On Apr 8, 2010, at 2:02 PM, Dave Ketchum wrote:
On Apr 7, 2010, at 8:29 PM, robert bristow-johnson wrote:
On Apr 7, 2010, at 6:25 PM, Dave Ketchum wrote:
This is some thought about keeping it simple, yet doable.
I will lean toward Ranked Pairs with margins,
not sure what "with margins" does. i'll read below...
vs comparing per winning votes - each has backers.
Juho just explained it, so now i know (earlier i had wondered if
"margins" was a normalized or percentage beat strength). i've always
thought that the Tideman RP was *only* framed in terms of margins. i
do not know why anyone would back the "winning votes" metric for beat
strength.
but amending toward other types of Condorcet should be doable.
Voting: Voter can rank one or more candidates. Equal ranking
permitted. Counters care only which of any pair of candidates
ranks higher, not how voter decides on ranking. Write-ins
permitted (if few write-ins expected, counters may lump all such
as if a single candidate - if assumption correct the count
verifies it; if incorrect, must recount).
My thoughts are that supporting write-ins is worthy and doable, but
am not excited about how.
i've thunked about it. just like supporting third parties and
independent candidates (who get ballot access) is important (and is,
indeed, why we are promoting preferential voting in the first place),
for the same reason, i think the ability to deal with write-ins (but
just *one* write-in per ballot) is also important.
of course, nearly always, the aggregate "write-in" will not win, and
then it doesn't matter who the folks are that are written in. (one
point about the 2000 Bush v. Gore in Florida is that there were some
idiots that checked a listed candidate and then also wrote that very
same name in the write-in slot. the machine counter rejected these
ballots as overvotes, but when the media hand-recounted Florida
statewide and determined that, due to these overvotes, Gore won by 172
out of 5 million.) the issue is how to make legal verbiage for how to
deal with the case where "write-in" wins. then, of course, a hand
recount is necessary (perhaps assisted by the scanning machines) and a
straight-forward procedure must have language that does not *assume*
at the outset who the winning write-in is, but accomplishes the task.
Counting: Besides the N*N matrix, I would add an N array to
optimize this. Count each ranked candidate in the array. Later
the array will be added into the matrix rows as if the ranked
candidates won in every one of their pairs. This is correct for
pairs with no ranking, and for pairs with one ranked.
already this is complicated and someone in the "One person, one
vote" crowd (the anti-IRVers) in Burlington would say that you're
trying to pull one over on them.
Ranked pairs need not be that complicated.
So far we are just doing counting. After that, time to think of CW
and cycles, and for methods such as ranked pairs mattering.
Think of 10 candidates and a bunch of bullet voters. For each
ballot its candidate needs counting in each of its 9 pairs. I would
have the counters count such in its element in the N array, with all
of the N array added into the N*N matrix in one step later.
Try a voter ranking two of the 10 (should be common for those not
doing bullet). Stepping two entries in the N array will get 16
entries in the matrix properly stepped. One entry will get stepped
as if each ranked higher, so I have the counters adjusting for this
as part of counting that ballot.
you don't need to be adjusting any other counts in the N*(N-1)/2
pairs, which is my preference of expressing the "N*N matrix". i
still find the "N*N matrix" to be a useless visual tool. i want to
see N*(N-1)/2 pairs of numbers. that's how you visualize in a
glance how Condorcet decides an election. this "N*N matrix", such
as it is, is just useless.
Your preference over the matrix is interesting, provided you keep
your location of pairs understandable and get the same winner as the
matrix would achieve.
is this example (the 2009 Burlington mayoral race) clear enough?:
M 4064
K 3477
< 587>
M 4597 K 4313
W 3664 W 4061
< 933> < 252>
M 4570 K 3944 W 3971
S 2997 S 3576 S 3793
<1573> < 368> < 178>
M 6263 K 5515 W 5270 S 5570
H 591 H 844 H 1310 H 721
<5672> <4671> <3960> <4849>
it's just a visual rearrangement of the N*N matrix. it comes out as a
triangle, instead of a square matrix (with a blank diagonal), and you
can see clearly who beats who, and if the candidates are nicely
ordered (as they were in 2009, it's unambiguously M>K>W>S>H) in a
Condorcet sense.
the number in <brackets> is the margin and is the first number that
Tideman RP looks at to determine which (remaining) pair is committed
to (and removed) in each scan. the clearest expression of voter
preference is that M is preferred over H. the least clear is that W
is preferred over S. what is interesting (and i believe a source to
the lack of voter confidence in the election result) is that the
second weakest defeat is what decided the election. but you can look
at that triangle and immediately see who beats who and how badly were
they beaten.
the N*(N-1)/2 pairs is all of the numerical information you need,
and all you need to do is compute, for each pair, the abs value of
the difference of vote counts. so you have 3 numbers for each of
the pairs. then there are N*(N-1)/2 - 1 scans, each identifying
and removing the remaining pair with the highest abs(diff). then,
with each of these removed pairs in order of their removal,
construct the "who beats who" paths out of nodes starting with the
highest abs(diff), and as Tideman sez, ignoring pairs that are
contradictions, resulting in a cycle after "committing" (i would
use that word instead of "locking") to the race pairs previously
accepted.
that is simpler. both as language of law, and to program into a
computer.
Your many scans are something I would avoid.
seems to me that a computer can deal with the pairwise tallies (and
computing the margin for each) could do N*(N-1)/2 - 1 scans in a
hurry. it's O(N^2) (N is the number of candidates, not voters). as
far as scanning ballots at the precinct level, each ballot is scanned
once and the computer tallies N*(N-1)/2 numbers for each scanned ballot.
For some of this computers vs humans matters - counting the bullet
votes is easier for humans to get right by doing a single entry in
the N array vs 9 in the matrix for each such ballot, while a
computer only needs simple programming.
but can we depend on people doing bullet voting. in the worst case
(regarding complexity), every ballot would have every candidate
ranked, and the right procedure would work the best in the worst case.
In looking for the CW I eliminate a candidate via each carefully
selected comparison - 9 comparisons if 10 candidates. That does not
tell if there is a CW, so scan that candidate's pairs - if any
losers that is a cycle member and we now know all the cycle members
it loses to. Scan those members' pairs for the other members they
lose to, continuing until we know all the interesting pairs.
Most cycles only have three members. Anyway, we have the pairs that
compose the cycle and can use whatever method we choose to sort them
out.
in the case of a 3-cycle, it should be obvious which pair race should
be discounted (the one with the smallest margin), Tideman and Schulze
agree which pairing is ignored, and once that happens, it's clear who
the winner is.
it sure seems to me that the existing Tideman RP is simpler and at
least as "meaningful" in reflecting voter preference.
Most methods only care about cycle members and their relationships.
I find the members and only consider methods interested in such -
and do not argue here about which of such methods to actually use.
what's nice about Tideman or Schulze, is that the methods do not care
about who the Smith set is or even if there *is* a cycle. the method
just grinds away and mindlessly picks the winner according to the
rules (and i think the Tideman rules are more transparent than the
Schulze rules however both can be given concise language, even though
i didn't know that about Schulze until Markus posted his proposed
legislation).
--
r b-j [email protected]
"Imagination is more important than knowledge."
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