Voting Rules for
 Accurate
Democracy

Different uses for voting need different types of voting.

Plurality rules, Majority rules, Condorcet completion rules and others.

Compare One-Winner
Voting Systems

Criticisms of Condorcet rules based on behaviors of voters, politicians and governments.
Compare single-winner voting systems:
Condorcet efficiency
Utility efficiency
Nearness to the center

Condorcet Efficiency

Even though Condorcet winners can beat each of the other candidates in one-on-one elections, most voting rules do not always elect them. Given 100 elections with no voting cycles, what percentage of the 100 Condorcet winners will each voting system elect? This number is a voting system's "Condorcet efficiency". To estimate the efficiency of each voting system, several political scientists have used computers to simulate groups of voters.

[Samuel Merrill III coined this term and defined it: "The Condorcet efficiency of a voting procedure is the proportion or percentage of a class of elections (for which a Condorcet candidate exists) in which the voting system chooses the Condorcet candidate as winner." (Merrill: Glossary)]

 Table 1. Condorcet Efficiencies
in computer simulated elections with 4 candidates and 4 issues
data from Chamberlin, Cohen, and Coombs, 1984

21 Voters
Voting Rule Impartial culture Candidate Dispersion
Low
Candidate Dispersion
Medium
Candidate Dispersion
High
Coombs 93 96 98 99
Hare (IRV) 92 72 75 90
Borda 86 83 83 92
Plurality 69 59 53 77
 

1000 Voters
Voting Rule Impartial culture Candidate Dispersion
Low
Candidate Dispersion
Medium
Candidate Dispersion
High
Coombs 91 81 99 99
Hare (IRV) 92 32 60 84
Borda 89 85 86 97
Plurality 69 27 33 70

 

Table 2. Condorcet Efficiencies
in computer simulated elections with 5 candidates and 1000 voters
from Merrill, 1988, page 24.

% with a CW Random society D=2 D=4 D=2 D=4 D=2 D=4 D=2 D=4
    Dispersion = 0.5 Dispersion = 1.0
    C = 0.5 C = 0.0 C = 0.5 C = 0.0
Voting system Random society D=2 D=4 D=2 D=4 D=2 D=4 D=2 D=4
Plurality 60 21 28 27 42 57 67 61 81
Runoff 82 31 44 39 62 80 87 79 96
Hare (IRV) 88 34 50 38 72 78 86 83 97
Approval 67 73 76 75 82 74 78 81 84
Borda 85 84 87 86 94 86 89 89 92
Coombs 90 90 91 90 94 97 97 95 97
Utility max. 78 80 85 83 86 83 88 88 90
% elections with a CW 76 98 98 98 99 99 99 99 99
The bottom row gives the percent of elections which had a Condorcet winner.  

Impartial culture. An impartial culture is a model of an electorate in which all preference orders (for a set of candidates) are equally likely.

"Spatial model" refers to simulations with a bell-curved distribution of voters on each issue. A "dispersion" of 1.0 (or medium) means the average distance between candidates' opinions is as wide as the average distance between voters' opinions; 0.5 means the candidates tend to be more moderate than the voters. The latter corresponds to the assumption that most candidates seek the large group of voters in the middle of the bell curve. Low dispersion = 0.4 and high = 1.5.

C = 0.5 means there is some correspondence between a voter's position on one issue and his position on others; C = 0.0 means there is no relationship between issues.
D is the number of issues simulated.

The last line shows what percentage of Election which have a Condorcet Winner.

Plurality has the worst scores. Runoff and IRV also do poorly in some situations. Often IRV's flaw results from the squeeze effect. The Condorcet-completion rules by Black, Copeland, Dodgson, Kemeny, Schulze, Tideman and others have Condorcet efficiencies of 100% as does LOR which elects the Condorcet winner when there is one, else the IRV winner.

Manipulation of any rule can hide Condorcet winners. A rule's resistance to manipulation is a key to its Condorcet efficiency in policy votes.

Merrill explores Condorcet efficiencies in more complex situations (Merrill, page 39). IRV's chance of electing the Condorcet candidate drops in a polarized society. Its efficiency rises with rising voter uncertainty about candidates' positions on issues but it remains lower than most other rule's. The efficiency of IRV and other non-Condorcet rules drops as the number of candidates increases. Obviously, the elections in which IRV picks the Condorcet winner are a subset of those in which LOR does. voter uncertainty, pre-election polls and "strategic voting" in which each person uses polling information, optimizing his ballot to elect candidates he likes and block those he dislikes.

Surveys and actual elections reveal some randomness, some clusters of like-minded voters and some agreement on the candidates' relative positions left to right. A mixture of random and a normally-distributed voters approximates the observed patterns. But just as random and spatial models lead to different results, so the actual data differs from both of them. Tideman reportedly found that even plurality rule picked the Condorcet winner in 95% of three-candidate elections. He used survey data to simulate rank-order ballots. (Merrill, page 70) This does not recommend plurality since its efficiency drops as the number of contestants rises and all other systems scored higher. Chamberlin and Featherston found similar results when they simulated ballots to resemble the distribution and clustering they found in the APA electorate. So the pattern of opinion dispersion affects Condorcet efficiencies. But the relative standing of the voting systems does not change.

Condorcet efficiency has great importance because they tend to be the median candidates and a happy result for the greatest number of voters. This is not necessarily the greatest total happiness as utility voting systems attempt to define it.

