[EM] Recursion for the Daring!

[Forest Simmons]

For now, let's call the method "Recursive Protection Against Cursed
Burial," or RPACB for short. Also for now, let's leave open the meaning of
"worst".

RPACB:
Elect the ballot Condorcet candidate if there is one,
Else recursively elect the RPACB winner of the ballot set restricted to the
candidates that pairwise defeat the worst Smith candidate.

I hope that wasn't too anticlimactic after all of the suspenseful
preparation for the mysterious technique  called "recursion"!

-Forest

El mar., 11 de ene. de 2022 7:19 p. m., Forest Simmons <
forest.simmons21 at gmail.com> escribió:

> When trying to explain a multistep procedure, sooner or later you reach
> the point where the pattern should be evident to your audience ... that's
> the point where you say, "... and so forth."
>
> That's fine for informal interactive communication like this EM List, but
> when you need to pin down unambiguosly exactly what you mean ... for
> example, when giving instructions to a computer (or writing a proposal into
> law), ... that's where recursion may be the ideal tool for the job.
>
> The case I'm thinking about now is our Q&D/C approximation to TACC, ...
> can we improve the approximation while preserving the succinct concision?
>
> Here's the Q&D version:
>
> Elect the "best" candidate that pairwise defeats the "worst" Smith
> candidate.
>
> A concise eleven word description of an amazingly burial resistant
> Condorcet method.
>
> I purposely left "best" and "worst" undefined, because we're still
> exploring the possibilities. In TACC (Total Approval Chain Climbing)
> "worst" refers to least total approval .... the "climbing" is climbing up a
> List of candidates previously sorted into approval order ... preferably
> explicit approval, but implicit approval is next best ... the best we can
> do when restricted to the Universal Domain  rules of RCV, i.e. ordinal
> preference information only. [It's an ironically sad oxymoron ... " ..
> restricted to Universal ..."]
>
> There is nothing corresponding to "best" in TACC, because TACC just keeps
> on going until there is only one candidate left.  In Q&D we have been using
> the same score system for "best" and "worst" ... highest "basic score"* and
> lowest basic score, respectively. But there is no compelling reason for
> adhering slavishly to that custom. In fact, this opening is our big fat
> opportunity for incorporating recursion into Q&D!
>
> *["Basic score" is a flexible version of implicit approval.  It means the
> number of ballots on which the candidate being scored is ranked above at
> least one candidate. The flexibility comes with the option of recalculating
> the score when restricting to a subset, like the Smith set, of continuing
> (i.e. un-disqualified) candidates.]
>
> So here's a first draft of our recursive version of Q&D:
>
> Elect the "recursively determined best" candidate that pairwise defeats
> the "worst" Smith candidate.
>
> The word count goes up from eleven to thirteen ... not too shabby!
>
> Nitty gritty gory details coming soon!
>
> [But now that you know how recursion works, you probably don't need me to
> spell it out.]
>
> -Forest
>
> El lun., 10 de ene. de 2022 9:51 p. m., Forest Simmons <
> forest.simmons21 at gmail.com> escribió:
>
>> Warning ... recursion is like spicy Thai or Mexican food ... if you are
>> not used to it, it can make your head explode!
>>
>> But when safely in the hands of responsible people it can give valuable
>> information about conceptual complexity.
>>
>> Recursively defined procedures, while maximally succinct and conceptually
>> simple (in the minds of people used to them)  are often computationally
>> "hard", involving an impractically large number of simple, but highly
>> repetitive operatioms.
>>
>> So we have a dilemma: how do we shield the tender minded from recursive
>> descriptions while sharing their beautiful simplicity with the initiated.
>>
>> In ancient times secretive, exclusive societies like the Pythagoreans or
>> the Masons organized according to various levels of what might be called
>> "security clearance" in modern parlance.
>>
>> In modern schools, prerequisites  must be met before admission to an
>> advanced course.
>>
>> On the EM list, inquisitive but sensible people just give up if the going
>> gets too tough too soon; "Obviously this post was not intended for the
>> general public."
>>
>> I'm going to try to make this simple enough that someone with no previous
>> experience with recursion can see what is going on.
>>
>> The tricky (and clever) part of recursion is judicious use of what seems
>> to be circular definition, but is in reality a kind of convergent inwardly
>> spiraling reasoning.
>>
>> If your first grade teacher asked you to define the number six, you might
>> say that it was the number just before seven, which would be true, but not
>> as helpful as saying the number just after five.
>>
>> In the seventh grade a smart alec student might say that six is the
>> number  that you get if you subtract it from 18 and divide the result by
>> two.
>>
>> This answer would highly displease your third grade teacher as an example
>> of circular definition. But the 7th grade algebra teacher would say, "This
>> kid has talent."
>>
>> If you do a directed graph of the function
>>
>> x---->f(x), where f(x) =(18-x)/2,
>>
>> the picture you get is a spiral flow converging to the number 6 on the
>> number line.  If you graph y=f(x) on an (x, y) coordinate system and
>> connect points (x, y) to (y, f(y)) with arrows, you with get a "web"
>> diagram of the same underlying spiral structure converging to the point
>> (6,6) on the graph of y=x.
>>
>> So apparent circularity in definition may lead unambiguously (in some
>> cases) to a well-defined number.
>>
>> In the algebra example, it is sufficient for the slope of the graph of f
>> to be strictly less than one, in absolute value.
>>
>> For recursively defined procedures it is sufficient that any self
>> referential part of the procedure be restricted to a set of lesser
>> cardinality than the "main" set being dealt with.
>>
>> If you want to recursively define a procedure on a set of size n,
>> temporarily remove a member of the set. Then "recursively"apply the
>> procedure to the smaller set of size (n-1), and then re-incorporate the
>> member that was removed.
>>
>> For example, to alphabetize a random list of words, remove any word from
>> the list, recursively alphabetize the rest of the list, and then finally
>> place the word that was removed into its proper place.
>>
>> This is not in any way an efficient procedure, but it is amenable to
>> logically precise, concise, conceptually simple verbiage.
>>
>> Even the lexicographical order itself that is needed for placing the
>> temporarily removed (last) word into its proper place is easily defined
>> recursively:
>>
>> (For simplicity assume no other alpha numeric characters like hyphens,
>> spaces, numerals, etc.)
>>
>> Find the first place (from the left end) of the two words where they do
>> not have the same letter. If that place is at the first letter of the word,
>> then precedence goes to the word whose initial letter is closer to the
>> front of the alphabet. Otherwise, temporarily remove the common initial
>> string of letters and apply the procedure recursively to the remaining
>> sub-strings of letters.
>>
>> To Be Continued....
>>
>>
>>
>>
>>
>>
>>
>>
-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://lists.electorama.com/pipermail/election-methods-electorama.com/attachments/20220112/388a2bb2/attachment.html>