# [EM] Two more 3-slot FBC/ABE solutions

C.Benham cbenhamau at yahoo.com.au
Tue Jan 24 09:53:12 PST 2012

```Following on from my recent definition of  the APPMM criterion/set, I'd
like to propose two not bad 3-slot methods that meet the FBC..

Recall that I defined the APPMM criterion thus:

> *If the number of ballots on which some set S of candidates is voted
> strictly above all the candidates outside S is greater than the number
> of ballots on which any outside-S candidate is voted strictly above
> any member of S, then the winner must come from S.*

The APPMM set is the set of candidates not disqualified by the APPMM
criterion.

APMM//TR:

* Voters fill out 3-slot ratings ballots. Default rating is Bottom
(signifying least preferred and not approved.) The other slots are Top
(signifying most preferred) and Middle.

From the set of candidates not disqualified by the APPMM criterion,
elect the one with the most Top ratings.*

APMM//CR:

* Voters fill out 3-slot ratings ballots. Default rating is Bottom
(signifying least preferred and not approved.) The other slots are Top
(signifying most preferred) and Middle.

From the set of candidates not disqualified by the APPMM criterion,
elect the one with the highest  Top minus Bottom ratings score.*

So far I can't see that these are technically any better  than my
earlier suggestion of  TTPBA//TR, and unlike that method they fail the
"Tied at the Top Pairwise Beats All" criterion.

But like that method they meet the Plurality and  Mono-add-Plump
criteria, and also have no problem with Kevin's bad MMPO example.

I'm happy for APMM//CR to be also called APMM//Range. This method is
more Condorcetish than APMM//TR, for example:

49: C>B
27: A>B
24: B>A

B>A 73-27,  B>C 51-49,  A>C 51-49.

APMM//TR elects A, while  APMM//CR elects B (like TTPBA//TR).

I am sure that APMM//TR has no defection incentive in the "Approval Bad
Example", and the other method also does in the example normally given.

Of course some other points-score scheme (perhaps giving greater weight
to to Top Ratings) is possible.

Chris Benham

```