# Comparison of electoral systems

(Redirected from Voting system criterion)

A major branch of social choice theory is devoted to the comparison of electoral systems, otherwise known as social choice functions. Viewed from the perspective of political science, electoral systems are rules for conducting elections and determining winners from the ballots cast. From the perspective of economics, mathematics, and philosophy, a social choice function is a mathematical function that determines how a society should make choices, given a collection of individual preferences.

This article discusses methods and results of comparing different systems. There are two broad ways to compare voting systems:

1. Metrics of voter satisfaction, based on models of how voters and candidates behave (possibly including empirical data).

## Evaluation by metrics

### Models of the electoral process

Voting methods can be evaluated by measuring their accuracy under random simulated elections aiming to be faithful to the properties of elections in real life. The first such evaluation was conducted by Chamberlin and Cohen in 1978, who measured the frequency with which certain non-Condorcet systems elected Condorcet winners.[1]

#### Condorcet jury model

The Marquis de Condorcet viewed elections as analogous to jury votes where each member expresses an independent judgement on the quality of candidates. Candidates differ in terms of their objective merit, but voters have imperfect information about the relative merits of the candidates. Such jury models are sometimes known as valence models. Condorcet and his contemporary Laplace realized that in such a model, voting theory could be reduced to probability by finding the expected quality of each candidate.[2]

The jury model implies several natural concepts of accuracy for voting systems under different models:

1. If voters' evaluations have errors following a normal distribution, the ideal procedure is score voting.
2. If only ranking information is available, and there are many more voters than candidates, any Condorcet method will converge on a single Condorcet winner, who will have the highest probability of being the best candidate.[3]

However, Condorcet's model is based on the extremely strong assumption of independent errors, i.e. voters will not be systematically biased in favor of one group of candidates or another. This is usually unrealistic: voters tend to communicate with each other, form parties or political ideologies, and engage in other behaviors that can result in correlated errors.

#### Black's spatial model

Duncan Black proposed a one-dimensional spatial model of voting in 1948, viewing elections as ideologically driven.[4] His ideas were later expanded by Anthony Downs.[5] Voters' opinions are regarded as positions in a space of one or more dimensions; candidates have positions in the same space; and voters choose candidates in order of proximity (measured under Euclidean distance or some other metric).

Spatial models imply a different notion of merit for voting systems: the more acceptable the winning candidate may be as a location parameter for the voter distribution, the better the system. A political spectrum is a one-dimensional spatial model.

#### Neutral models

Neutral voting models try to minimize the number of parameters and, as an example of the nothing-up-my-sleeve principle. The most common such model is the impartial anonymous culture model (or Dirichlet model). These models assume voters assign each candidate a utility completely at random (from a uniform distribution).

#### Comparisons of models

Tideman and Plassmann conducted a study which showed that a two-dimensional spatial model gave a reasonable fit to 3-candidate reductions of a large set of electoral rankings. Jury models, neutral models, and one-dimensional spatial models were all inadequate.[6] They looked at Condorcet cycles in voter preferences (an example of which is A being preferred to B by a majority of voters, B to C and C to A) and found that the number of them was consistent with small-sample effects, concluding that "voting cycles will occur very rarely, if at all, in elections with many voters." The relevance of sample size had been studied previously by Gordon Tullock, who argued graphically that although finite electorates will be prone to cycles, the area in which candidates may give rise to cycling shrinks as the number of voters increases.[7]

#### Utilitarian models

A utilitarian model views voters as ranking candidates in order of utility. The rightful winner, under this model, is the candidate who maximizes overall social utility. A utilitarian model differs from a spatial model in several important ways:

• It requires the additional assumption that voters are motivated solely by informed self-interest, with no ideological taint to their preferences.
• It requires the distance metric of a spatial model to be replaced by a faithful measure of utility.
• Consequently, the metric will need to differ between voters. It often happens that one group of voters will be powerfully affected by the choice between two candidates while another group has little at stake; the metric will then need to be highly asymmetric.

It follows from the last property that no voting system which gives equal influence to all voters is likely to achieve maximum social utility. Extreme cases of conflict between the claims of utilitarianism and democracy are referred to as the 'tyranny of the majority'. See Laslier's, Merlin's and Nurmi's comments in Laslier's write-up.[8]

James Mill seems to have been the first to claim the existence of an a priori connection between democracy and utilitarianism – see the Stanford Encyclopedia article.[9]

### Comparisons under a jury model

Suppose that the i th  candidate in an election has merit xi (we may assume that xi ~ N (0,σ2)[10]), and that voter j 's level of approval for candidate i may be written as xi + εij (we will assume that the εij are iid. N (0,τ2)). We assume that a voter ranks candidates in decreasing order of approval. We may interpret εij as the error in voter j 's valuation of candidate i and regard a voting method as having the task of finding the candidate of greatest merit.

Each voter will rank the better of two candidates higher than the less good with a determinate probability p (which under the normal model outlined here is equal to ${\displaystyle {\tfrac {1}{2}}\!+\!{\tfrac {1}{\pi }}{\textrm {tan}}^{-1}{\tfrac {\sigma }{\tau }}}$ , as can be confirmed from a standard formula for Gaussian integrals over a quadrant[citation needed]). Condorcet's jury theorem shows that so long as p > 12, the majority vote of a jury will be a better guide to the relative merits of two candidates than is the opinion of any single member.

Peyton Young showed that three further properties apply to votes between arbitrary numbers of candidates, suggesting that Condorcet was aware of the first and third of them.[11]

• If p is close to 12, then the Borda winner is the maximum likelihood estimator of the best candidate.
• if p is close to 1, then the Minimax winner is the maximum likelihood estimator of the best candidate.
• For any p, the Kemeny-Young ranking is the maximum likelihood estimator of the true order of merit.

