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Condorcet Efficiency

[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)]

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

“Random society” or “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 normal bell-curve 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.

A sim maker's choice of models effects the results. Disregard research that does not use realistic data. Remember: “Garbage in, garbage out.“ To learn about life, use the most normal, lifelike sim.

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 the winners 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

[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.]