Cardinal voting systems

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Cardinal voting methods, aka evaluative, rated, graded, or range methods, are one of the major classes of voting. They are ones in which the voter can evaluate each candidate independently on the same scale to cast a Cardinal ballot. Unlike some ranked systems, a voter can give two candidates the same rating or not use some ratings at all if they desire, and skipped ratings can affect the result.

Cardinal voting is when each voter can assign a numerical score to each candidate. Strictly speaking, cardinal voting can pass more information than the ordinal (rank) voting. This can clearly be seen by the fact that a rank can be derived from a set of numbers provided there are more possible numbers than candidates. However, some information is not used in the sense that voters can't express maximally strong preferences between every pair of candidates when there are more than two candidates (see Rated pairwise preference ballot).

A distinction should be made between the "pure" cardinal methods Approval Voting and Score Voting, and "semi-cardinal" methods, such as STAR Voting and all other cardinal methods. Most of this article discusses the properties that pure cardinal methods pass.

In pure Cardinal voting, if any set of voters increase a candidate's score, it obviously can help him, but cannot hurt him. That is a restatement of monotonicity. It is a stricter requirement than Independence of Irrelevant Alternatives so it is satisfied as well. As such, a voter's score for candidate C in no way affects the battle between A vs. B. Hence, a voter can give their honest opinion of C without fear of a Wasted vote or hurting A. There is never incentive for favorite betrayal by giving a higher score to a candidate who is liked less.

While in all systems all votes are actually counted, there is a psychological effect to the feeling that the vote “does not count” in a Wasted vote situation. Pure cardinal voting is likely to maximize the number of people who vote for a candidate to become the representative. This is expected to have a knock-on effect of better acceptance of results and higher voter turnout.

Vote aggregation and tallying methods

Cardinal voting is called Score voting when a sum or average is used to tally votes to find the utilitarian winner. It is typical to use a sum. Averages will give a differing result in systems where there is a no opinion option for each candidate meaning that the average is done over a differing number of voters for each candidate.

In multi-member systems the aggregation method can be split into the winner selection and the ballot reweighting methods. Optimal systems, however, combine these.

Majority criterion

The median can also be used to aggregate a cardinal ballot in Majority judgment systems. The use of the median is intended to further diminish the effects of strategic voting. Majority judgment voting satisfies the majority criterion for rated ballots, stated as "if one candidate is preferred and max-scored by a majority (more than 50%) of voters, then that candidate must win". It should be noted that Instant-runoff voting satisfies the majority criterion, which is stronger. While it might sound like this is always a good requirement of a voting system, consider a polarized scenario where 51% prefer one candidate and hate the other while the remaining 49% is just the opposite. If there was a third candidate who 100% would be satisfied with they would not be elected in a system which satisfied the majority criterion (though they would be elected in a system which satisfied the Condorcet criterion if 4% or more of the majority expressed no preference between the consensus candidate and their favorite candidate, and these were the only major candidates). Satisfying the majority criterion reduces incentive for compromise and lowers Bayesian Regret.

Gradation and range

The range (scale) does not matter for aggregation by sum, average or median. This can be demonstrated by showing that there is always a mapping to the desired range which preserves the results. Simply put, voting in the range [0,1] or [0,100] or even [-42,7] is irrelevant. However, there could be psychological effect to the voter when voting.

However, the gradation or the number of choices within the range does matter. This is where cardinal voting gets its name, the cardinality of a set of numbers is a measure of the number of elements of the set. For cardinal voting to contain more information than ordinal voting, the number of gradations must be greater than the number of candidates. This is clear since this is the only way a clear ordering can be determined from a cardinal value. Further gradation would result in better discernment of the amount to which each candidate is preferred. However, it becomes increasingly difficult to determine by the voter how different ratings would translate into winning candidates. Score voting, Cardinal aggregated by sum, is unbiased relative to polarization if the gradation is sufficiently large.

