@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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