[EM] Improved BTR-IRV, Benham

[Forest Simmons]

When I first encountered IRV, it seemed rather redundant to include the
check at each step for a majority winner ...  because skipping that check
would make no difference in the final outcome ... in fact, at no later
stage would the majority winner of an earlier stage become the candidate
with the fewest transferred votes.

 The succinct technical writer would just omit that check from the loop.

But I wasn't aware at the time, that the single biggest practical drawback
of IRV was the logistical cost of its many passes through the ballot set to
carry out all of the vote transfers.

[I'm sure that there are purists who consider the majority-check shortcut
to be an unprincipled violation of the explicit accounting of vote
transfers ... while eliminating candidates one by one.]

Similarly, the Undefeated Pairwise Winner Check at each stage of BTR-IRV,
Benham is mathematically redundant, but it serves the same important
purpose of reducing the number of required passes through the ballots (for
the vote transfers).

Another major reduction in passes through the ballot set ... while actually
enhancing the quality of the outcome ... can be achieved by finishing each
elimination step with a mop up of the milquetoast/ darkhorse candidates
that might otherwise slip through and needlessly gum up the works:

This mop-up is accomplished by the pairwise loser at each stage taking down
with it the candidates that are too weak to defeat it pairwise.

Here's a summary of Improved  BTR-IRV, Benham.

1. Check for an undefeated candidate among those remaining ... (and
announce the victory party if there is one).

2. Find the pairwise loser PL, between the two remaining candidates with
the fewest transferred votes.

3. Eliminate this PL along with every candidate that does not defeat it
pairwise.

4. If more than one candidate remains, repeat these steps.

This is a very good method that can be given full support without apology
or embarrassment.

In fact, in this context the only additional improvement worth making would
be to replace the phrase "with the fewest transferred votes" in step 2,
with the phrase "nearest the unfavored end of the agenda."

That step would make the method efficiently precinct summable with one pass
through the the ballots.

[It would also confer monotonicity, provided the agenda itself was
generated monotonically. In a separate thread I have given three simple
examples of how to do this.]

But that would just be the frosting on the cake ... a step that can wait as
long as it takes for the public to make the psychological adjustment to
.... well, to simply use exactly the same kind of agenda that small
Deliberative Assemblies (governed by protocols similar to Robert's Rules of
Order), have been using for many centuries.

This Improved BTR-IRV has better burial resistance than either ordinary
BTR-IRV  or plain old vanilla IRV itself.

Furthermore, it can never elect a candidate covered by one of the other
candidates ... an embarrassing defect of all other extant election methods
... except Copeland, which has other more serious problems.

Here's a burial resistance example:

40 A>B(Sincere A>C)
35 B>C
25 C>A

I have shown how this example arises naturally from three factions
concentrated near the vertices of a generic scalene triangle ... say three
villages trying to settle on a shared storage location.

Alternative C is the sincere pairwise undefeated alternative, but has been
subverted by faction A in an attempt to create a cycle that its superior
size might benefit from.

[[It is much more likely that a Rock Paper Scissors cycle be created
artificially like this, than by random accident.]

Ordinary IRV eliminates the smallest faction first, and transfers the votes
thusly:

65 A>B
35 B>A

So the manipulating faction is highly rewarded by a resounding victory!

Ordinary BTR-IRV compares the B and C factions pairwise... finding C to be
the pairwise loser to be eliminated ... leading to the same final runoff
between the A and B factions as before ... again rewarding A for its
perfidy!

But under the auspices of Improved BTR-IRV, when C is eliminated, it takes
down A with it ... since C defeats A, 60 to 40 ... leaving only B ... not a
happy outcome for A.

If they were used to voting under Improved BTR-IRV rules, they would have
anticipated this disappointment, and avoided it by never venturing their
shenanigans ... which would have rewarded the sincere undefeated pairwise
candidate ... in the example, the village opposite the longest side of the
triangle!

This example shows how opportunistic candidates can be thwarted from their
unscrupulous manipulations by a good voting method like Improved BTR-IRV
(and very few others).

This example also shows that the mop-up that finishes each elimination step
has more benefit than just filtering out darkhorse candidates ... it is
also effective at catching sneaky unscrupulous manipulators.

Was this explanation helpful?

-Forest
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