[EM] Method definitions

Markus Schulze schulze at sol.physik.tu-berlin.de
Sun May 28 08:16:53 PDT 2000

Dear participants,

a few month ago there was a discussion whether Condorcet
implicitely presumes that every voter makes a complete
ranking of all candidates. On 8 March 2000, I wrote that
in the paper "Sur les Elections / On elections" (Journal
d'Instruction Sociale, vol. 1, p. 25-32, 1793) Condorcet
explicitely discusses the possibility of equal rankings.
Now I have found a translation of this paper.

McLean translates the important part of this paper as
follows (Iain McLean, Fiona Hewitt, "Condorcet," Edward
Elgar Publishing, 1994, page 236-238):
>    For a man who chooses alone and who wants to adhere
> to a strict procedure, an election is the result of a
> series of judgements comparing all the candidates two
> by two. The candidates to whom he restricts his choice
> are, in this case, the men he considers worthy of the
> place.
>    Similarly, an election vote is the result of those
> judgements which obtain a majority. In each judgement,
> an individual weighs up the reasons for preference
> between two candidates. In an election, each person's
> votes for or against a candidate represent these reasons,
> and are counted rather than weighed up. When a man
> compares two individuals and prefers the second to the
> first, and then, on comparing the second with a third,
> prefers the latter, it would be selfcontradictory if he
> did not also prefer the third to the first. If, however,
> on making a direct comparison of the first and the third,
> he found reasons for preferring the first, he would then
> have to examine this judgement, balance the reasons
> behind it with those behind his other judgements which
> cannot exist alongside this new one and sacrifice the
> one he considers least probable.
>    In an election between three candidates, the three
> judgements which obtain majority support when the
> candidates are compared two by two may sometimes be
> inconsistent, even when the individual judgements of
> each voter involve no contradiction. This can easily
> be shown through examples and can be explained by the
> simple fact that the majority in favour of each of
> these accepted propositions is not made up of the same
> individuals. We must therefore reject the proposition
> with the smallest majority and retain the other two.
> Thus, whenever we are able to obtain only a fairly large
> probability, we must reject a proposition which is
> probable in itself if it excludes another which is more
> probable.
>    An election result incorporating a proposition which
> runs contrary to one of the majority judgements may still
> be very probable. In fact, this only occurs in situations
> where it is certain that the majority has been mistaken
> at least once. The probability of the result being
> accurate is then equal to the probability that the
> majority has made only one mistake regarding a certain
> proposition.
>    There is no need to point out that the contradiction
> which can occur between majority judgements when there
> are three candidates must be even more likely when there
> are a greater number of candidates; or that, in such
> cases, several of these judgements may contradict one
> another, meaning that the majority must necessarily have
> made more than one mistake. The consequences of this are
> the same.
>    The list of candidates who are put forward for election
> must be determined if all the members of an assembly are
> to give a complete vote by comparing the same candidates
> two by two, or to list them in order of merit, which comes
> to the same thing. That is, everyone must know the names
> of the candidates between whom the voters will be choosing.
>    However, it is not necessary for everyone to compare
> all the candidates or to form a complete list. A voter may
> for various reasons regard a certain number of candidates
> as equal to one another, either after considering their
> attributes or because he does not know the candidates and
> is therefore unable, or unwilling, to judge them.
>    This condition in no way restricts the voters' freedom,
> since it simply requires everyone to decide which candidates
> he wishes to choose between. The list of all those put
> forward in this way would then present each voter with the
> names of all the candidates between whom the other voters
> wanted the election to be conducted, and he would then
> have complete freedom to decide how he could share in this
> judgement: which candidates he wanted to rank in order of
> merit and which to reject entirely by placing them after
> all the others.
>    Any election method in which the votes given are
> incomplete will produce results which contradict the will
> which the majority would have had if complete votes had
> been collected.
>    The results of these incomplete votes will of course have
> some degree of probability of being correct, but it would be
> similar to that of a proposition which has been only half
> examined. In fact, we should support a probable proposition
> only when we have discovered the impossibility of
> incorporating new information, and as long as this
> impossibility lasts.
>    However, we would be just as far from fulfilling our aim
> if we forced each voter to express, not the complete vote
> which he actually forms, but a complete vote in an absolute
> sense; that is, if we forced him to establish an order of
> preference between all the candidates, including those he
> does not know. Clearly, he would then rank the latter at
> random and his vote could result in the election of a
> candidate who would not otherwise have had sufficient
> support. In the first case, we are neglecting judgements
> which should have been assessed, and in the second, we are
> assessing judgements which have not been given. In the first
> case, we are acting as if we had randomly excluded a certain
> number of voters, and in the second as if we were randomly
> giving some of them double the number of votes.
>    In theory, therefore an election procedure should be as
> follows: after having determined the list of acceptable
> candidates, each voter should express his complete will,
> whether of preference or indifference.
>    A table of majority judgements between the candidates
> taken two by two would then be formed and the result - the
> order of merit in which they are placed by the majority -
> extracted from it. If these judgements could not all exist
> together, then those with the smallest majority would be
> rejected.
>    This is exactly the same procedure as that followed by
> any individual who wants to make a considered choice by
> using a general, regular method which applies to all
> situations.
>    Since there is only one way of obtaining a true decision,
> the procedures used by a deliberative assembly should be as
> close as possible to those used when an individual examines
> a question for himself.
>    This principle can have other important applications. In
> this case, it allows us to develop an election method which
> is reasonably natural and as perfect as the nature of things
> permits.


The term "majority / majorite" (instead of the usually used
term "plurality / pluralite") is also used in the French
original of this paper.


McLean comments on Condorcet's paper as follows (page 48):
> 'On elections' is hard to evaluate because it is extremely
> scrappy. But it clearly marks a return to earlier themes of
> probability and enlightenment, and hence away from citizenship
> and democracy; it may also mark some disenchantment with
> Condorcet's ideal voting scheme. Had Condorcet lived, he
> might have resolved the internal contradictions in his
> thought, although the tensions are so great that even he may
> not have been able to accomplish this.

Markus Schulze
schulze at sol.physik.tu-berlin.de
schulze at math.tu-berlin.de
markusschulze at planet-interkom.de

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