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

[Steve Eppley]

@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
------------------
On 3/27/2018 4:45 PM, Ross Hyman wrote:
> 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-