Robert F. Bordley constructed a 'utilitarian' model which is a slight variant of Condorcet's jury model.[12] He viewed the task of a voting method as that of finding the candidate who has the greatest total approval from the electorate, i.e. the highest sum of individual voters' levels of approval. This model makes sense even with σ2 = 0, in which case p takes the value ${\displaystyle {\tfrac {1}{2}}\!+\!{\tfrac {1}{\pi }}{\textrm {tan}}^{-1}{\tfrac {1}{n-1}}}$  where n is the number of voters. He performed an evaluation under this model, finding as expected that the Borda count was most accurate.

### Simulated elections under spatial models

A simulated election can be constructed from a distribution of voters in a suitable space. The illustration shows voters satisfying a bivariate Gaussian distribution centred on O. There are 3 randomly generated candidates, A, B and C. The space is divided into 6 segments by 3 lines, with the voters in each segment having the same candidate preferences. The proportion of voters ordering the candidates in any way is given by the integral of the voter distribution over the associated segment.

The proportions corresponding to the 6 possible orderings of candidates determine the results yielded by different voting systems. Those which elect the best candidate, i.e. the candidate closest to O (who in this case is A), are considered to have given a correct result, and those which elect someone else have exhibited an error. By looking at results for large numbers of randomly generated candidates the empirical properties of voting systems can be measured.

The evaluation protocol outlined here is modelled on the one described by Tideman and Plassmann.[6] Evaluations of this type are commonest for single-winner electoral systems. Ranked voting systems fit most naturally into the framework, but other types of ballot (such a FPTP and Approval voting) can be accommodated with lesser or greater effort.

The evaluation protocol can be varied in a number of ways:

• The number of voters can be made finite and varied in size. In practice this is almost always done in multivariate models, with voters being sampled from their distribution and results for large electorates being used to show limiting behaviour.
• The number of candidates can be varied.
• The voter distribution could be varied; for instance, the effect of asymmetric distributions could be examined. A minor departure from normality is entailed by random sampling effects when the number of voters is finite. More systematic departures (seemingly taking the form of a Gaussian mixture model) were investigated by Jameson Quinn in 2017.[13]

#### Evaluation for accuracy

m
method
3 6 10 15 25 40
FPTP 70.6 35.5 21.1 14.5 9.3 6.4
AV/IRV 85.2 50.1 31.5 21.6 12.9 7.9
Borda 87.6 82.1 74.2 67.0 58.3 50.1
Condorcet 100.0 100.0 100.0 100.0 100.0 100.0

One of the main uses of evaluations is to compare the accuracy of voting systems when voters vote sincerely. If an infinite number of voters satisfy a Gaussian distribution, then the rightful winner of an election can be taken to be the candidate closest to the mean/median, and the accuracy of a method can be identified with the proportion of elections in which the rightful winner is elected. The median voter theorem guarantees that all Condorcet systems will give 100% accuracy (and the same applies to Coombs' method[14]).

Evaluations published in research papers use multidimensional Gaussians, making the calculation numerically difficult.[1][15][16][17] The number of voters is kept finite and the number of candidates is necessarily small.

The computation is much more straightforward in a single dimension, which allows an infinite number of voters and an arbitrary number m of candidates. Results for this simple case are shown in the first table, which is directly comparable with Table 5 (1000 voters, medium dispersion) of the cited paper by Chamberlin and Cohen. The candidates were sampled randomly from the voter distribution and a single Condorcet method (Minimax) was included in the trials for confirmation.

m
method
10
FPTP 0.166
AV/IRV 0.058
Borda 0.016
Condorcet 0.010

The relatively poor performance of the Alternative vote (IRV) is explained by the well known and common source of error illustrated by the diagram, in which the election satisfies a univariate spatial model and the rightful winner B will be eliminated in the first round. A similar problem exists in all dimensions.

An alternative measure of accuracy is the average distance of voters from the winner (in which smaller means better). This is unlikely to change the ranking of voting methods, but is preferred by people who interpret distance as disutility. The second table shows the average distance (in standard deviations) minus ${\displaystyle {\sqrt {\tfrac {2}{\pi }}}}$  (which is the average distance of a variate from the centre of a standard Gaussian distribution) for 10 candidates under the same model.

#### Evaluation for resistance to tactical voting

James Green-Armytage et al. published a study in which they assessed the vulnerability of several voting systems to manipulation by voters.[18] They say little about how they adapted their evaluation for this purpose, mentioning simply that it "requires creative programming". An earlier paper by the first author gives a little more detail.[19]

The number of candidates in their simulated elections was limited to 3. This removes the distinction between certain systems; for instance Black's method and the Dasgupta-Maskin method are equivalent on 3 candidates.

The conclusions from the study are hard to summarise, but the Borda count performed badly; Minimax was somewhat vulnerable; and IRV was highly resistant. The authors showed that limiting any method to elections with no Condorcet winner (choosing the Condorcet winner when there was one) would never increase its susceptibility to tactical voting. They reported that the 'Condorcet-Hare' system which uses IRV as a tie-break for elections not resolved by the Condorcet criterion was as resistant to tactical voting as IRV on its own and more accurate. Condorcet-Hare is equivalent to Copeland's method with an IRV tie-break in elections with 3 candidates.

