the maximum allowed score and the lowest receives the minimum, preserving relative score differences. A ballot scoring A:3, B:2, C:1 on a 0-to-5 scale becomes A:5, B:2.5, C:0 after normalization.[1](https://electowiki.org/wiki/Cardinal_voting_systems)
**The universal paradox.** [Strategic voting](https://www.edgepedia.org/strategic-voting) is not unique to rated systems. [Ceteris paribus](https://www.edgepedia.org/ceteris-paribus), a voter may obtain a preferred outcome by voting strategically, that is, not according to true preferences, and all known voting procedures suffer from this paradox.[8](https://www.lse.ac.uk/cpnss/assets/documents/voting-power-and-procedures/workshops/2010/duBaffy2010-Felsenthal.pdf)
## By the numbers
Simulation figures circulate in the election-methods community for how cardinal methods perform at approximating the electorate's overall satisfaction. According to Electowiki, score voting has the lowest Bayesian Regret, a measure of how much voters collectively regret the outcome compared with the best achievable one, among all common single-winner election methods that have been tested, with STAR voting not included in such studies; the same source reports that STAR voting was found to have the highest Voter Satisfaction Efficiency rating overall.[1](https://electowiki.org/wiki/Cardinal_voting_systems) These figures should be read as community estimates rather than settled results.
## Practical use and adoption
**United States.** Approval voting is currently used in [Fargo, North Dakota](https://www.edgepedia.org/fargo-north-dakota), and St. Louis, Missouri.[4](https://www.eac.gov/sites/default/files/electionofficials/Final_Alternative_Voting_Methods_in_the_United_States_508.pdf) In 2020, 68% of St. Louis voters approved Proposition D, which implemented a nonpartisan approval-voting primary for city offices in which voters may approve any number of candidates and the top two by approvals advance to the general election; St. Louis first used the system in its 2021 elections and again in a 2022 special election.[4](https://www.eac.gov/sites/default/files/electionofficials/Final_Alternative_Voting_Methods_in_the_United_States_508.pdf) In the same period, voters in Seattle, Washington rejected a ballot measure to adopt approval voting in November 2022 while approving a ranked-choice-voting proposal in the same election.[4](https://www.eac.gov/sites/default/files/electionofficials/Final_Alternative_Voting_Methods_in_the_United_States_508.pdf)
**Cumulative voting** is used in dozens of U.S. jurisdictions, often adopted in response to Voting Rights Act challenges, and was used to elect the [Illinois House of Representatives](https://www.edgepedia.org/illinois-house-of-representatives) from 1870 to 1980.[4](https://www.eac.gov/sites/default/files/electionofficials/Final_Alternative_Voting_Methods_in_the_United_States_508.pdf) More broadly, more than 200 of the over 8,000 U.S. election jurisdictions use an alternative voting method for some or all elections.[4](https://www.eac.gov/sites/default/files/electionofficials/Final_Alternative_Voting_Methods_in_the_United_States_508.pdf)
**Professional associations.** [Approval voting](https://www.edgepedia.org/approval-voting) is the most frequently adopted alternative to plurality rule and is used by several professional associations.[2](https://doi.org/10.1257/jep.9.1.3)
## History
From the election of [Pope Celestine V](https://www.edgepedia.org/pope-celestine-v) in 1294 through 1621, forty-one popes were elected using a system of approval voting in which cardinals could approve candidates without limit, though in practice they were largely judicious with their approval votes, typically voting for only one.[5](https://electionscience.org/education/early-history)
The modern method dates to the 1970s, when approval voting was invented as a non-ranked alternative to plurality rule.[2](https://doi.org/10.1257/jep.9.1.3) Its formal analysis is attributed to Steven Brams and Peter Fishburn in 1978 and 1983.[8](https://www.lse.ac.uk/cpnss/assets/documents/voting-power-and-procedures/workshops/2010/duBaffy2010-Felsenthal.pdf) [Majority judgment](https://www.edgepedia.org/majority-judgment) was proposed by Balinski and Laraki in 2007 and elaborated in 2010.[3](https://plato.stanford.edu/entries/voting-methods/)
## References
1. [Cardinal voting systems, Electowiki](https://electowiki.org/wiki/Cardinal_voting_systems)
2. [An Introduction to Vote-Counting Schemes, Journal of Economic Perspectives](https://doi.org/10.1257/jep.9.1.3)
3. [Voting Methods, Stanford Encyclopedia of Philosophy](https://plato.stanford.edu/entries/voting-methods/)
4. [Alternative Voting Methods in the United States, U.S. Election Assistance Commission](https://www.eac.gov/sites/default/files/electionofficials/Final_Alternative_Voting_Methods_in_the_United_States_508.pdf)
5. [The Early History of Approval Voting, Center for Election Science](https://electionscience.org/education/early-history)
6. [Cardinal Systems, Springer book chapter](https://link.springer.com/chapter/10.1007/978-3-319-09810-4_7)
7. [Elections and Strategic Voting: Condorcet and Borda, Dasgupta & Maskin](https://scholar.harvard.edu/files/maskin/files/elections_and_strategic_voting_condorcet_and_borda_dasgupta_11.04.2019.pdf)
8. [Review of Paradoxes Afflicting Various Voting Procedures, Felsenthal, LSE](https://www.lse.ac.uk/cpnss/assets/documents/voting-power-and-procedures/workshops/2010/duBaffy2010-Felsenthal.pdf)
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*Topic: Encyclopedia › Society and history › Politics and government › Elections and representation › Electoral systems and principles › Voting systems › Cardinal and rated systems › Cardinal and rated voting systems (overview)*
*Initially written Sep 17, 2026 · Reviewed: Sep 19, 2026; Oct 25, 2026 · Edited: Sep 19, 2026; Oct 25, 2026 · Last review: Oct 25, 2026*
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