Comparison of electoral systems

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

Metrics of voter satisfaction, either through simulation or survey.
[|Adherence to logical criteria].
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.

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 demonstrated that, in such a model, voting theory could be reduced to probability by finding the expected quality of each candidate.
The jury model implies several natural concepts of accuracy for voting systems under different models:

If voters' evaluations have errors following a normal distribution, the ideal procedure is score voting.
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.

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

proposed a one-dimensional spatial model of voting in 1948, viewing elections as ideologically driven. His ideas were later expanded by Anthony Downs. 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.
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. The most common such model is the impartial anonymous culture model. These models assume voters assign each candidate a utility completely at random.

Comparisons of models

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. They looked at Condorcet cycles in voter preferences 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.

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.
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.

Comparisons under a jury model

Suppose that the i^th candidate in an election has merit x_i, and that voter j 's level of approval for candidate i may be written as x_i + ε_ij. We assume that a voter ranks candidates in decreasing order of approval. We may interpret ε_ij 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.
Condorcet's jury theorem shows that so long as p > , 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.

If p is close to, 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. 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 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, 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.
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 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 were investigated by Jameson Quinn in 2017.
Evaluation for accuracy

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.
Evaluations published in research papers use multidimensional Gaussians, making the calculation numerically difficult. 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 of the cited paper by Chamberlin and Cohen. The candidates were sampled randomly from the voter distribution and a single Condorcet method was included in the trials for confirmation.

	10
FPTP	0.166
AV/IRV	0.058
Borda	0.016
Condorcet	0.010

The relatively poor performance of the Alternative vote 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. 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 minus for 10 candidates under the same model.