#### Evaluation for the effect of the candidate distribution

x
m
0 0.25 0.5 1 1.5
3 87.6 87.9 88.9 93.0 97.4
6 82.1 80.2 76.2 71.9 79.9
10 74.1 70.1 61.2 47.6 54.1
15 66.9 60.6 46.4 26.6 30.8
25 58.3 47.0 26.3 8.1 10.1
40 50.2 33.3 11.3 1.5 2.1

Some systems, and the Borda count in particular, are vulnerable when the distribution of candidates is displaced relative to the distribution of voters. The attached table shows the accuracy of the Borda count (as a percentage) when an infinite population of voters satisfies a univariate Gaussian distribution and m candidates are drawn from a similar distribution offset by x standard distributions. Red colouring indicates figures which are worse than random. Recall that all Condorcet methods give 100% accuracy for this problem. (And notice that the reduction in accuracy as x increases is not seen when there are only 3 candidates.)

Sensitivity to the distribution of candidates can be thought of as a matter either of accuracy or of resistance to manipulation. If one expects that in the course of things candidates will naturally come from the same distribution as voters, then any displacement will be seen as attempted subversion; but if one thinks that factors determining the viability of candidacy (such as financial backing) may be correlated with ideological position, then one will view it more in terms of accuracy.

Published evaluations take different views of the candidate distribution. Some simply assume that candidates are drawn from the same distribution as voters.[16][18] Several older papers assume equal means but allow the candidate distribution to be more or less tight than the voter distribution.[20][1] A paper by Tideman and Plassmann approximates the relationship between candidate and voter distributions based on empirical measurements.[15] This is less realistic than it may appear, since it makes no allowance for the candidate distribution to adjust to exploit any weakness in the voting system. A paper by James Green-Armytage looks at the candidate distribution as a separate issue, viewing it as a form of manipulation and measuring the effects of strategic entry and exit. Unsurprisingly he finds the Borda count to be particularly vulnerable.[19]

#### Evaluation for other properties

• As previously mentioned, Chamberlin and Cohen measured the frequency with which certain non-Condorcet systems elect Condorcet winners. Under a spatial model with equal voter and candidate distributions the frequencies are 99% (Coombs), 86% (Borda), 60% (IRV) and 33% (FPTP).[1] This is sometimes known as Condorcet efficiency.
• Darlington measured the frequency with which Copeland's method produces a unique winner in elections with no Condorcet winner. He found it to be less than 50% for fields of up to 10 candidates.[17]

### Experimental metrics

The task of a voting system under a spatial model is to identify the candidate whose position most accurately represents the distribution of voter opinions. This amounts to choosing a location parameter for the distribution from the set of alternatives offered by the candidates. Location parameters may be based on the mean, the median, or the mode; but since ranked preference ballots provide only ordinal information, the median is the only acceptable statistic.

This can be seen from the diagram, which illustrates two simulated elections with the same candidates but different voter distributions. In both cases the mid-point between the candidates is the 51st percentile of the voter distribution; hence 51% of voters prefer A and 49% prefer B. If we consider a voting method to be correct if it elects the candidate closest to the median of the voter population, then since the median is necessarily slightly to the left of the 51% line, a voting method will be considered to be correct if it elects A in each case.

The mean of the teal distribution is also slightly to the left of the 51% line, but the mean of the orange distribution is slightly to the right. Hence if we consider a voting method to be correct if it elects the candidate closest to the mean of the voter population, then a method will not be able to obtain full marks unless it produces different winners from the same ballots in the two elections. Clearly this will impute spurious errors to voting methods. The same problem will arise for any cardinal measure of location; only the median gives consistent results.

The median is not defined for multivariate distributions but the univariate median has a property which generalizes conveniently. The median of a distribution is the position whose average distance from all points within the distribution is smallest. This definition generalizes to the geometric median in multiple dimensions. The distance is sometimes described as a voter's 'disutility' from a candidate's election, but this identification is purely arbitrary.

If we have a set of candidates and a population of voters, then it is not necessary to solve the computationally difficult problem of finding the geometric median of the voters and then identify the candidate closest to it; instead we can identify the candidate whose average distance from the voters is minimized. This is the metric which has been generally deployed since Merrill onwards;[20] see also Green-Armytage and Darlington.[19][16]

The candidate closest to the geometric median of the voter distribution may be termed the 'spatial winner'.

## Evaluation by real elections

Data from real elections can be analysed to compare the effects of different systems, either by comparing between countries or by applying alternative electoral systems to the real election data. The electoral outcomes can be compared through democracy indices, measures of political fragmentation, voter turnout,[21][22] political efficacy and various economic and judicial indicators. The practical criteria to assess real elections include the share of wasted votes, the complexity of vote counting, proportionality, and barriers to entry for new political movements.[23] Additional opportunities for comparison of real elections arise through electoral reforms.

A Canadian example of such an opportunity is seen in the City of Edmonton (Canada), which went from first-past-the-post voting in 1917 Alberta general election to five-member plurality block voting in 1921 Alberta general election, to five-member single transferable voting in 1926 Alberta general election, then to FPTP again in 1959 Alberta general election. One party swept all the Edmonton seats in 1917, 1921 and 1959. Under STV in 1926, two Conservatives, one Liberal, one Labour and one United Farmers MLA were elected.

## Comparison of single-winner voting methods

### Logical criteria for single-winner elections

Traditionally the merits of different electoral systems have been argued by reference to logical criteria. These have the form of rules of inference for electoral decisions, licensing the deduction, for instance, that "if E and E ' are elections such that R (E,E '), and if A is the rightful winner of E , then A is the rightful winner of E ' ".

The criteria are as debatable as the voting systems themselves. Here we briefly discuss the considerations advanced concerning their validity, and then summarize the most important criteria, showing in a table which of the principal voting systems satisfy them.

### Result criteria (absolute)

We now turn to the logical criteria themselves, starting with the absolute criteria which state that, if the set of ballots is a certain way, a certain candidate must or must not win.

