[EM] "Condorcet winner" versus "winner of Condorcet's method" (was Re: 2018 Chess Candidates Tournament)

[robert bristow-johnson]

---------------------------- Original Message ----------------------------

Subject: [EM] "Condorcet winner" versus "winner of Condorcet's method" (was Re: 2018 Chess Candidates Tournament)

From: "Steve Eppley" <SEppley at alumni.caltech.edu>

Date: Wed, March 28, 2018 11:52 am

To: election-methods at lists.electorama.com

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> @Ross Hyman: Ding Liren was not a Condorcet

> winner in that chess tournament, because a

> Condorcet winner is an alternative that

> defeats all other alternatives pairwise. 

> Ding Liren didn't defeat all other players;

> he won only one game.
yes, i would not call that the CW.


> Some people might prefer a weaker,

> non-standard definition of Condorcet winner:

> a candidate that's undefeated pairwise. (Like

> Ding Liren.)  In public elections the two

> definitions (if implemented by two voting

> methods) would behave the same with regard to

> the incentives on candidates, potential

> candidates, voters, parties, donors, etc.,

> because ties are rare when there are many

> voters, as there are in public elections.

 
and the reason for that is that with an electorate of decent size (like at least hundreds of voters) the probability of a tie in any pairing of candidates is very low.


> Don't be misled the way many people have

> been, especially mathematicians not familiar

> with the social choice theory literature. 

> They wrongly believe "Condorcet winner" means

> the winner according to Condorcet's method,

> and thus that Condorcet's method simply

> elects the candidate that defeats all others

> pairwise, and is indecisive when no such

> candidate exists.  "Condorcet winner" is a

> term of art (a.k.a. jargon).  Unlike Borda

> winner, which is not a term of art and merely

> means the winner according to Borda's method,

> and Black's method, which is not a term of

> art and merely means the winner according to

> Black's method, etc.

>

> Because sometimes there is no candidate that

> defeats all others pairwise, the confusion

> has caused a number of writers to wrongly

> claim Condorcet's method is often indecisive

> and therefore unsuitable for elections.
i just read what i see here and what i see in the EM wiki and in Wikipedia.  i hadn't thunk there was a "Condorcet's method" but that there are a few decisive methods that are "Condorcet compliant", which means these methods
will elect the CW **if** a CW exists (and i really think that in most public elections with a ranked-order ballot, that a CW will exist virtually all of the time, and most of the time, i'll bet that the IRV method will also elect the CW, but not always).
> (In simulations with random
voting, the frequency
> of scenarios in which no candidate defeats

> all others increases asymptotically to 100%

> as the number of candidates increases to

> infinity, and as the number of voters

> increases.)  But the voting method Condorcet

> promoted in his famous 1785 essay is very

> decisive:

>

> CONDORCET'S METHOD (copied from page lxviii

> of his 1785 essay):

>

> Here's its 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."

>

> The phrase "this general rule, for all the

> times when one is forced to elect" meant he

> was referring to a very decisive voting method.

>

> A "proposition" is a pairwise statement like

> "x should finish ahead of y."  It has the

> plurality if the number of voters who agree

> with it exceeds the number of voters who

> agree with the opposite proposition.

>

> "Taking successively commencing with the

> largest" means considering the propositions

> one at a time, from largest to smallest.

> (Like MAM and Tideman's Ranked Pairs do. 

> However, MAM and Ranked Pairs measure size in

> different ways: MAM measures the size of the

> majority, whereas Ranked Pairs subtracts the

> size of the opposing minority from the size

> of the majority.

 
that's RP-margins.  there is also RP-winningVotes.  how does this method from Condorcet differ from RP-winningVotes?


>  The word "plurality" can

> mean either of those: either the larger

> count, or the difference between the larger

> count and the opposing count.)

>

> The "result" is an order of finish, like "x

> finishes ahead of y, y finishes ahead of z,

> etc."  It's a collection of pairwise results,

> each of which is obtained either directly

> from a proposition that has a plurality, or

> transitively from a combination of pairwise

> results obtained directly.  An example of a

> pairwise result obtained transitively is the

> pairwise result "x finishes ahead of z"

> obtained transitively from "x finishes ahead

> of y" and "y finishes ahead of z."  By

> definition, an order of finish is an

> ordering, and is thus transitive and acyclic.

>

> "Without regard for the less probable

> propositions that follow" means disregarding

> propositions that conflict (cycle) with the

> results already obtained from propositions

> that have larger pluralities.
I cannot see how that differs from Ranked Pairs.


>  For example,

> disregarding "z should finish ahead of x"

> after having obtained the pairwise results

> that "x finishes ahead of y" and "y finishes

> ahead of z."

>

> Note: No language in the definition of

> Condorcet's method refers to an alternative

> that defeats all others pairwise. (Nor to an

> alternative that's undefeated pairwise.) 

> Although it can be deduced that Condorcet's

> method will elect an alternative that defeats

> all others, it will also elect an alternative

> even when no alternative defeats all

> others... in other words it's very decisive. 
so this historical "Condorcet's method" always elects a single-winner and, **if** a pairwise champion exists, it will elect that pairwise champion.  so "Condorcet's method" is Condorcet-compliant.


> People who write about "Condorcet completion"

> rules -- first check whether there exists an

> alternative that defeats all others and then,

> if no such alternative exists, proceed in

> some other way to find the winner -- have

> misunderstood Condorcet's method,

 
or, perhaps we haven't heard of Condorcet's "method".  but if they apply "Condorcet completion" rules to another Condorcet-compliant method that doesn't need completion rules (such as RP or Schulze), i think that reflects the same
misunderstanding.

> which is

> already "complete" (very decisive when there

> are many voters, because when there are many

> voters it's rare that any two majorities are

> the same size, and rare that any pairings are

> ties).
yup.
thanks for the information, Steve.


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