@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.

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.

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. (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):

"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."

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. 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. 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.

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, 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).

Some prominent authors have wrongly claimed

Condorcet's method is Maxmin (elect the

candidate whose largest defeat is the

smallest), which is equivalent to

successively deleting the smallest majority

until a candidate is undefeated pairwise.

With Maxmin, an alternative defeated pairwise

by all others (a.k.a. "Condorcet Loser") can

finish in first place, because all of its

defeats could be small majorities, and thus

could be deleted. But with Condorcet's

method a Condorcet Loser, if one exists,

always finishes in last place. None of its

defeats conflict with any other pairwise

results, so none of its defeats will be

disregarded and thus all other candidates

will finish ahead of it.

--Steve

------------------

I think chess is an example where Borda is

preferable to Condorcet. Grand masters are

criticized for excessively drawing. If the

winner was chosen by Condorcet, this would

exasperate the problem by further

incentivizing draws. Ding Liren is the

Condorcet winner because he did not lose a

single game. But he drew 13 times out of

14 games.

-snip-

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