[EM] Improved TACC (Total Approval Chain Climbing)
Forest Simmons
forest.simmons21 at gmail.com
Fri Nov 12 17:10:49 PST 2021
I need to simplify this definition so that incessant (& annoying)
references to forms of the word "survivor" are not needed... let's see if a
recursive definition might help:
The input to this method M consists of an agenda A of alternatives in
approval order along with a table T of their pairwise win/loss/tie
relationships. The output M(A,T) of the method will be some alternative Y
from the agenda A.
1. Identify the alternative X closest to the good (high approval) end of
the agenda that covers every worse (lower approval) alternative on the
agenda.
2. Eradicate from both the agenda A and from the win/loss/tie table T ...
every alternative (including X) that does not beat X pairwise.
3. IF this elimination step exhausts the agenda A, then set the output Y
equal to X.
4. ELSE apply this method M recursively to elect Y= M(A, T).
Now (por favor) disregard the previous definition with all of its typos,
thinkos, y otros defectos!
Promised examples tomorrow!
-FWS
El jue., 11 de nov. de 2021 8:50 p. m., Forest Simmons <
forest.simmons21 at gmail.com> escribió:
> A few years ago we applied to the "total approval order" a "chain
> climbing" idea of Jobst Heitzig for monotonically extracting a member of
> the Banks set from an ordered list of alternatives ... monotonic in the
> sense that if you move the winner W higher up the list while preserving the
> relative order of the other alternatives, or if uniquely W acquires a new
> pairwise victory, then the alternative W still wins the chain climbing
> competition.
>
> In general, an alternative is a Banks alternative if it is the head of a
> maximal totally ordered chain of alternatives ... where the order relation
> in question is that of pairwise defeat... total order means that (1) each
> member of the chain pairwise defeats every member of the chain that comes
> after it and (2) is defeated by every member that precedes it. "Maximal"
> means (3) there is no room to expand the chain while maintaining the total
> order conditions (1) and (2).
>
> In practice it is not necessary to find an entire maximal chain; it
> suffices to reach the point where no alternative can be added to the front
> end of the chain ... since it is only the winner, i.e. the head of the
> chain, that we are interested in ... at least in our single winner context.
>
> In the original version of TACC we initialize our chain with the least
> total approval alternative .... then while there remains any un-tested
> alternative, we test the lowest total approval previously untested
> alternative X against the current members of the chain one-by-one as it (X)
> "climbs" up the chain. If it makes it to the head of the chain without
> being bumped off (pairwise defeated and discarded) along the way, it (X)
> gets incorporated into the chain as the current head or champion. The last
> alternative incorporated into the chain (as champion/winner) directly beats
> every other member of the chain ... and each non-chain member was bumped by
> a chain member, so the champion beats each alternative in a "short
> beatpath" of two or fewer steps.
>
> Now here is the new & improved version in succinct form:
>
> While ...
> any undiscarded alternative remains (i.e. "survives") ...
> discard all (previous) survivors that do not pairwise beat the survivor X
> that has the highest total approval among those survivors that are not
> beaten pairwise by any survivor with lower total approval.
> EndWhile
>
> Elect the final survivor.
>
>
> This method definition is not easy to digest in one or two cursory
> readings, but it is all there.
>
> Working through a few concrete examples will easily clear up any confusion
> (and expose any potential bugs) ... which we will attempt tomorrow!
>
> -FWS
>
> -FWS
>
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