# [EM] Plurality crireion (was "There's indecisiveness, and then there's indecisiveness")

Chris Benham chrisbenham at bigpond.com
Tue Jun 7 08:38:17 PDT 2005

```Mike,
You wrote (Mon.Jun.6):

> Now, about the Plurality Criterion:
>
> Doesn't it say: X must not win if the number of people voting Y in 1st
> place is greater than the number of peope voting X over Y?

No. The "pairwise version" says that  X must not win if  there are more
voters that rank X above all the other candidates than there are voters
that rank Y over *any*  candidate.

> But will failure of the Plurality Criterion cause a strategy problem
> for voters?

Possibly only the relatively mild one of  giving them a random-fill
incentive, but not all voting method problems are strategy problems. (IMHO).

> No one has made a case that the Plurality Criterion is essential, or
> that failing it is a grave problem.

I thought I did that here:
http://lists.electorama.com/pipermail/election-methods-electorama.com/2005-March/014958.html

> "Plurality:  if some candidate x has more first-preference votes than
> some other candidate y has votes in total, then x's probability
> of election must be greater than y's."
>
> In his papers, Woodall likes to economise on axioms; so doesn't
> include the common-sense axiom that a ballot that leaves one candidate
> unranked should be treated/regarded  the same as a ballot that ranks
> this candidate last and all the other candidates the same.
> His  "votes in total" refers to explicit rankings in any position.
>
> Incorporating this axiom, a "version" of the Plurality criterion  I
> like is
>
> "If  some candidate x is ranked in first-place on more ballots than
> candidate y is ranked above equal-last, then y can't win".
>
> A Woodall example:
> 11  ab
> 07  b
> 12  c
>
> so A is barred by C's  "plurality" over A.
> A>B 11-7  (m3)
> B>C 18-12 (m6)
> C>A 12-11 (m1)
> Margins elects A, while WV elects B.
>
> It took me a long time before I could see any particular point to this
> criterion, but now I rate it very highly.
> Obviously if candidate x has a "plurality" over candidate y, then x
> pairwise beats y. If y is elected, then those voters that prefer x to y
> will have a virtually  *unanswerable* complaint:  "How can you justify
> x losing to y? Maybe x losing to some z can be justified, but not to y!"
>
> If some pairwise method that fails Plurality has just replaced FPP,
> and  x's supporters are not great fans of the new method and x comes
> FP-first,
> then it is easy to imagine that a riot could be fomented.

That said, my current favourite single-winner rankings method is
CDTT,IRV, which also fails the Plurality criterion. When the method
meets Later-no-Harm while failing
Later-no-Help all the voters have an incentive (normally, and in the
zero-info. case) to fully rank, thus making a failure of  Plurality in
practice very unlikely.
A possible "real-world" horror scenario is that a lot of voters truncate
anyway, acting on the advice of parties whose agenda is to bring the
method into disrepute so that they are in a better position to move to
have it scrapped.

In the  49A, 24B, 27C>B scenario both CDTT,IRV and MMPO,FPP elect C,
which is a serious violation of  the Plurality criterion.

Chris  Benham

```