[EM] Probability of losing a sincere Cond. Winner

[Elisabeth Varin/Stephane Rouillon]

Adam --

I have very well understood you define a strong CW as having
more than 50% votes against any other candidate.

I made more comparative study of both ideal scenarios....

With my scenario, I would need to track and convince 11% of the 18%
potential truncaters in order to steal the strongly supported CW with (wv)
(more than half).

With your scenario, you would need to track and convince 47% of the 49%
potential truncaters in order to steal the poorly supported CW with (rm)
and 48% of the 49% potential truncaters (almost all of them) in order to
steal the poorly supported CW with (margins).

It modifies the odds of an unsincere truncation, don't you think so?

But it dazzled me: almost all of the potential truncaters needed to steal a strongly supported CW. You are right he is strong!

But again it depends on the criteria: if we compare the strongness of the CW to
its stealer opponent using the other criterias, it is reversed.
Your "strong" CW has only a sloppy 2% margin advance over its potential stealer.
Other margins in the scenario are greater.
My "weak" CW has 10 points of advance before losing its winning votes duel.
Others advance are the same.

Your "strong" CW can get stolen by removing a huge support he had against a third
candidate. My "weak" CW can get defeated by removing some of the direct support
he had over its potential stealer.
How would these scenarios behave when candidates and voters increase in number?
I do not know yet.

But it seems poorly-supported and strongly supported CW is a relative definition
that can depend on the criteria too...

Steph.

Adam--

I fully agree with your way of seeing things.
It is only about a matter of proportion were we do not see things the same.
I will use samples of what follows.

But if you shift the percentages
around a little, or reduce the degree of truncation a bit in the edge
factions, then it breaks down.  Truncation only works strategically in
winning votes when there is a lot of truncation (indifference) to start,
and even then it only works in specific cases with fractured electorates,
where the Condorcet winner has pretty poor support from other factions.

With 3 candidates, I have to agree with you, even if I haven't yet done the calculus... It seems
easier to build cases against margins than against winning votes. I do not know for relative
margins...

For more candidates, I do not know. But I think we have a problem of reference.
You seem to be interested into american presidential election, with typically a small
number of candidates (3 non-negligeable last time). I like it, but an electoral system
should work well when multiple candidates run for office. Usually I think the more
alternative a voter is offered, the more different debates and positions are offered
and the healthier the democracy (if this does not paralyse the system by vote-splitting).
French got 17 official candidates last time. If I remember well, 15 were non-negligeable
(it means by rallying their vote to the third runner, they could have changed the outcome
of the election). In such a case, with 17 candidates, please do not tell me that truncation
from start (sincer preferences) would not be the natural way to vote... (less than 60% of
voters go vote already, and we only ask them to pick one name!)

I admit my sincere Condorcet Winner has a "poor support" (thanks for answering with a good
example for what I asked Mike) compared to your sincere Condorcet Winner.
But what is more realistic? Identifying and convincing 18% of the voters to tilt their preferences
from poll predictions as in my case, or 49% in your case. As you said my case is very unstable,
I bet I would convince nobody. But what about yours? Even if I admit it is very more probable
than mine, how probable is it? 1% of the cases? We definitively need to compute this number,
even for the not so simple 3 candidates case with a small number of voters. Compared to the polls
precision of 9 different ballots (A, B, C, ABC, ACB, BAC, BCA, CAB , CBA) I think you would still
convince nobody. Finally, with 4 candidates or more, forget polls precision, forget identifying voters
with the same rankings, forget strategy at all. It has no mathematical relevance. It is too small cases.

Since sincere preferences seem the most probable input, I think we should consider an equal probability
of the 9 different ballots. I hope we can create a stealing CW case with few voters, because only with
4 voters we have 6561 equiprobable simulation cases...

The next thing we will have to do is look at what happens when the number of candidates or/and voters
inceases. Would the ratio (CW can be stolen cases / voting simulations) increase or diminish in each case?

This will be true in a huge range of cases -
basically, any time the second-place candidate's support for the winner
allows the winner to beat another edge candidate.  You can fudge the
numbers in the above example around a large amount and get the same results
- provided you don't give the George faction an absolute majority of course.

To summarize, I believe your "huge range of cases" will be incredibly
small compared to the overall range of possibilities...

And polls will not be sufficient to make it trustable enough, unless
they can garantee less than 5 % errors and their result is your example.

With your ideal example for winning votes a 6% error per kind of ballot
could twist the result against truncaters wishes:
43%: George>Al>Ralph

6%: Al>George>Ralph

18%: Al>Ralph>George

33%: Ralph>Al>George
So in your ideal case, yes you could convince some people. But even
then, after looking at the real numbers, in most of the cases it would
not change the outcome... (George would have won and still does, or Al
would have won and still does) after polls errors corrections.

I do not think preventing unsincere truncation should be the main issue.
To me, optimal fairness of the representation of the wish of the people
should.

Unsincere truncation will almost never happen. Sincere truncation will
almost always happen.
Steph.
Adam Tarr a écrit :
>This appears to be an example that illustrates a
more stable outcome is

>achievable

>by counting equal ranked options 1/2 vote each.
Matt, we went over this before.  By adding 1/2 of a vote for each
side, you

turn winning votes into margins.  It's not a compromise in-between
the two

at all - it becomes margins exactly.  Adding half-votes in margins
does not

change the outcome at all.  So since Stephane cooked up his example

specifically to show when winning-votes encourages truncation (and
margins

does not) it's not surprising that adding half-votes "solves" the problem.
The reality, though, is that it's significantly easier to come up with

examples where truncation is encouraged in margins-based methods. 
Here's

the example Stephane came up with, casted into percentages and given
some

familiar names:
36% George

9% Al>George>Ralph

18% Al>Ralph>George

18% Ralph>Al>George

19% Ralph
Ralph is the Condorcet winner, but truncation by the middle 18%, in
winning

votes, will give the election to Al.  But if you shift the percentages

around a little, or reduce the degree of truncation a bit in the edge

factions, then it breaks down.  Truncation only works strategically
in

winning votes when there is a lot of truncation (indifference) to start,

and even then it only works in specific cases with fractured electorates,

where the Condorcet winner has pretty poor support from other factions.
Contrast that to this case, which is not hard to come up with at all:
49%: George>Al>Ralph

12%: Al>George>Ralph

12%: Al>Ralph>George

27%: Ralph>Al>George
Al is the Condorcet winner.  Now, if the George voters truncate
in margins,

they win the election.  This will be true in a huge range of cases
-

basically, any time the second-place candidate's support for the winner

allows the winner to beat another edge candidate.  You can fudge
the

numbers in the above example around a large amount and get the same
results

- provided you don't give the George faction an absolute majority of
course.
Winning votes gives the election to Al, even if George voters truncate. 
If

you add half-votes, then you run into the same problems you have with

margins -- naturally, since adding half-votes turns winning votes into
margins.
-Adam
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