[EM] Condorcet exegesis

[Richard Lung]

Condorcet pairing

Suppose a preferential (non-binary) election, by 100 voters for three 
candidates, A, B and C, contesting two seats, produces this result:

ABC

463420

A and B are elected on over one third of the votes each (by Droop quota).

Suppose, however, that this contest result is contested by Condorcet 
pairing:

ABACBC

663456443466

A gets all C vote and C gets all A vote. But B is denied its 
proportionate share of representation. A and C are the two-seat 
Condorcet winners. They are implicitly elected each on single majorities 
of over half the votes, 50+ votes each, from an electorate of 100 
voters. At any rate, that is the minimum democratic threshold (even 
Kenneth Arrow insists on). – Binary choice is “The tyranny of the 
(single) majority.” (John Stuart Mill; Lani Guinier.)

For the sake of argument, consider the logic behind alternate pairings. 
Each pairing is a provisional exclusion count, removing each candidate, 
in turn, as a third preference. Thus the A-B pairing amounts to 
A>B>C.The A-C pairing implies A>C>B. The B-C pairing implies B>C>A. 
Condorcet pairing elects A and C on the contradiction that preferences 
A>B>C and C>B>A are both correct.

On the basis of an original triple preference (non-binary) count, A>B>C 
happens to be correct.

The question is: how does one arrive at such a preferential count? (That 
is what I’ve been trying to explain, against some personal animosity.) 
It depends on a multi-majority (proportional) count, as distinct from a 
single majority count, summing a (multi-) preference vote, as distinct 
from a binary choice.

Traditional single transferable vote is sufficiently robust to achieve 
this, with its rational election count. STV does contain a residual 
irrational element, in a sort of last past the post exclusion count. But 
the exclusion count can be calculated on similar rational grounds, as 
the election count. Thus, a binomial STV.

  Regards,

Richard Lung.



On 06/09/2022 11:28, Colin Champion wrote:
> Perhaps some people will be interested in another conclusion I came to 
> from reading Condorcet's Essai. He proposed a method of breaking 
> cycles which generated a lot of confusion until Peyton Young glossed 
> it as a garbled account of the Kemeny-Young method. His reading has 
> been widely accepted; Tideman (in his 2006 book) declared that 
> "Condorcet's intent is decoded to my satisfaction" by Young.
>
> Condorcet described his method twice: forwards in the Preliminary 
> Discourse and backwards in the body of the work. Young only discusses 
> the backwards version. In both cases Condorcet starts from a list of 
> pairwise comparisons, sorted by margin. In the backwards version he 
> writes: "We will successively discard from the contradictory set the 
> preferences which have the smallest majority, and elect the candidate 
> preferred by those which remain". Presumably he stops discarding when 
> the residue is consistent; the flaw is that by this point there may 
> not be enough comparisons left to determine a unique winner. Young 
> noticed this and remarked that "It seems more likely that Condorcet 
> meant to *reverse*, rather than to *delete* the weakest proposition". 
> This is nonsense: no one writes "delete" when they mean "swap", and 
> Young's reading doesn't fit the forwards version.
>
> The forwards statement is clearer: "We thus obtain the following 
> general rule, that whenever we are required to elect a candidate, we 
> must take in turn all the pairwise preferences which have majority 
> support, starting with the largest majorities, and make a decision 
> according to these initial preferences as soon as they imply one, 
> without worrying about the less probable later preferences." In other 
> words, given a list sorted in decreasing order of margin, take an 
> initial part which is small enough to be consistent but large enough 
> to determine a unique winner. This has a corresponding flaw, which is 
> that as you work through the list, you may be forced to include a 
> comparison which contradicts those already present before reaching the 
> point at which you have a winner. But there's nothing here which you 
> can interpret as meaning "swap" rather than something else.
>
> It seems to me as clear as daylight that Condorcet had an incomplete 
> grasp of Tideman's Ranked Pairs. Tideman recognised the risk that a 
> new pair may contradict the ones already in the list, and he saw what 
> to do about it, namely throw it away.
>
> CJC
>
> ----
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