Cardinal voting systems
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 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. Unlike ordinal voting, Arrow's Impossibility Theorem does not apply to cardinal methods. Furthermore, all cardinal methods satisfy the participation criterion.
In 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. Cardinal voting maximizes the number of people who vote for a candidate to become the representative. This is expected to have a knockon effect of better acceptance of results and higher voter turnout.
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Vote Aggregation and Tallying Methods[edit  edit source]
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.
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, stated as "if one candidate is preferred by a majority (more than 50%) of voters, then that candidate must win". It should be noted that Instantrunoff voting also satisfies this criterion. 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 an equal preference for the consensus candidate and their favorite candidate). Satisfying the majority criterion reduces incentive for compromise and lowers Bayesian Regret.
In multimember systems the aggregation method can be split into the winner selection and the ballot reweighting methods. Optimal systems, however, combine these.
Gradation and Range[edit  edit source]
The Range 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 Ordinal voting both have natural proextremist polarization bias, conversely, Approval has procentrist bias. Political polarization is generally viewed as divisive and undesirable so forcing the electorate towards a moderate candidate should be in the general good. All majoritarian systems are polarizing and are therefor not necessarily desirable.
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[edit  edit source]
Score voting has the lowest Bayesian Regret among all common singlewinner 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[edit  edit source]
Method  Aggregation  Gradation 

Score Voting  Sum  > 2 
Approval Voting  Sum  Binary 
STAR voting  Sum, then top two runoff  > 2 
Median Ratings  Median  > 2 
Majority Choice Approval  Median  Binary 
Majority Approval Voting  Median  Binary 
MultiMember Methods[edit  edit source]
Bloc Methods[edit  edit source]
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 05. 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[edit  edit source]
Sequential Cardinal Systems elect winners one at a time in sequence using a candidate selection method and a reweighting mechanism. The singlewinner 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 

Reweighted Range Voting  > 2  Sum  Jefferson method 
Sequential proportional approval voting  Binary  Sum  Jefferson method 
Sequentially Spent Score  > 2  Sum  Vote Unitarity 
Allocated Score  > 2  Sum  Allocate 
Sequential Monroe  > 2  Highest Sum in a Hare Quota  Allocate 
Sequential Ebert  Binary  Sum  Ebert's Method

Optimal Proportional Methods[edit  edit source]
Optimal Systems 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 permuting to all possible winner sets not a maximization algorithm. This makes such systems computationally expensive.