Majority criterion (MC)
Will a candidate always win who is ranked as the unique favorite by a majority of voters? This criterion comes in two versions:
1. Ranked majority criterion, in which an option which is merely preferred over the others by a majority must win. (Passing the ranked MC is denoted by "yes" in the table below, because it implies also passing the following:)
2. Rated majority criterion, in which only an option which is uniquely given a perfect rating by a majority must win. The ranked and rated MC are synonymous for ranked voting methods, but not for rated or graded ones. The ranked MC, but not the rated MC, is incompatible with the IIA criterion explained below.
Mutual majority criterion (MMC)
Will a candidate always win who is among a group of candidates ranked above all others by a majority of voters? This also implies the majority loser criterion – if a majority of voters prefers every other candidate over a given candidate, then does that candidate not win? Therefore, of the methods listed, all pass neither or both criteria, except for Borda, which passes Majority Loser while failing Mutual Majority.
Condorcet criterion
Will a candidate always win who beats every other candidate in pairwise comparisons? (This implies the majority criterion, above.)
Condorcet loser criterion (cond. loser)
Will a candidate never win who loses to every other candidate in pairwise comparisons?

### Result criteria (relative)

These are criteria that state that, if a certain candidate wins in one circumstance, the same candidate must (or must not) win in a related circumstance.

Independence of Smith-dominated alternatives (ISDA)
Does the outcome never change if a Smith-dominated candidate is added or removed (assuming votes regarding the other candidates are unchanged)? Candidate C is Smith-dominated if there is some other candidate A such that C is beaten by A and every candidate B that is not beaten by A etc. Note that although this criterion is classed here as nominee-relative, it has a strong absolute component in excluding Smith-dominated candidates from winning. In fact, it implies all of the absolute criteria above.[specify]
Independence of irrelevant alternatives (IIA)
Does the outcome never change if a non-winning candidate is added or removed (assuming voter preferences regarding the other candidates are unchanged)?[24] For instance, plurality rule fails IIA; adding a candidate X can cause the winner to change from W to Y even though Y receives no more votes than before.
Local independence of irrelevant alternatives (LIIA)
Does the outcome never change if the alternative that would finish last is removed? (And could the alternative that finishes second fail to become the winner if the winner were removed?)
Independence of clone alternatives (cloneproof)
Does the outcome never change if non-winning candidates similar to an existing candidate are added? There are three different phenomena which could cause a method to fail this criterion:
Spoilers
Candidates which decrease the chance of any of the similar or clone candidates winning, also known as a spoiler effect.
Teams
Sets of similar candidates whose mere presence helps the chances of any of them winning.
Crowds
Additional candidates who affect the outcome of an election without either helping or harming the chances of their factional group, but instead affecting another group.
Monotonicity criterion (monotone)
If candidate W wins for one set of ballots, will W still always win if those ballots change to rank W higher? (This also implies that you cannot cause a losing candidate to win by ranking them lower.)
Consistency criterion (CC)
If candidate W wins for one set of ballots, will W still always win if those ballots change by adding another set of ballots where W also wins?
Participation criterion (PC)
Is voting honestly always better than not voting at all? (This is grouped with the distinct but similar Consistency Criterion in the table below.[25])
Reversal symmetry (reversal)
If individual preferences of each voter are inverted, does the original winner never win?

### Ballot-counting criteria

These are criteria which relate to the process of counting votes and determining a winner.

Polynomial time (polytime)
Can the winner be calculated in a runtime that is polynomial in the number of candidates and linear in the number of voters?
Resolvable
Can the winner be calculated in almost all cases, without using any random processes such as flipping coins? That is, are exact ties, in which the winner could be one of two or more candidates, vanishingly rare in large elections?
Summability (summable)
Can the winner be calculated by tallying ballots at each polling station separately and simply adding up the individual tallies? The amount of information necessary for such tallies is expressed as an order function of the number of candidates N. Slower-growing functions such as O(N) or O(N2) make for easier counting, while faster-growing functions such as O(N!) might make it harder to catch fraud by election administrators.[citation needed]

### Strategy criteria

These are criteria that relate to a voter's incentive to use certain forms of strategy. They could also be considered as relative result criteria; however, unlike the criteria in that section, these criteria are directly relevant to voters; the fact that a method passes these criteria can simplify the process of figuring out one's optimal strategic vote.

Later-no-harm criterion, and later-no-help criterion
Can voters be sure that adding a later preference to a ballot will not harm or help any candidate already listed?[26]
No favorite betrayal (NFB)
Can voters be sure that they do not need to rank any other candidate above their favorite in order to obtain a result they prefer?[27]

### Ballot format

These are issues relating to the expressivity or information content of a valid ballot. One of the most important is how much information a ballot can express. A cardinal ballot provides more information than an ordinal ballot: a cardinal ballot contains information about the strength of preferences, whereas an ordinal ballot only expresses whether a preference exists. Plurality voting provides even less information than either (as it only orders a single candidate above the rest).

### Stronger

A criterion A is "stronger" than B if satisfying A implies satisfying B. For instance, the Condorcet criterion is stronger than the majority criterion, because all majority winners are Condorcet winners. Thus, any voting method that satisfies the Condorcet criterion must satisfy the majority criterion.

### Compliance of selected single-winner methods

The following table shows which of the above criteria are met by several single-winner methods.

