[EM] Best Deterministic Ranked Preference Method?

[Forest Simmons]

Kevin,

Thanks for holding my feet to the fire!

...Weinstein approval comment in line below. We seem to understand it
differently in two ways...

On Thursday, December 24, 2020, Kevin Venzke <stepjak at yahoo.fr> wrote:

> Hi Forest, two replies in one:
>
> >On Wednesday, December 23, 2020, Kevin Venzke <stepjak at yahoo.fr> wrote:
> >>
> >> I thought it was odd that if one votes A>B (approving both) and B
> pairwise defeats A, then A can't get credit for this
> >>vote. So I took the liberty of devising another method called BTP
> (P=preferred). In BTP, instead of checking for losses
> >>to all the approved candidates, you check for losses to the candidates
> strictly preferred to the candidate you're
> >>scoring. This revised method turns out to be incredibly similar to
> MinMax(WV). It's not so obvious to me yet why it
> >>should be, but for now I'm amazed.
> >>
> >We're getting close with BTP!
> >
> >My tweak: x gets a point from every ballot B on which it is both ranked
> AND it pairwise beats every alternative
> >ranked strictly above it on that ballot B.
> >
> >More generally, ... a point from every ballot on which it is both
> approved and it beats every alternative ranked above it.
> >
> >(No alternative gets a point from any ballot on which it is not approved,
> i.e. not ranked in the case of implicit approval)
> >
> >How about that?
>
> I call this BTPA. (But I used beat-or-tie, not strictly beat.) It is quite
> different from BTP or BTA, and appears extremely similar to Condorcet//DSC,
> especially with 3 candidates. For my taste this is not very good. It's also
> pretty close to fpA-fpC though, with 3 candidates.
>
> Basically with BTPA your score as a candidate is capped by your implicit
> approval, and you can be deprived of it by weaker candidates who beat you
> pairwise. In contrast it seems to be important to BTP's properties that
> candidates can score off of ballots that didn't vote for them, which only
> voted for candidates that were beaten by them.
>
> Also, it seems to me to be an advantage of BTP over BTA or BTPA that you
> don't need any implicit approval concept.
>
> ---
>
> Regarding HBH and Weinstein, I'm having a lot of trouble getting this to
> work reasonably. For one thing, I can't find a way to interpret
> single-round Weinstein DSV Approval so that it is monotone. Also, if
> approval can extend into the bottom rank, it violates Plurality a lot. I
> can disallow that, but this creates a lot more monotonicity issues.
>
> (One could suggest that the approval cutoff be placed prior to the rank
> that exceeds 50% of the random ballot odds, but this doesn't seem workable
> because it means a majority favorite can't receive his votes.)
>
> Here's a Weinstein mono-raise example for your consideration.
>
> 0.311: D>C>A>B approves DCA

But .311+ .186 +.270 > .50 so only D & C can be approved. "the total
probability of approved alternatives on a ballot cannot exceed 50 percent"

> 0.270: A>B>C>D approves AB
> 0.186: C>B>D>A approves CBD

But .186 +(.143+ .087) + .311 > .50 so only C and B can be approved.

> 0.143: B>D>A=C approves BD
> 0.087: B>D>A=C approves BD
>
> D wins. Now change the 0.143 bloc to vote D>B at the top instead of B>D.


> In my version this transposition is actually two moves because equal
> ranking must be allowed for random ballot probabilities to split (between B
> and D in this case). So raising D leads to B=D, and it takes another move
> to lower B.
>

That said, it seems like the approval cutoff rising on a faction where D
was approved and B not, could lead to D falling out of approval, etc.

>
> I find that this causes the .311 bloc to stop approving A (since D carries
> more weight now), and the .270 bloc to start approving C (since B carries
> less weight). So now C wins.
>
> Kevin
>

I'm starting to think of the points awarded as a kind of updating of
approvals in light of the pairwise information; an alternative that beats
every alternative ranked above it is a threat to be reckoned with, so
perhaps the new info might justify lowering the approval cutoff adjacent to
the threat ... below it if the threat is ranked ... above it otherwise.

Running with this idea ... run one pass of a bubble sort from the bottom
ranked (but not truncated) alternative upward (Llull applied to the ballot
order of the ranked alternatives). This Llull winner becomes the
(inclusive) approval cutoff.

A refinement of this idea starts with a (tentative) explicit approval
cutoff.  If the Llull winner falls below the tentatve cutoff, then the new
cutoff is immediately above the Llull winner.  In every case the new cutoff
is obtained by moving from the tentative cutoff to a position immediately
adjacent to the Lull winner (stopping as soon as adjacency has been
achieved).

This seems to be taking us back to Rob LeGrand ...

Joyeux Noel!

Forest
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