[EM] Random Ballot Favorite Chain Climbing/Building

[Forest Simmons]

It turns out that what we want is chain building by fitting each newly
drawn ballot favorite into the chain where-ever possible without destroying
its total order relation. (A chain is a totally ordered set.)

So chain building can ascend or descend ad libitum ... as they say, "Just
follow your nose"

Also it simplifies everything conceptually and computationally if we
replace and shuffle the ballots between all consecutive drawings.

Remember that the proto example was the candidate cycle ABCA with
respective first place probabilities of p=fpA, q=fpB, and r=fpC.

The possible maximal chains are ...
A>B, B>C, and C>A,
with probabilities proportional to ...
pq, qr, and rp, respectively,
when chains are built ad libitum.

The condition for A winning is
pq>max(qr,rp),
which (turns out to be) equivalent to
(p-r)>max(q-p, r-q),
within this probability context of p+q+r=1.

This last inequality is our fpA-fpC condition for A winning in our proto
example.

Next, a simple introduction to the method suitable for a voters' pamphlet
...







On Thu, Oct 20, 2022, 1:24 PM Forest Simmons <forest.simmons21 at gmail.com>
wrote:

> One important (but easy) correction:
>
> In order to make this method Monotone, we have to start the chain from the
> bottom of the list ListF. That's what puts the "Climbing" in Random Ballot
> Favorite Chain Climbing!
>
> A comment on exposition for public consumption ... no mention of
> Condorcet  Smith, Landau, or Banks should be included in the method
> description, any more than a brief introduction to IRV needs to explain
> what to do if the last three remaining candidates have the same first place
> transferred vote totals.  Every generic public ballot set will have a Banks
> member with at least one first place vote, so no need to get people worried
> right off the bat about what to do in the impossibly rare contrary case.
> Stick with generic conditions in the voters' pamphlets ... just make sure
> that the rare exceptional possibilities are covered in the published
> official legal definition, as well as the RAQ's (Rarely Asked Questions) if
> not the FAQ's!
>
> I only mentioned it at all because of its earlier mention in related EM
> list threads.
>
> -Forest
>
> On Thu, Oct 20, 2022, 11:48 AM Forest Simmons <forest.simmons21 at gmail.com>
> wrote:
>
>> First a preliminary procedure to make sure no single candidate defeats
>> every member of the support of the random ballot favorite:
>> As long as there is such a candidate, retain only candidates of this
>> índole, recalibrating between elimination steps.
>>
>>
>> Next: a non-deterministic lottery method ... Random Ballot Favorite Chain
>> Climbing (RBFCC):
>>
>> Shuffle the ballots into some random order B1, B2, B3, ... and let ListF
>> be a list of the candidates in the order induced by the first choices of
>> the respective ballots in their order ... i.e. according to the order of
>> their first appearance as a first choice on a ballot in the sequence B1,
>> B2, B3, ...
>>
>> Now, Chain climb the list ListF by initializing the set variable CHAIN as
>> the empty set, and then ....
>> While some member of ListF defeats every member of CHAIN, add the first
>> such candidate into CHAIN. EndWhile
>>
>> The head of the completed chain is the RBFCC (random trial) winner.
>>
>> Next, for each candidate X, let RBFCC(X) be the winning probability for X
>> under this lottery.
>>
>> Finally, elect argmax RBFCC(X).
>>
>> Note that this method is Banks efficient, and obviously reduces to
>> "fpA-SumfpC" in the eponymous three candidate case.
>>
>> On a practical note, should the computation of the RBFCC probabilities be
>> intractable for some ballot set, then repeated trials in a MonteCarlo
>> simulation of the lottery can be used to determine argmax RBFCC(X) with
>> arbitrarily low error probability epsilon.
>>
>> Is this the simplest formulation of what we've been looking for?
>>
>> It doesn't seem like an easy method to "game".
>>
>> Other comments? Questions?
>>
>> Who can write this up in a way that Joe Q Public can easily relate to?
>>
>> -Forest
>>
>>
>>
>>
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