Criterion

Method
Maj­ority Maj.
loser
Mutual
maj.
Cond­orcet Cond.
loser
Smith/
ISDA
LIIA IIA Clone­proof Mono­tone Partic­ipation Reversal
symmetry
Poly­time Resolv­able Summ­able Later-
no-harm
Later-
no-help
Ballot
type
Appr­oval Yes Yes[citation needed] No No[a][b] No Yes[citation needed] Yes Yes[c] Yes Yes Yes Yes O(N) Yes O(N) No Yes Appr­ovals
Borda count No Yes No No[a] Yes No No No[broken anchor] Teams Yes Yes Yes O(N) Yes O(N) No Yes Ran­king
Bucklin Yes Yes Yes No No No No No No Yes No No O(N) Yes O(N) No Yes Ran­king
Cope­land Yes Yes Yes Yes Yes Yes No No[broken anchor][a] Teams,
crowds
Yes No[a] Yes O(N2) No O(N2) No[a] No Ran­king
IRV (AV) Yes Yes Yes No[a] Yes No[a] No No[broken anchor] Yes No No No O(N2) Yes[d] O(N!)[e] Yes Yes Ran­king
Kemeny–Young Yes Yes Yes Yes Yes Yes Yes No[broken anchor][a] Spoil­ers Yes No[a] Yes O(N!) Yes O(N2)[f] No[a] No Ran­king
Highest median/Majority
judgment
[g]
Rated[h] Yes[i] No[j] No[a][b] No No[a] Yes Yes[c] Yes Yes No[k] Dep­ends[l] O(N) Yes O(N)[m] No[n] Yes Scores[o]
Mini­max Yes No No Yes[p] No No No No[broken anchor][a] Spoil­ers Yes No[a] No O(N2) Yes O(N2) No[a][p] No Ran­king
Plura­lity/FPTP Yes No No No[a] No No[a] No No[broken anchor] Spoil­ers Yes Yes No O(N) Yes O(N) N/A[q] [q] Single mark
Score voting No No No No[a][b] No No[a] Yes Yes[c] Yes Yes Yes Yes O(N) Yes O(N) No Yes Scores
Ranked pairs Yes Yes Yes Yes Yes Yes Yes No[broken anchor][a] Yes Yes No[a][k] Yes O(N3) Yes O(N2) No[a] No Ran­king
Runoff voting Yes Yes No No[a] Yes No[a] No No[broken anchor] Spoil­ers No No No O(N)[r] Yes O(N)[r] Yes Yes[s] Single mark
Schulze Yes Yes Yes Yes Yes Yes No No[broken anchor][a] Yes Yes No[a][k] Yes O(N2)[28] Yes O(N2) No[a] No Ran­king
STAR
voting
No[t][citation needed] Yes No[u] No[a][b] Yes No[a] No No No Yes No[clarification needed] Depends[v] O(N) Yes O(N2) No[clarification needed] No[clarification needed] Scores
Sortition, arbitrary winner[w] No No No No[a] No No[a] Yes Yes No Yes Yes Yes O(1) No O(1) Yes Yes None
Random ballot[x] No No No No[a] No No[a] Yes Yes Yes Yes Yes Yes O(N) No O(N) Yes Yes Single mark
Comparison of voting systems
Criterion: Majority Majority loser criterion Mutual majority criterion Condorcet winner Condorcet loser Smith ISDA LIIA Cloneproof Monotone Participation Reversal Later-no-harm Later-no-help Polynomial time Resolvability
Schulze Yes Yes Yes Yes Yes Yes Yes No Yes Yes No Yes No No Yes Yes
Ranked pairs Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes No Yes No No Yes Yes
Tideman alternative Yes Yes Yes Yes Yes Yes Yes No Yes No No No No No Yes Yes
Kemeny–Young Yes Yes Yes Yes Yes Yes Yes Yes No Yes No Yes No No No Yes
Copeland Yes Yes Yes Yes Yes Yes Yes No No Yes No Yes No No Yes No
Nanson Yes Yes Yes Yes Yes Yes No No No No No Yes No No Yes Yes
Black Yes Yes No Yes Yes No No No No Yes No Yes No No Yes Yes
Instant-runoff voting Yes Yes Yes No Yes No No No Yes No No No Yes Yes Yes Yes
Borda count No Yes No No Yes No No No No Yes Yes Yes No Yes Yes Yes
Baldwin Yes Yes Yes Yes Yes Yes No No No No No No No No Yes Yes
Bucklin Yes Yes Yes No No No No No No Yes No No No Yes Yes Yes
Plurality Yes No No No No No No No No Yes Yes No Yes Yes Yes Yes
Coombs Yes Yes Yes No Yes No No No No No No No No No Yes Yes
Minimax Yes No No Yes No No No No No Yes No No No No Yes Yes
Anti-plurality No Yes No No No No No No No Yes Yes No No No Yes Yes
Dodgson Yes No No Yes No No No No No No No No No No No Yes

## Practical factors

The concerns raised above are used by social choice theorists to devise systems that are accurate and resistant to manipulation. However, there are also practical reasons why one system may be more socially acceptable than another, which fall under the fields of public choice and political science.[8][16] Important practical considerations include:

• Ease of explanation. Some voting rules are difficult to explain to voters in a way they can intuitively understand, which may undermine public trust in elections.[8][failed verification] For example, while Schulze's rule performs well by many of the criteria above, it requires an involved explanation of beatpaths.
• Ease of voting. Different kinds of ballots may be easier to fill out; for example, studies generally find that voters generally consider ranked voting to be complex and confusing when compared to rated voting or plurality voting.

Other considerations include barriers to entry to the political competition[29] and likelihood of gridlocked government.[30]

## Comparison of multi-winner systems

Multi-winner electoral systems at their best seek to produce assemblies representative in a broader sense than that of making the same decisions as would be made by single-winner votes. They can also be route to one-party sweeps of a city's seats, if a non-proportional system, such as plurality block voting or ticket voting, is used.

