[EM] “Monotonic” Binomial STV

[Kristofer Munsterhjelm]

On 01.03.2022 01:58, Forest Simmons wrote:
> "...It follows that if the abstentions add up to a 
>  quota, a seat is not taken...."
> 
> Kind of like NOTA ... none of the above.
> 
> I'm trying to think how I would design a method in the spirit of
> Binomial STV .... elections vs exclusions ... preferences vs reverse
> preferences.
> 
> Perhaps some variant of Bucklin that gradually collapses ballot rankings
> inward (ER Whole?) when not enough top or bottom votes exist to meet
> quotas for further inclusion or exclusion ... taking special care to
> insure both monotonicity and clone independence in the process, if possible.

Symmetric weighted positional methods could be considered to work this
way. The three-candidate single-winner Binomial STV works like fpA/lpA;
you could also imagine fpA-lpA, which is really the weighted positional
method (1, 0, 0, ...., -1): first preferences plus zero times second
preferences plus zero times ... plus -1 times last preferences.

And if the weight vector is symmetric (e.g. x_1, x_2, ..., x_k, -x_k,
-x_2, ..., -x_1), then it could be considered to factor in both
inclusion (the positive weights) and exclusion (the negative ones). Then
Borda could be considered a system of this form, for instance.

> I think collapsing has more potential for monotonicity than does
> elimination, and I'm glad that Binomial stv keeps all of the players in
> the game until the final count, like Bucklin does.

I have been considering generalizations of fpA-fpC to multiple
candidates along these lines, although I'm missing significant pieces of
the puzzle. Suppose we want to pass single-candidate DMTBR, no matter
how many candidates run. Then an obvious way is to somehow reduce the
n-candidate election into a three-candidate election.

One way of doing that would be, for a candidate A, to collapse every
other candidate into either B or C. Let the sets c_B and c_C be the
candidates that are meant to be collapsed to B or C respectively. Then
modify a ballot by removing every candidate but the highest ranked
candidate in c_B, and then relabeling that candidate B; and removing
every candidate but the highest rated candidate in c_C, and relabeling
that candidate C.

(This looks like elimination, but it's more like relabeling every
candidate in c_B to B and then removing redundant later ranks, e.g.
X>Y>A>Z>W becomes B>B>A>B>C, and then B>A>C.)

If the assignment of candidates into c_B and c_C are held constant, then
the transformation is monotone, because ranking some candidate in c_B
higher either has no effect or ranks B higher on the transformed ballot
(all else equal), and the same holds for c_C. And since fpA-fpC itself
is monotone for three candidates, it can't produce a monotonicity
failure, either.

If A is a Condorcet winner with >1/3 fpp, then no matter how the other
candidates are partitioned into c_B and c_C, A keeps this property. And
if the burial doesn't change the partition of the other candidates, then
it also passes single-candidate DMTBR, because burying A under some
candidate X is weaker than burying A under B in the three-candidate
election, which three-candidate DMTBR already protects against.

So one could imagine a method where f(A, {B, C, D...}, {E, F, G...}) is
A's score with c_B being {B, C, D, ...} and c_C being {E, F, G, ...},
and then letting A's score be the min (or max, or sum, or some other
operator) of a number of these collapsed-election functions.

But that's the significant piece I lack: I don't know how to compose
these to preserve DMTBR, because the relative values of the various f
functions may change as a consequence of burial; and if f doesn't
contain an f for every possible partition, it's not obvious *which*
partitions belong in there, and how to make sure the selection thereof
can't be exploited in a way that fails single-candidate DMTBR. And if
the partition depends on the outcome so far, it becomes more like
elimination and may lead to path dependence problems again, just like
ordinary elimination.

-km