[EM] Can anyone help with straight-ahead Condorcet language?

[Steve Eppley]

On 9/12/2021 11:49 AM, robert bristow-johnson wrote:
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> I am actually fiddling around with creating plausible language for RP. But right now I am trying to show to legislators how *simple* in concept Condorcet is. So I am less concerned with the fallback language in case there is no CW. 

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Here's simple language to explain the concept:

    Count all the head-to-head majorities using the information in the voters' orders of preference.

    Construct the order of finish by processing the majorities one at a time, from largest majority to smallest majority, placing each majority's more-preferred candidate ahead of their less-preferred candidate in the order of finish.

I also recommend providing two simple examples: The first example with a Condorcet Winner and three candidates (perhaps named Left, Center and Right).  The second example with no Condorcet Winner and three candidates (perhaps named Rock, Scissors and Paper).

If one believes it's essential to include the rock-paper-scissors exception in the "simple concept" language, here's more complete language:

    [...]

    Construct the order of finish by processing the majorities one at a time, from largest majority to smallest majority, placing each majority's more-preferred candidate ahead of their less-preferred candidate in the order of finish (unless their less-preferred candidate has already been placed ahead of their more-preferred candidate).

Condorcet himself did NOT define his voting method as "First check whether a candidate defeats all others head-to-head, etc."  Here's what he actually wrote in his 1785 essay, after his meandering analysis of some 3-candidate cyclic examples:

    Il résulte de toutes les réflexions que nous venon de faire,
    cette règle génerale, que toutes les fois qu'on est forcé d'élire,
    il faut prendre successivement toutes les propositions qui ont
    la pluralité, en commençant par celles qui ont la plus grande,
    & prononcer d'après le résultat que forment ces premières
    propositions, aussi-tôt qu'elles en forment un, sans avoir égard
    aux propositions moins probables qui les suivent.

In case your French is rusty, here's a literal translation to English:

    The result of all the reflections that we have just done,
    is this general rule, for all the times when one is forced to elect:
    one must take successively all the propositions that have
    the plurality, commencing with those that have the largest,
    and pronounce the result that forms from these first
    propositions, as soon as they form it, without regard
    for the less probable propositions that follow them.

Here's how I interpret the terms in Condorcet's definition:

By "for all the times when one is forced to elect" Condorcet meant this is his voting rule for any single-winner election.

By "propositions" Condorcet meant propositions of the form "x shall finish ahead of y."  Votes that rank x over y constitute support for "x shall finish ahead of y" and opposition to "y shall finish ahead of x."

By "propositions that have the plurality" he meant the propositions supported by a relative majority. (Which could be less than half the votes if some voters express indifference.  His essay assumed no indifference.)

To "take successively" a collection means to take one thing at a time, in some order.  This has two possible interpretations: (1) Each thing may be one item (one proposition) in the collection, or (2) each thing could be a subset of the collection if there's a way to order the possible subsets so that the subsets can be taken one at a time.  The simpler and more natural interpretation is one proposition at a time, and that's how I interpret it.  It follows that "commencing with those that have the largest" means "from largest majority to smallest majority."

By "less probable propositions that follow" Condorcet meant propositions with smaller pluralities. (Either less support, or less support-minus-opposition.)  Because their pluralities are smaller, they follow later in the order of succession (which I usually call the order of precedence).  Condorcet's majority rule heuristic was: The larger the number of people who think x is better than y, the more likely it is that x is better than y.

By "pronounce the result that forms from these first propositions" I think it's clear Condorcet meant to include results implied by transitivity.  For example, if the two largest majorities support "Scissors shall finish ahead of Paper" and "Rock shall finish ahead of Scissors," he would place Scissors ahead of Paper and Rock ahead of Scissors in the order of finish.  An order of finish is transitive, and by transitivity he has also placed Rock ahead of Paper, "without regard for the less probable" "Paper shall finish ahead of Rock" proposition that follows.

With those interpretations, it's straight-forward to translate the English literal translation of Condorcet's method to the simple concept language I suggested above, repeated here for convenience:

    Construct the order of finish by processing the majorities one at a time, from largest majority to smallest majority, placing each majority's more-preferred candidate ahead of their less-preferred candidate in the order of finish (unless their less-preferred candidate has already been placed ahead of their more-preferred candidate).

--Steve