### Metrics for multi-winner evaluations

Evaluating the performance of multi-winner voting methods requires different metrics than are used for single-winner systems. The following have been proposed.

• Condorcet Committee Efficiency (CCE) measures the likelihood that a group of elected winners would beat all losers in pairwise races.[31]
• The Gallagher Index and Loosemore–Hanby index (LH) measure proportionality between seat share and party vote share. Gallagher generally uses overall voting party percentages or votes compared to seat percentages to assess proportionality so ignores presence of districts if any.
• Wasted votes measure the fraction of electorate not represented by any representative.

### Criterion tables

The following table shows which of the above criteria are met by several multiple winner methods.

Criterion

Method
Proportional Mono­tone Consis­tency Partic­ipation No
Favorite
Betrayal
Semi­honest Universally Liked Candidates
Monroe's method Yes Depends[32] Yes No No
Chamberlin-Courant[33]
Ebert's Yes No Yes
Psi Yes Yes Yes No No No
Harmonic Yes Yes Yes No No No
Sequential Proportional Approval Yes Yes No No No No
Re-weighted Range Yes Yes No No No No
Proportional Approval Yes Yes Yes No No No
Single Transferable Vote Yes No No No No No Yes
CPO-STV Yes No No No No No Yes
Schulze STV Yes Yes No No No No Yes
Phragmen
Quota Borda System
STV with Borda Elimination[34]
QPQ[35]
Expanding Approvals Rule[36]
Sequential STV[37]
Single non-transferable vote No Yes Yes Yes No No — (not proportional)
Limited vote No Yes Yes Yes No[38] No — (not proportional)
Cumulative voting No Yes Yes Yes No[38] No — (not proportional)
Minmax Approval[39]
Sortition, Arbitrary Winner No Yes Yes Yes Yes Yes — (not proportional)
Multiple Random Ballots style="background:#9EFF9E;vertical-align:middle;text-align:center;" class="table-yes"|Yes Yes Yes Yes Yes Yes

## Notes

1. Condorcet, Smith and Independence of Smith-dominated alternatives criteria are incompatible with Independence of irrelevant alternatives, Consistency, Participation, Later-no-harm, Later-no-help, and Favorite betrayal[clarification needed] criteria.
2. ^ a b c d In Approval, Range, and Majority Judgment, if all voters have perfect information about each other's true preferences and use rational strategy, any Majority Condorcet or Majority winner will be strategically forced – that is, win in all of one or more strong Nash equilibria. In particular if every voter knows that "A or B are the two most-likely to win" and places their "approval threshold" between the two, then the Condorcet winner, if one exists and is in the set {A,B}, will always win. These methods also satisfy the majority criterion in the weaker sense that any majority can force their candidate to win, if it so desires. Laslier, J-F (2006), "Strategic approval voting in a large electorate" (PDF), IDEP Working Papers (405), Marseille, France
3. ^ a b c Approval voting, range voting, and majority judgment satisfy IIA if it is assumed that voters rate candidates individually and independently of knowing the available alternatives in the election, using their own absolute scale. For this to hold, in some elections, some voters must use less than their full voting power or even abstain, despite having meaningful preferences among the available alternatives. If this assumption is not made, these methods fail IIA, as they become more ranked than rated methods.
4. ^ If the number of candidates grows faster than the square root of the number of voters, this may not be the case, as ties at any point in the process, even between two non-viable candidates, could affect the final result. If the rule for resolving such ties involves no randomness, though, the method does pass the criterion. [citation needed]
5. ^ The number of piles that can be summed from various precincts is floor ((e−1) N!) − 1.
6. ^ Each prospective Kemeny-Young ordering has score equal to the sum of the pairwise entries that agree with it, and so the best ordering can be found using the pairwise matrix.
7. ^ Bucklin voting, with skipped and equal-rankings allowed, meets the same criteria as Majority Judgment; in fact, Majority Judgment may be considered a form of Bucklin voting. Without allowing equal rankings, Bucklin's criteria compliance is worse; in particular, it fails Independence of Irrelevant Alternatives, which for a ranked method like this variant is incompatible with the Majority Criterion.
8. ^ Majority Judgment does not always elect a candidate preferred over all others by over half of voters; however, it always elects the candidate uniquely top-rated by over half of voters.
9. ^ Majority Judgment may elect a candidate uniquely least-preferred by over half of voters, but it never elects the candidate uniquely bottom-rated by over half of voters.
10. ^ Majority Judgment fails the mutual majority criterion, but satisfies the criterion if the majority ranks the mutually favored set above a given absolute grade and all others below that grade.
11. ^ a b c In Majority Judgment, Ranked Pairs, and Schulze voting, there is always a regret-free semi-honest ballot for any voter, holding other ballots constant. That is, if they know enough about how others will vote (for instance, in the case of Majority Judgment, the winning candidate and their winning median score), there is always at least one way for them to participate without grading any less-preferred candidate above any more-preferred one. However, this can cease to hold if voters have insufficient information.
12. ^ Majority judgment can actually pass or fail reversal symmetry depending on the rounding method used to find the median when there are even numbers of voters. For instance, in a two-candidate, two-voter race, if the ratings are converted to numbers and the two central ratings are averaged, then MJ meets reversal symmetry; but if the lower one is taken, it does not, because a candidate with ["fair","fair"] would beat a candidate with ["good","poor"] with or without reversal. However, for rounding methods which do not meet reversal symmetry, the odds of breaking it are comparable to the odds of an irresolvable (tied) result; that is, vanishingly small for large numbers of voters.
13. ^ Majority Judgment is summable at order KN, where K, the number of ranking categories, is set beforehand.
14. ^ Though Majority Judgment does not pass this or similar criteria, there are other similar median methods, such as those based on Bucklin voting, which can meet a related, weaker criterion: ranking an additional candidate below the median grade (rather than your own grade) of your favorite candidate, cannot harm your favorite. Balinski, M., and R. Laraki. "A Theory of Measuring, Electing, and Ranking." Proceedings of the National Academy of Sciences 104, no. 21 (2007): 8720.
15. ^ In fact, Majority Judgment ballots use ratings expressed in "common language" rather than numbers, that is, each rating has an absolute meaning.
16. ^ a b A variant of Minimax that counts only pairwise opposition, not opposition minus support, fails the Condorcet criterion and meets later-no-harm.
17. ^ a b Since plurality does not allow marking later preferences on the ballot at all, it is impossible to either harm or help a favorite candidate by marking later preferences, and so it trivially passes both Later-No-Harm and Later-No-Help. However, because it forces truncation, it shares some problems with methods that merely encourage truncation by failing Later-No-Harm. Similarly, though to a lesser degree, because it doesn't allow voters to distinguish between all but one of the candidates, it shares some problems with methods which fail Later-No-Help, which encourage voters to make such distinctions dishonestly.
18. ^ a b Once for each round.
19. ^ That is, second-round votes cannot help or harm candidates already eliminated.
20. ^ STAR voting will elect a majority candidate X if X is in the runoff, and X's voters can guarantee they make the runoff by strategically giving the highest score to X and the lowest score to all opponents. However, if there are two or more opponents that get any points from X's voters, these opponents could shut X out of the runoff. Thus, STAR fails the majority criterion.
21. ^ As with the majority criterion, STAR voting fails the mutual majority criterion. However, the more candidates are in the mutual majority set, the greater the chance that at least one of them will be in the runoff, and thus be guaranteed to win.
22. ^ STAR does not define a full outcome ordering, only a winner. With any number of candidates besides 3, the winner cannot stay the same if the ballots are reversed.
23. ^ Sortition, uniformly randomly chosen candidate is winner. Arbitrary winner, some external entity, not a voter, chooses the winner. These methods are not, properly speaking, voting methods at all, but are included to show that even a non-voting method can still pass some of the criteria.
24. ^ Random ballot, uniformly randomly chosen ballot determines winner. This and closely related methods are of mathematical interest because they are the only possible methods which are truly strategy-free, that is, your best vote will never depend on anything about the other voters. However, this method is not generally considered as a serious proposal for a practical method.

