# [EM] "Mutual Plurality" criterion suggestion

Greg Dennis greg.dennis at voterchoicema.org
Fri May 11 03:14:23 PDT 2018

```Chris, do you have a precise definition of "irrelevant ballot"? Just a
ballot that expresses indifference between the smallest mutual majority set?

On Sun, May 6, 2018, 12:09 PM Kristofer Munsterhjelm <km_elmet at t-online.de>
wrote:

> On 05/06/2018 05:58 PM, Chris Benham wrote:
> >> *If there exists one or more sets S of at least one candidate that is
> >> voted above (together in any order) all other
> >> candidates on a greater number of ballots than any outside-S candidate
> >> is voted above any member of S (in any positions)
> >> then the winner must come from the smallest S.*
> >>
> > On 7/05/2018 12:21 AM, Kristofer Munsterhjelm wrote:
> >> Isn't the set of all candidates always a Mutual Plurality set, in a
> >> vacuously true sense?
> >
> > I meant to imply that if there aren't any "other candidates" then the
> > "set" doesn't exist.  Maybe:
> >
> > *If there exists one or more subsets S of at least one candidate that is
> > voted above (together in any order)  all of the (one or more) outside-S
> > candidates on a greater number of ballots than any outside-S candidate
> > is voted above any member of S (in any positions) then the winner
> > must come from the smallest S.*
> >
> > But as I initially defined it, then I suppose yes. But that doesn't much
> > matter. All methods might then elect from at least one Mutual Plurality
> > set, but only those who elect from the smallest one meet the criterion.
>
> I think the original definition works, as the same thing happens for
> mutual majority. Every method elects from some solid coalition that has
> greater than majority support (namely, the coalition of all candidates),
> but the method only passes the mutual majority criterion if it elects
> from the smallest such set. In some situations, that smallest set *is*
> the set of all candidates, which means there's no special case logic
> needed for such a case; a method that passes mutual majority in the
> "proper" cases is then free to choose any candidate to be elected
> without violating the criterion.
>
> I agree, though. It doesn't much matter, beyond in an elegance of
> definition sense.
>
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