If it were impossible to rank options equally, then the combination of a
global quorum and an an elimination of unacceptable option (options to
which the default is preferred by a majority) would have essentially the
same effect as a per-option quorum.
This is easy to see. Every ballot would rank an option A as either
better than D or worse than D, so if Z is the total number of ballots
cast, then Z = V(A,D) + V(D,A), for all options A not equal to D
(actually, more generally Z = V(A,B)+V(B,A) for all B != A). If we say
that if Z<Q, the election is void, we only have to worry about cases
when Z>Q. If A is to be considered acceptable, then V(A,D) > V(D,A).
But if we assume that A would be rejected by a per-option quorum, then
we have R>V(A,D), or 2R > 2V(A,D) > V(D,A) + V(A,D) = Z >= Q, or 2R >
Q. But we choose R and Q, so we can easily set Q = 2R if we want.
Then the only way that the difference between per-option quorum and
global quorum would make a difference is if the total votes were between
R and 2R. Arguably, that's a small enough turnout that we'd want to
reject the vote as too few voters anyway.
However, we allow truncated ballots and equally-ranked options. None of
the examples I've seen so far have used this as a reason to argue
against per-option quorum, or even taken them into account in the
discussion.
Imagine a vote along the lines of:
100 ballots of the form:
[1] Red
[ ] Blue
[ ] Default
100 ballots of the form:
[1] Red
[ ] Blue
[1] Default
25 ballots of the form:
[ ] Red
[1] Blue
[ ] Default
with an R of 105.
The defeats matrix looks like
Red Blue Default
Red --- 200 100
Blue 25 --- 25
Def 0 100 ---
In this example, Red is the IDW, and absolutely no one thought that Red
was worse than the default option. Yet Red is rejected because fewer
than 105 people thought it was better than the default option, so
default wins.
Is this the "expected" behavor?
Let's say we have the following ballots:
100: Red>Blue>Default
100: Red=Default>Blue
25: Blue>Red=Default
Then Red is still the IDW because it defeats default 100:0 and defeats
blue 200:25. Yet Red is eliminated because only 100 voted for it over
the default, and Blue wins.
We allow options to be equally ranked to give voters a way of saying
that option A is equally acceptable to them as option B -- they would be
equally happy with either outcome.
But there is no way to say that one would be equally satisfied with
option A or the default. Ranking an option A as equally acceptable with
the default option penalizes A, since that ranking does not count
towards option A's quorum requirement.
We've made three changes to SSD:
1) We've instituted a per-option quorum, requiring a minimum number of
votes for a particular option over the default in order to be considered.
2) We've instituted the ability to rank options as equal on a ballot.
3) We are using the ranking relative to the default option as proxy for
an approval ballot, and only considering options that are approved by
the majority.
Our analysis of 1 and 3 have been based on the law of the exclused
middle: for any ballot, either A>D or D>A. We haven't considered the
effects of 2.
I think that the combination of all three changes has unforseen effects.