## References

1. ^ a b c d Chamberlin, John R.; Cohen, Michael D. (December 1978). "Toward Applicable Social Choice Theory: A Comparison of Social Choice Functions under Spatial Model Assumptions". American Political Science Review. 72 (4): 1341–1356. doi:10.2307/1954543. ISSN 0003-0554. JSTOR 1954543. S2CID 155928569.
2. ^ M. J. A. N. de Caritat, Marquis de Condorcet. His book was published in 1785. The title may be translated as "Essay on the application of probability theory to majority voting".
3. ^ For Condorcet and Laplace see G. G. Szpiro, "Numbers Rule" (2010).
4. ^ Black, Duncan (1948). "On the Rationale of Group Decision-making". Journal of Political Economy. 56 (1): 23–34. doi:10.1086/256633. ISSN 0022-3808. S2CID 153953456.
5. ^ Anthony Downs, "An Economic Theory of Democracy" (1957).
6. ^ a b Tideman, T. Nicolaus; Plassmann, Florenz (2012), Felsenthal, Dan S.; Machover, Moshé (eds.), "Modeling the Outcomes of Vote-Casting in Actual Elections", Electoral Systems, Berlin, Heidelberg: Springer Berlin Heidelberg, pp. 217–251, doi:10.1007/978-3-642-20441-8_9, ISBN 978-3-642-20440-1, retrieved 2023-04-02
7. ^ Tullock, Gordon (1967). "The General Irrelevance of the General Impossibility Theorem". The Quarterly Journal of Economics. 81 (2): 256–270. doi:10.2307/1879585. JSTOR 1879585.
8. ^ a b c Jean-François Laslier (2011). And the loser is... Plurality Voting. ISBN 978-3-642-42955-2. ISSN 2267-828X. Wikidata Q108664719. {{cite book}}: |journal= ignored (help)
9. ^ Ball, Terence and Antis Loizides, "James Mill", The Stanford Encyclopedia of Philosophy (Winter 2020 Edition), Edward N. Zalta (ed.).
10. ^ The notation is defined in the article Normal distribution§Notation. The assumption of normality is convenient, and provides a generative model suitable for use in simulations, but Condorcet's and Young's results do not rely on it, being derived from pure probability theory.
11. ^ Young, H. P. (1988). "Condorcet's Theory of Voting". American Political Science Review. 82 (4): 1231–1244. doi:10.2307/1961757. ISSN 0003-0554. JSTOR 1961757. S2CID 14908863.
12. ^ Bordley, Robert F. (March 1983). "A Pragmatic Method for Evaluating Election Schemes through Simulation". American Political Science Review. 77 (1): 123–141. doi:10.2307/1956015. ISSN 0003-0554. JSTOR 1956015. S2CID 147689260.
13. ^ Quinn, Jameson (2017-02-10). "Voter Satisfaction Efficiency FAQ". GitHub Pages. Retrieved 2019-07-24.
14. ^ Grofman, Bernard; Feld, Scott L. (December 2004). "If you like the alternative vote (a.k.a. the instant runoff), then you ought to know about the Coombs rule". Electoral Studies. 23 (4): 641–659. doi:10.1016/j.electstud.2003.08.001.
15. ^ a b Tideman, T. Nicolaus; Plassmann, Florenz (March 2014). "Which voting rule is most likely to choose the "best" candidate?". Public Choice. 158 (3–4): 331–357. doi:10.1007/s11127-012-9935-y. ISSN 0048-5829. S2CID 189841413.
16. ^ a b c d Darlington, Richard B. (2018). "Are Condorcet and minimax voting systems the best?". arXiv:1807.01366v8 [physics.soc-ph].
17. ^ a b Darlington, Richard B. (2016). "Minimax is the best electoral system after all". arXiv:1606.04371 [stat.ME].
18. ^ a b Green-Armytage, James; Tideman, T. Nicolaus; Cosman, Rafael (January 2016). "Statistical evaluation of voting rules" (PDF). Social Choice and Welfare. 46 (1): 183–212. doi:10.1007/s00355-015-0909-0. ISSN 0176-1714. S2CID 253842768.
19. ^ a b c Green-Armytage, James (January 2014). "Strategic voting and nomination". Social Choice and Welfare. 42 (1): 111–138. doi:10.1007/s00355-013-0725-3. ISSN 0176-1714. S2CID 253847024.
20. ^ a b Merrill, Samuel (February 1984). "A Comparison of Efficiency of Multicandidate Electoral Systems". American Journal of Political Science. 28 (1): 23–48. doi:10.2307/2110786. JSTOR 2110786 – via JSTOR.
