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# Per-item "quorum" and truncated ballots

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.
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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.
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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.
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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       ---

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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.
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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

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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.
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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.
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We've made three changes to SSD:

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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.
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2) We've instituted the ability to rank options as equal on a ballot.

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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.
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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.
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I think that the combination of all three changes has unforseen effects.

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