[EM] Plurality criterion and the "sincere CW"

C.Benham cbenham at adam.com.au
Sun Jun 30 09:00:34 PDT 2019

Juho (and interested others),

The Plurality criterion was coined in 1994 by Douglas Woodall. Quoting 
him exactly from then:

> The following rather weak property was formulated with single-seat 
> elections in mind, but it makes sense also for multi-seat elections 
> and, again, it clearly holds for STV .
> Plurality. If some candidate /a/ has strictly fewer votes in total 
> than some other candidate /b/ has first-preference votes, then /a/ 
> should not have greater probability than /b/ of being elected.
> *
> *
No mention of any "implicit approval cutoff".  I know that at the time 
Woodall was only thinking about strict rankings from the top with 
truncation allowed.
If equal-first ranking is allowed, then for the purpose of this 
criterion we should be using the fractional (summing to 1) 
interpretation of the number of
"first-preference votes".

Juho seems to think that the Plurality criterion is a "feature" or 
strategy device  that somehow encourages truncation.  It isn't and doesn't.

If the method uses one of the traditional Condorcet algorithms that are 
almost the same as each other (Smith//MinMax, Schulze, River, Ranked Pairs)
and uses Winning Votes as the measure of pairwise defeat strength, then 
the method meets Plurality and also has, at least in the zero-info case, 
a weak
*random-fill incentive.

IRV, and IRV modified to meet Smith by before each elimination checking 
to see if there is pairwise-beats-all candidate among those remaining, 
both meet
the Plurality criterion.  In those methods do the voters have any have 
any incentive "not to rank the candidates of the competing groupings" ?  
No they

So what is the point of the Plurality criterion?  To my mind it is 
simply about not offending obvious fairness and common-sense.

Juho, try to imagine that you have no interest in or knowledge about 
voting algorithms, you've never thought about the split-vote problem. 
You are accustomed
to voting in plurality elections (or even perhaps Approval elections) 
and you've never been interested in doing anything other than voting for 
your sincere
favourite, who regularly wins.  You are content with the current voting 
method and can't see any point in changing it.

Now imagine some voting-reform movement succeeds and the new method is, 
say, MinMax(Margins).  You hear that voters can now rank more
than one candidate and you simply seek assurance that you will be 
allowed to go on voting as before and you assume that the government must
more-or-less know what it's doing and assume the method won't in any way 
be less fair than before.

In this election your favourite is A.
46: A
44: B>C
10: C

It is announced that the winner is B.  At first you think "A got more 
first-preference votes than B, it must have something to do with some 
second preference votes", but then you notice that B got the same number 
of second-preference votes as A (zero), and then you ask "How on earth
did this crazy new method elect B over my favourite A, who very clearly 
got more "votes" (marks next to his name on the paper ballots) all of which
were first-preference votes!"

On hearing the reply "Oh, that's because B was the fewest votes shy of 
being the Condorcet winner" do you (a) say "Oh how silly of me, obviously
that's fair!" or  (b) say .. something much less understanding and 
accepting ?

This scenario also works if the old method was IRV.  You might also 
notice that this first MMM election scenario is also a massive egregious 
of the Later-no-Help criterion (because if the B voters had truncated 
then B wouldn't have won).  Do you like that criterion?

If the old method had been Approval, you would then presumably be 
understanding and resigned if it is announced that C won.
In fact electing A is a failure of the Minimal Defense criterion. Do you 
like that one?   So methods that meet both MD and Plurality (such as Winning
Votes and Smith//implicitA) must elect C.

> ... methods might not elect the best winner (sincere Condorcet winner). 
If voters decline to (or don't bother to) express some or all of their 
very weak (possibly light-minded) pairwise preferences by truncating, 
then I don't
classify that as "insincere" voting.  Since therefore there could be 
several (or even many) alternative "sincere voting" profiles it follows 
that there
could be more than one "sincere CW".  It seems obvious to me that the 
one of of those that is based on only the relatively strong pairwise 
will have a higher "social utility" than one based on all pairwise 
preferences which include a lot of very weak ones.

49: A>>>B>C
03: B>A>>>C
48: C>>>B>A

Say these are the sincere preferences. If the voters care to express all 
their pairwise preferences then the "sincere CW" is B, but if they choose
what I consider to be an alternative way of sincere voting and truncate 
where that will only "conceal" some weak pairwise preferences then
an alternative "sincere CW" is (the apparently higher Social Utility 
candidate) A.

In fact if the method used was the tweaked IRV  method with an explicit 
approval cutoff that I recently suggested and the cast votes were
49: A>>B
03: B>A>>
48: C>>B

then only C would be disqualified (because A both pairwise beats C and 
is more approved than C) and then B is eliminated and A wins.
I doubt that there would much blood flowing in the streets caused by the 
failure to elect the voted CW (B).

As consolation for not meeting the Condorcet criterion we would have a 
method much more resistant to Burial strategy than any Condorcet
method (and maybe more appealing to people who like IRV).

Juho Laatu* juho.laatu at gmail.com 
/Sat Jun 29 07:43:29 PDT 2019/

> P.S. I don't like the plurality criterion. It actually sets an 
> implicit approval cutoff at the end of the listed candidates. The 
> worst part of that idea is that it encourages voters not to rank the 
> candidates of the competing groupings. That (potentially huge amount 
> of missing information) is not good for ranked methods. If voters 
> learn to use that feature, methods might not elect the best winner 
> (sincere Condorcet winner). 

The following rather weak property was formulated with single-seat 
elections in mind, but it makes sense also for multi-seat elections and, 
again, it clearly holds for STV .
Plurality. If some candidate a has strictly fewer votes in total than 
some other candidate b has first-preference votes, then a should not 
have greater probability than b of being elected.

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