The other extreme case of gradation is Approval Voting, for which the voter is given only a binary (yes/no) choice. This is then the same as the typical plurality voting system except more than one choice can be made. Plurality and some ordinal voting methods have natural pro-extremist polarization bias, conversely, Approval has pro-centrist bias. Political polarization is generally viewed as divisive and undesirable so forcing the electorate towards a moderate candidate should be in the general good. In addition, some cardinal proponents argue that all majoritarian systems are polarizing and are therefore not necessarily desirable.

Incentive to reach out to minority groups

It is worth noting why Approval Voting does not lead to a tyranny of a centrist majority situation. There is difference between a tendency towards a moderate or compromise candidate and a majority candidate. For example, if there is a small group in desire of representation then the candidates would gain approval if they could add the desires of this group to their platform. This means issues that are neutral to the centrist majority and highly relevant to a small group are important for candidates to understand. Additionally, if the overlap of votes is released then the candidates can study the results to determine which candidates represented an isolated group. For example, if there were a candidate who only received votes because of a particular issue, then all candidates would be wise to integrate this issue into their platform for the next election to be more competitive. However, a case can be made that candidates are incentivized to make promised to special interest groups which benefit the few a lot but do not hurt the majority enough for them to get mobilized. In many instances, like with tax code, this effect lowers the total prosperity of the society at large. This effect certainly exists in other systems and it has not been empirically shown that it is more problematic in Approval Voting.

Determining relative accuracy or utility between voting methods

Score voting has the lowest Bayesian Regret among all common single-winner election methods which have been tested. (STAR Voting has not been included in Bayesian Regret studies to date.) Bayesian Regret is a measure of how the second order consequences of using a system affects the population. It can be thought of as the quantifiable amount of “expected avoidable human unhappiness.” It draws its merit from utilitarianism which intends to optimize for the total amount across the population. This is opposed by the theory of majority rule which intends to optimize only for the majority.

Voter Satisfaction Efficiency (VSE) is a newer model which has been used to evaluate voting method utility. VSE is an inverse of Bayesian Regret, with higher scores representing better utility. STAR Voting was found to have the highest Voter Satisfaction Efficiency rating overall.

Single-member methods

Main article: Single-member district
Method Aggregation Gradation
Score voting Sum or average > 2
Approval voting Sum Binary
STAR voting Sum, then top two run-off > 2
Reciprocal Score Voting Sum > 2
Majority Judgment Median > 2
Chiastic Score Voting Highest intersection > 3
Majority Choice Approval Median Binary
Majority Approval Voting Median Binary

Multi-member methods

Bloc methods

Bloc methods find the candidate set with the most support or the most votes overall. The number of seats up for election is determined and the top candidates are elected to fill those seats.

  • Bloc Approval Voting: Each voter chooses (no ranking) as many candidates as desired. Only one vote is allowed per candidate. Voters may not vote more than once for any one candidate. Add all the votes. Elect the candidates with the most votes until all positions are filled.
  • Bloc Score Voting: Each voter scores all the candidates on a scale with three or more units. Starting the scale at zero is preferable. Add all the scores. Elect the candidates with the highest total score until all positions are filled.
  • Bloc STAR Voting: Each voter scores all the candidates on a scale from 0–5. All the scores are added and the two highest scoring candidates advance to an automatic runoff. The finalist who was preferred by (scored higher by) more voters wins the first seat. The next two highest scoring candidates then runoff, with the finalist preferred by more voters winning the next seat. This process continues until all positions are filled.

Sequential proportional methods

Sequential cardinal Systems elect winners one at a time in sequence using a candidate selection method and a reweighting mechanism. The single-winner version of the selection is applied to find the first winner, then a reweighting is applied before the selection of the next and subsequent winners. A reweighting is applied to either the ballot or the scores for the ballot itself. The purpose of the reweighting phase is to ensure that the Hare Quota Criterion is met to ensure proportional election outcomes.