21. ^ Lijphart, Arend (March 1997). "Unequal Participation: Democracy's Unresolved Dilemma". American Political Science Review. 91 (1): 1–14. doi:10.2307/2952255. JSTOR 2952255. S2CID 143172061.
22. ^ Blais, Andre (1990). "Does proportional representation foster voter turnout?". European Journal of Political Research. 18 (2): 167–181. doi:10.1111/j.1475-6765.1990.tb00227.x.
23. ^ Tullock, Gordon (1965). "Entry Barriers in Politics". The American Economic Review. 55 (1/2). American Economic Association: 458–466. ISSN 0002-8282. JSTOR 1816288. Retrieved 2024-04-03.
24. ^ Vasiljev, Sergei (April 1, 2008), Cardinal Voting: The Way to Escape the Social Choice Impossibility, SSRN eLibrary, SSRN 1116545. Note that in practice, voters could change their votes depending on who is in the race (especially in cardinal voting methods). However, this possibility is ignored, because if it were accounted for, no deterministic method could possibly pass this criterion.
25. ^ Consistency implies participation, but not vice versa. For example, range voting complies with participation and consistency, but median ratings satisfies participation and fails consistency.
26. ^ Woodall, Douglas (December 1994), "Properties of Preferential Election Rules", Voting Matters (3)
27. ^ Small, Alex (August 22, 2010), "Geometric construction of voting methods that protect voters' first choices", arXiv:1008.4331 [cs.GT]
28. ^ Sornat, Krzysztof; Vassilevska Williams, Virginia; Xu, Yinzhan (2021-07-18). "Fine-Grained Complexity and Algorithms for the Schulze Voting Method". Proceedings of the 22nd ACM Conference on Economics and Computation. ACM. pp. 841–859. arXiv:2103.03959. doi:10.1145/3465456.3467630. ISBN 978-1-4503-8554-1.
29. ^ Tullock, Gordon (1965). "Entry Barriers in Politics". The American Economic Review. 55 (1/2): 458–466. JSTOR 1816288.
30. ^ McGann, Anthony J., and Michael Latner. "The calculus of consensus democracy: Rethinking patterns of democracy without veto players." Comparative Political Studies 46.7 (2013): 823-850.
31. ^ Diss, Mostapha; Doghmi, Ahmed (2016). "Multi-winner scoring election methods: Condorcet consistency and paradoxes". Public Choice. 169 (1–2): 97–116. doi:10.1007/s11127-016-0376-x. ISSN 0048-5829. S2CID 73721191.
32. ^ Pereira, Toby (2016-02-17). "Proportional Approval Method using Squared loads, Approval removal and Coin-flip approval transformation (PAMSAC)". arXiv:1602.05248 [cs.GT].
33. ^ Diss, Mostapha; Kamwa, Eric; Tlidi, Abdelmonaim (2018). "The Chamberlin-Courant Rule and the k-Scoring Rules: Agreement and Condorcet Committee Consistency". SSRN Electronic Journal. Elsevier BV. doi:10.2139/ssrn.3198184. ISSN 1556-5068.
34. ^ Geller, Chris. "Single transferable vote with borda elimination: a new vote counting system" (PDF).
35. ^ Woodall, Douglas R. (October 2003). "QPQ, a quota-preferential STV-like election rule" (PDF). Voting Matters (17). Retrieved 2024-04-03.
36. ^ Aziz, Haris; Lee, Barton E. (2019-08-09). "The expanding approvals rule: improving proportional representation and monotonicity". Social Choice and Welfare. 54 (1). Springer Science and Business Media LLC: 1–45. arXiv:1708.07580. doi:10.1007/s00355-019-01208-3. ISSN 0176-1714.
37. ^ Hill, I. D.; Gazeley, Simon (June 2005). "Sequential STV - a further modification" (PDF). Voting Matters (20). Retrieved 2024-04-03.
38. ^ a b Amy, Douglas J.; Behind the Ballot Box: A Citizen's Guide to Voting Systems, p. 120 ISBN 0275965864
39. ^ Brams, Steven J.; Kilgour, D. Marc; Sanver, M. Remzi (2007-04-28). "A minimax procedure for electing committees" (PDF). Public Choice. 132 (3–4). Springer Science and Business Media LLC: 401–420. doi:10.1007/s11127-007-9165-x. ISSN 0048-5829.