Comparison of voting rules
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] The Marquis de Condorcet viewed elections as analogous to jury votes where each member expresses an independent judgement on the quality of candidates.
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.
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.
Duncan Black proposed a one-dimensional spatial model of voting in 1948, viewing elections as ideologically driven.
These models assume voters assign each candidate a utility completely at random (from a uniform distribution).
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.
[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.
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
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.
By looking at results for large numbers of randomly generated candidates the empirical properties of voting systems can be measured.
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.
The candidates were sampled randomly from the voter distribution and a single Condorcet method (Minimax) was included in the trials for confirmation.
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.
This is unlikely to change the ranking of voting methods, but is preferred by people who interpret distance as disutility.
James Green-Armytage et al. published a study in which they assessed the vulnerability of several voting systems to manipulation by voters.
The conclusions from the study are hard to summarise, but the Borda count performed badly; Minimax was somewhat vulnerable; and IRV was highly resistant.
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.
[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.
[19] 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.
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 same problem will arise for any cardinal measure of location; only the median gives consistent results.
Traditionally the merits of different electoral systems have been argued by reference to logical criteria.
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.
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.
Evaluating the performance of multi-winner voting methods requires different metrics than are used for single-winner systems.