System Gradation Selection Reweight PR Philosophy Party List Case
Reweighted Range Voting > 2 Sum Thiele Interpretation Highest averages method
Single distributed vote > 2 Sum Thiele Interpretation Highest averages method
Sequential proportional approval voting Binary Sum Thiele Interpretation Highest averages method
Sequentially Spent Score > 2 Sum Vote Unitarity Hamilton method
Allocated Score > 2 Sum Monroe interpretation Hamilton method
Sequential Monroe > 2 Highest Sum in a Hare Quota Monroe interpretation Hamilton method
Sequential Phragmen Binary Sum Phragmén interpretation ??
Sequential Ebert Binary Sum Phragmén interpretation ??

Optimal proportional methods

Optimal methods select all winners at once by optimizing a specific desirable metric for proportionality. First a "quality function" or desired outcome is determined, and then an algorithm is used to determine the winner set that best maximizes that outcome. In most systems this is done by trying every possible winner set rather than by more complex optimization algorithms. This makes such systems computationally expensive.


Many cardinal voting methods fail the later-no-harm criterion because they tend to use all of the information on a voter's ballot at once to find a consensus or Utilitarian winner. While later-no-harm is considered an important property by most supporters of STV, other election method supporters disagree: it has been considered to be "quite unreasonable" and "unpalatable", and in general too uncompromising.[1]

Many consider the majority criterion essential to a voting system or even democracy itself, and argue that cardinal methods' failure to pass this criterion is a flaw of these methods. Others believe utilitarianism is more democratic and representative, and argue that majoritarianism is just a low resolution version of utilitarianism.

Some proponents of majority rule argue that voters cannot express absolute utilities because there is no way to settle on a common utility scale. As a consequence, there may be multiple honest ballots, and even if the cardinal voting method in question passes independence of irrelevant alternatives, the voters may normalize to different scales if a candidate enters or exits the election. These proponents argue that the presence of irrelevant candidates may thus change the outcome of the election, even for methods nominally passing IIA. Balinski and Laraki argued that a common language is required to avoid the implications of Arrow's impossibility theorem, and designed Majority judgment to use scales based on such common language.[2]


Connection to majority rule

A score for a candidate can be thought of in the Pairwise counting context as "in a Head-to-head matchup between this candidate and a candidate I don't support at all, I would give (score/max score) votes to this candidate." For example, a voter who scores a candidate at 80% (i.e. 4 out of 5) would give such a candidate 80% or 0.8 votes to beat a candidate they completely oppose.

The connection between cardinal methods and Category:Majority rule-based voting methods can most clearly be seen when looking at a runoff where a scale of 0 to 1 (with decimal values allowed) is used, because in such situations, if every voter uses only the min or max scores and show all of their ranked preferences, then the pairwise preferred candidate will have the same points-based margin as they would a votes-based margin in an FPTP-based runoff. Also see the utility article for discussion on this.

Approval rating

The concept of an approval rating, sometimes just called approval, is often used to count the vote/score totals for each candidate. When the score scale being used only allows for nonnegative scores, then a candidate's approval rating is just the percentage of their number of votes/points divided by the maximally attainable votes/score any candidate can get. The maximally attainable votes/score is whatever total votes/score a candidate would get if all voters theoretically gave them the maximal support/score. So for example, if every voter gives a candidate a 5 on a scale from 0 to 10, they'd have a 50% approval rating. The notion of an approval rating makes it easier to see the connection between the vote/score totals in different rated methods, such as Approval voting and Score voting, and with different score scales.


Normalization refers to, on a rated ballot with more than two allowed scores, when a voter's ballot is modified such that the candidate they originally scored highest is given the highest allowed score, the candidate they originally scored lowest is given the lowest allowed score, and all other candidates's scores are kept in between so as to maintain the ratio of scores between each pair of candidates' scores (as best as possible). For example, on a scale from 0 to 5, with three candidates A, B, and C, if the voter scored them as A:3 B:2 C:1, then their normalized ballot would be A:5 B:3 (or B:4; technically it should be B:3.333, but that might not be allowed), and C:0.