Utility efficiency

The major competitor to Condorcet efficiency is utility efficiency. It attempts to measure how likely a voting system is to elect the candidate with supporters who feel strongly and opponents who don't much care. Many people are skeptical about trying to compare utility values from one voter to another and to hundreds of voters; so Condorcet efficiency remains the most widely accepted measure.

[footnote 1: Researchers attempt to make utility measure the "distance" between a candidate and a voter on an issue. They average the scores for all issues to determine the expected utility value of the candidate for that voter. The candidate's averaged utility score for all voters is said to be her social utility to the electorate. The highest candidate scores from each election in a series of elections are averaged to find the highest average possible. Then the social utility scores of winners under a voting rule are averaged and compared with the highest possible to give theorists a number for the utility of the rule's utility winners' efficiency as a percentage of the highest utility possible.

Following R.J. Weber (1977), most Researchers subtract a large number of utility points, equal to the score of a randomly selected candidate, from both the utility maximizer and the voting system's winners. The size of each score is reduced. But the difference between their scores remains the same. So the difference is now a larger percentage of a score. This exaggerates the differences between voting systems on utility efficiency. You must decide whether such exaggeration helps you see the differences or misleads your understanding of these differences. Get definitions from Merrill, Bordley, and Mueller.]

[footnote 2: There are several different conceptions of "distance": linear, square root, and logarithmic (Merrill page 42, Bordley), and no standard unit to measure interpersonal utility for all types of issues. For these reasons, many people are skeptical about the meaning, comparison, and statistical manipulation of interpersonal utilities.]

Table 3. Utility Efficiencies
in computer simulated elections with 5 candidates and 1000 voters.
from Merrill, page 35
System Random society D=2 D=4 D=2 D=4 D=2 D=4 D=2 D=4
    Dispersion = 0.5 Dispersion = 1.0
    C = 0.5 C = 0.0 C = 0.5 C = 0.0
Voting system Random society D=2 D=4 D=2 D=4 D=2 D=4 D=2 D=4
Plurality 70 -1 0 22 52 64 75 74 93
Runoff 81 28 47 48 75 86 92 88 98
Hare (IRV) 82 40 59 52 82 88 92 91 98
Approval 90 96 96 95 98 96 96 97 98
Borda 95 97 97 96 99 98 98 97 99
Coombs 87 92 92 92 94 96 96 96 98
 

Merrill concludes his chapter on utility efficiency saying that :
“The candidate with the maximum social utility is no more likely to be the Condorcet candidate than is the candidate selected by many if not most of the systems studied. That is to say, the Condorcet criterion and the criterion of maximizing social utility are in fact very different.

“Looked at from the other side of the coin, however, one sees that the Condorcet candidate generally has high social utility, although she may not have the highest of all candidates. This can be seen by comparing the social-utility efficiencies of the Black and Borda systems. The two systems differ only when there is a Condorcet candidate; [Black chooses the Condorcet candidate when there is one] the fact that the former has almost as high an efficiency as the latter indicates that the Condorcet candidate has relatively high social utility, although not as high as the Borda winner even when a Condorcet candidate exists.” (Merrill, page 37)

Whenever the two criteria indicate different winners, the Condorcet winner would beat the utility winner in a one on one election.

The problem with all utility voting systems is that a minority of voters can claim on their ballots that their candidate has a much higher utility value for them than any other candidate. With this claim they may be able to "steal" the election from a complacent majority.

Distribution of Winners

Chamberlin and Cohen's 1978 spatial-model simulations showed Condorcet picked the candidate "nearest" the "center of the electorate" 87% of the time. I think this suggests a political measure of political outcomes -- in contrast to the economic measure of utility. To measure the dispersions of voters and candidates and the distributions of winners and budget allocations assumes that each citizen has an equal right not only to vote but to be represented and to live under government programs compatible with the citizen's philosophy. A system that produced proportional outcomes would reduce majority domination of minorities and so make empire building unattractive. The majority would lose some of its autonomy for every increase in territory.

Table 4. Nearness to the Center of the Theoretical Electorate
4 candidates with low dispersion relative to 1000 voters
from Chamberlin and Cohen (1978)

Voting
system
Nearest
candidate
Second
Nearest
Third
Nearest
Furthest
candidate
Condorcet .87 .11 .02 .00
Borda .81 .17 .02 .00
Coombs .75 .20 .05 .00
IRV .33 .33 .29 .05
Plurality .23 .27 .12 .38

Condorcet has the narrowest distribution around the center. IRV has the second widest. LOR's distribution of winners will depend on the percentage of elections with natural or manipulated voting cycles. We know that natural cycles are rare in elections but they maybe common when enacting policies.

Perhaps Condorcet tends to elect high utility candidates because it directly compares every candidate with each of the others. Simulations by Bordley and Merrill both found Condorcet's rule picked winners a bit lower in utility than Borda which uses all information in one step. Condorcet certainly beats IRV which uses only first-choice information at each of several steps. Notice that plurality tends to elect the least-favorite candidate, the one toward one edge on a scattergram. That's because she has no competition for the voters in that area of the electorate. Meanwhile other candidates split-up the first-choice votes from the electorate's center.

You can run simulation experiments with PoliticalSim TM or just play with it enough to get a feel for the tendancies of a few major rules. Manipulations



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