Cardinal voting systems: Difference between revisions
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{{Wikipedia|Cardinal voting}}
'''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
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
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
==Vote
[[Cardinal voting]] is called [[Score
In multi-member systems the aggregation method can be split into the winner selection and the ballot reweighting methods. Optimal systems, however, combine these.
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 [[Instant-runoff 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.▼
=== Majority criterion ===
▲In multi-member systems the aggregation method can be split into the winner selection and the ballot reweighting methods. Optimal systems, however, combine these.
▲ The median can also be used to aggregate a cardinal ballot in [[:Category:Graded Bucklin methods|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]]
== Gradation and
The
However, the gradation or the number of choices within the range does matter. This is where [[
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 [[
=== 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
[[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|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.
[http://electionscience.github.io/vse-sim/VSE/ 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
:''Main article: [[Single-member district]]''
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! Method !! Aggregation !! Gradation
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| [[Score
|-
| [[Approval
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| [[STAR voting]] || [[STAR voting|Sum, then top two run-off]] || > 2
|-
| [[
|-
| [[Majority Judgment]]|| Median || > 2
|-
|[[Chiastic Score Voting]]
|Highest intersection
|> 3
|-
| [[Majority Choice Approval]]|| Median || [[Approval Voting|Binary]]
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|}
== [[Multi-
===[[Block voting|Bloc
Bloc
* '''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.
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* '''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 representation|
Sequential
{| class="wikitable"
|-
! System !! Gradation !! Selection !! Reweight [[Proportional representation | PR Philosophy]] !! [[Proportional_representation#Party_list_case | Party List Case]]
|-
| [[Reweighted Range Voting]] || > 2 || [[Utilitarian winner|Sum]] || Thiele Interpretation || [[Highest averages method]]
|-
| [[
|-
| [[
|-
| [[
|-
| [[
|-
| [[Sequential
|-
| [[
|-
| [[Sequential Ebert]] || [[Approval Voting|Binary]] || [[Utilitarian winner|Sum]] || [[Phragmén's Method | Phragmén interpretation]] || ??
|}
===
Optimal
* [https://rangevoting.org/QualityMulti.html Harmonic Voting]
* [[Proportional approval voting]]
* [[Phragmen's voting rules|Phragmén's
* [[Monroe's method]]
* [[Ebert's Method]]
* [[Phragmen's voting rules|max-Phragmén]]
* [[PAMSAC]]
== Criticism ==
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 [[Later-no-harm criterion#Criticism|too uncompromising]].<ref>Woodall, Douglas, Properties of Preferential Election Rules, [http://www.votingmatters.org.uk/ISSUE3/P5.HTM Voting matters - Issue 3, December 1994]</ref>
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.<ref name=":0">Balinski M. and R. Laraki (2007) «[https://www.pnas.org/content/pnas/104/21/8720.full.pdf A theory of measuring, electing and ranking]». Proceedings of the National Academy of Sciences USA, vol. 104, no. 21, 8720-8725.</ref>
== Notes ==
=== 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|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 ===
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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
:''
[[Arrow's impossibility theorem]] demonstrates the impossibility of designing
Furthermore, there are other [[Voting paradox| 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]].<ref>{{cite journal|last=Gibbard|first=Allan|author-link=Allan Gibbard|year=1973|title=Manipulation of voting schemes: A general result|url=http://www.eecs.harvard.edu/cs286r/courses/fall11/papers/Gibbard73.pdf|journal=Econometrica|volume=41|issue=4|pages=587–601|doi=10.2307/1914083|jstor=1914083}}</ref>. 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.<ref>{{cite journal | last=Dutta | first=Bhaskar | last2=Peters | first2=Hans | last3=Sen | first3=Arunava | title=Strategy-proof Cardinal Decision Schemes | journal=Social Choice and Welfare | publisher=Springer Science and Business Media LLC | volume=28 | issue=1 | date=2006-05-17 | issn=0176-1714 | doi=10.1007/s00355-006-0152-9 | pages=163–179|url=https://www.researchgate.net/publication/24064783_Strategy-proof_Cardinal_Decision_Schemes}}</ref>
==== Kotze-Pereira transformation ====
{{Main|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 [[
=== 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.
== References ==
▲It helps show the connection between different scales in a similar way to the [[Approval rating|approval rating]] concept.
<references/>
[[Category:Cardinal voting methods]]
[[Category:Ballot
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