It is argued that most voters in elections using rated ballots would normalize their ballots, as this maximizes their voting power in that election. This is in part because it maximizes their support for their favorite candidates and minimizes their support for their least favorite candidate. An argument to the contrary is that if, say, a voter hates all of the candidates but has only a slight preference for one of them, then giving full support to that candidate sends a signal that the candidate need not improve themselves in order to get more support from the voter, whereas giving that candidate a low score pressures them to do better in the future. In addition, voters with weak preferences who don't wish to overrule other voters who may have stronger preferences are unlikely to normalize, as in a ranked method they would often either not vote or rank many candidates equal to each other. A succinct way of summing it up is "if a voter doesn't give their favorite maximal support against their least favorite, then they don't deserve full voting power, or don't want it."

Normalization can also more broadly refer to when a voter's rated ballot is adjusted to fit between any two scores i.e. if the highest score a voter gave anyone was a 3 out of 5, then in some situations, it is desirable to ensure that after normalization, the voter's highest score given to any candidate will still be a 3 out of 5. This can be the case when a voter does not want to use all of their voting power.

Impossibility theorems

see also: Arrow's impossibility theorem, Gibbard's theorem

Arrow's impossibility theorem demonstrates the impossibility of designing a deterministic ordinal voting system which passes a set of desirable criteria. Since Arrow's theorem only applies to ordinal voting and not cardinal voting systems, several cardinal systems to pass all these criteria. The typical examples are score voting and majority judgment. Additionally, there are cardinal systems which fail one of Arrow's criteria, but not due to Arrow's theorem; for example, Ebert's method fails monotonicity. Subsequent social choice theorists have expanded on Arrow's central insight, and applied his ideas to specific cardinal systems.

Furthermore, there are other impossibility theorems which are different than Arrow's and apply to cardinal systems. The most relevant are Gibbard's theorem and the Balinski–Young theorem.

Gibbard's 1973 theorem holds that any deterministic process of collective decision making with multiple options will have some level of strategic voting.[3]. Later results show that even allowing for nondeterminism, only very particular methods are strategy-proof. For example, requiring weak unanimity and assuming voters do not give their utilities with infinite precision, the only strategy-proof cardinal method is random ballot.[4]

As a result of this much of the work of social choice theorists is to find out what types of strategic voting a system is susceptible to and the level of susceptibility for each. For example, single-member systems are not susceptible to free riding. The Balinski–Young theorem holds that a system cannot satisfy a quota rule while being both house monotone and population monotone. This is important because quota rules are used in most definitions of proportional representation and population monotonicity is intimately tied to the participation criterion.

Kotze-Pereira transformation

The KP transform converts rated ballots that allow for more than two scores into equivalent fractional rated ballots that allow for only two scores i.e. it transforms scored ballots into Approval ballots.

It helps show the connection between different scales in a similar way to the approval rating concept.

Scale invariance

Scale invariance is the property that multiplying all voters' scores by a constant value (i.e. a voter who scored a candidate a 9 out of 10 might have their score multiplied by 10 to yield a score of a 90 out of 100) shouldn't change the results of the voting method. The KP transform can be used to give scale invariance to many voting methods that fail it, such as RRV becoming scale invariant in the form of SPAV + KP.


  1. Woodall, Douglas, Properties of Preferential Election Rules, Voting matters - Issue 3, December 1994
  2. Balinski M. and R. Laraki (2007) «A theory of measuring, electing and ranking». Proceedings of the National Academy of Sciences USA, vol. 104, no. 21, 8720-8725.
  3. Gibbard, Allan (1973). "Manipulation of voting schemes: A general result" (PDF). Econometrica. 41 (4): 587–601. doi:10.2307/1914083. JSTOR 1914083.
  4. Dutta, Bhaskar; Peters, Hans; Sen, Arunava (2006-05-17). "Strategy-proof Cardinal Decision Schemes". Social Choice and Welfare. Springer Science and Business Media LLC. 28 (1): 163–179. doi:10.1007/s00355-006-0152-9. ISSN 0176-1714.