At 11:19 AM 2/2/2010, Warren Smith wrote:
see
puzzle 91:
http://rangevoting.org/PuzzlePage.html#p91
This is the puzzle:
Suppose voters in an N-candidate election submit a rank-order ballot
which ranks some subset of the N candidates in order, then leaves
the remaining candidates unranked. For example in a 5-candidate
election a legal vote would be "C>A>B>(all unranked candidates)."
How many different kinds of "total" should the election office in
each precinct publish in order to hold an election
* Using a Condorcet method based on the "pairwise matrix"?
* Using "instant runoff voting"?
* Using Borda count?
* Using range and approval voting?
(Give both a formula in terms of N, and compute the numbers when N=10.)
More specification of rules would need to be made for this to be
fully determined. Is a majority required to complete the election? If
it is, is it necessary to produce a reduced candidate set? By what
rules? Further, is any additional information to be collected? The
rank order on Range ballots may be important. I'd certainly want to
know if a Range winner was beaten in the rank comparisons, and by how much.
IRV, as suggested (not exactly "recommended") by Robert's Rules
requires a majority. As implemented in the U.S., generally, it's a
plurality method, cobbing together a plurality, often not a majority,
neglecting exhausted ballots.
The answer for Borda is also a bit ambiguous. How are empty ranks
treated? Borda count reduces to Range (N-1) if ranking freedom exists.
If the empty rank treatment is handled locally, which is likely, then
the number of totals is the same for Borda as for Range.
And the number of totals for Range and Approval are the same: a total
for each candidate, the simple sum of votes. If Range is average
Range, though, extra data would need to be transmitted, specifically
the number of voters with votes for each candidate.
I do not recommend average range for early implementation, but
transmitting the data for it would be a good idea.
Note that the data transmission for Plurality is the same as for
Approval, except with approval an additional number is also required,
probably: the number of voters.
If a majority is required, the number of marked ballots is necessary
for any method, not merely the totals for candidates, unless write-in
votes are not allowed and any extraneous marking on the ballot
completely invalidates it. (Not so with Robert's Rules. Blank ballots
don't count, they are "scrap paper," but any mark on a ballot that
might be interpreted as an attempt to vote causes the ballot to be
counted in the basis for a majority, even if the identity of the
candidate voted for is not discernable.)
Range presents a problem: If I vote 1 on a scale of 10 for a
candidate, this is pretty obviously a vote against the candidate,
unless, maybe, I don't vote higher than that for anyone! How can we
determine if a majority approved the result? I've suggested an
explicit approval cutoff, which also serves to give meaning to
midrange. It then means something like "I prefer the election of this
candidate to a runoff election being held." That would cause
categorization of ratings into approval ratings and disapproval
ratings. And the number of approvals would need to be reported separately.
If ranking information is irrelevant (it shouldn't be!) then Range
can be canvassed with totals at each rating for each candidate, plus
blanks. So if it is Range R (0 - R), then there are (R + 1) unique
ratings for each candidate plus the blank, so the number of rating
totals transmitted is N * (R + 2). In addition to this, we need the
number of voters who rated any candidate, one additional number, that
with the blank information, allows determining the number of voters
who rated an individual candidate. This is important for determining
"majority."
Bucklin was not mentioned. Bucklin is "instant runoff Approval," but
there are no eliminations, so simple totals for each rank suffice.
Bucklin was typically three-rank, with multiple approvals allowed in
the third rank. For a modern version, I'd simply allow multiple
approvals in each rank, it avoids discarding ballots with meaningful
information, and could cause no harm.
Bucklin could be canvassed all at once by transmitting, for M ranks,
N*M totals. However, for fast results and especially with hand
counting, the first rank information can be prepared (by sorting
ballots according to first rank, then specially treating any
"overvotes") and the totals transmitted. In many or most elections,
with relatively few candidates, that would determine the result. But
I consider it rude to ask voters to rank and then not count their
votes! So the process should then be repeated with each rank,
regardless. One other number is necessary, being the number of marked
ballots in the election (or however a "voter casting a ballot" is
defined, but it is not the number of "votes" in the event that
"overvoting" is allowed). That's the basis for majority, and is used
to determine if the process terminates.
Care should be taken with Bucklin to avoid counting ballots directly
and simply when they have more than one vote for a candidate. (i.e.,
the voter votes for the same candidate in first and second rank,
say). The rules should provide for interpretation of such a vote; it
clearly is a vote for the candidate, but in which rank? I could see
an argument for it being at the lower rank unless there is no other
first rank vote. In the end, it will typically matter less with
Bucklin, and the only thing to be strongly avoided is counting the vote twice!
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