[EM] Converting super-majorities to margins

[Rob Lanphier]

I think a generally useful definition of "super-majority" in
multi-candidate race is the pairwise winner with approval of the required
super-majority.  Approval on a ranked ballot could be expressed as a line,
all above which are considered approved, and all below which are not
(methods for doing this were discussed very early on on this, circa 96q1
or so).

So, for instance, for a two-thirds super-majority, the candiate would have
to achieve both of the following:
*  Simple majority pairwise victories over all opponents
*  Approval of two-thirds of voters.

Of course, super majorities could only be required in cases where the
status quo is an acceptable outcome, since that's what will result short
of anyone acheiving a super-majority.

Rob

On Sun, 9 Jan 2000, Blake Cretney wrote:

> Date: Sun, 09 Jan 2000 21:24:17 +0000
> From: Blake Cretney <bcretney at postmark.net>
> Reply-To: election-methods-list at eskimo.com
> To: Election Methods <election-methods-list at eskimo.com>
> Subject: [EM] Converting super-majorities to margins
> Resent-Date: Sun, 9 Jan 2000 13:22:38 -0800
> Resent-From: election-methods-list at eskimo.com
> 
> In my previous post I comment on the possibility of combining some
> sort of super-majority requirement with various single-winner methods.
>  At one point I mention that I assume that a qualifying super-majority
> should always have a greater margin of victory than a normal majority.
>  It isn't obvious, however, that this would be the case.  For example,
> one common super-majority requirement is a 2/3 vote.  However
> 
> 12 to 8 -> margin of 4
> has a greater margin than
> 6 to 3 -> margin of 3
> 
> But, 6 to 3 is a 2/3 vote, while 12 to 8 is not.
> 
> The general procedure I suggested was:
> 
> 1.  Order the options using a method without regard to SQ.
> 2.  Pick the highest ordered option with a path of super-majorities
> to the SQ.  For convenience, you can also use only super-majorities
> from a restricted set, as I suggest for Tideman.
> 
> This will work for the 2/3 requirement, but the procedure is
> complicated by the fact that two kinds of precedence are required.
> 
> As well, the method seems to have some internal conflict.  That is,
> if on the one hand, when using Tideman's locking procedure, we claim
> that 12-8 takes precedence over 6-3, on what basis do we decide that
> for a different purpose, 6-3 takes precedence over 12-8.  Tideman is
> based on the principle that we should be more confident in a majority
> with a higher margin.  If we discard this principle, why do we use
> Tideman?
> 
> It therefore seems desirable to convert the 2/3 requirement into a
> margin.  Fortunately, this is easy to do.  We define the required
> margin as the minimum margin necessary to ensure that an option does
> not win without having a 2/3 vote.  A convenient formula, is
> round(v/3), where "round" means round to nearest integer.
> 
> A side effect of this is that it means that contests with less than
> full participation will require more than 2/3 to qualify.  For
> example, 6-3 would not qualify for a vote of 21 ballots.  However,
> this can be viewed as desirable, as our confidence might diminish as
> participation drops.
> 
> ---
> Another case is where a quorum is desired.  For example, it may be required
> that at least 5 people participate in a vote for the SQ to be changed.  
> 
> Now, although quorums are often specified simply in terms of participation,
> it makes little sense to allow a vote to pass because of the presence of
> objections.  For example, with a quorum of 5, 
> 4 to 3 passes
> 4 to 0 fails
> 
> This absurdity is avoided because the 3 "no" voters would likely withdraw for
> the purposes of the vote, if this allowed it to be defeated by lack of quorum.
>  However, this kind of action is harder to manage in a ranked ballot, so I
> will make it a provision that "no" votes should never be counted towards
> making a quorum.
> 
> Clearly, "yes" votes should be considered in making up the quorum.  It is a
> little more difficult to decide what should be done with neutral votes.  That
> is, voters who rank two options as equal.  If the neutral voters really are
> neutral, they don't care how their vote is tallied, but it could make a
> difference to the outcome.  My choice is to count the neutral vote as half a
> vote towards making quorum.  Not only does this seem natural for a vote in
> between a "yes" and a "no", but it allows a quorum requirement to be easily
> converted to a margin requirement.  The formula is that the minimum margin is
> equal to two times the quorum, minus the total votes. m=2q-v
> 
> Let y, n, a be number of yes, no, and abstention votes
> Let v be total number of votes
> Let q be quorum required
> Let m be minimum margin required
> 
> Clearly,
> v=y+n+a
> m=2q-v
> Assume y-n>=m
>    y-n>=2q-v
>    y-n+v>=2q
>    y-n+y+n+a>=2q
>    2y+a>=2q
>    y+a/2>=q, which is how I said I wanted votes towards quorum to be tallied
> So, y-n>=m -> y+a/2>=q
> 
> The above argument can be reversed to show that y+a/2>q -> y-n>=m
> ---
> 
> Here's an example.  40 ballots.  The vote will require a quorum of 23 and a
> 2/3 vote.
> 
> The 2/3 vote gives us the requirement round(v/3)=round(40/3)=13
> 
> The quorum required gives m=2q-v=46-40=6.  Since 13 is also required for a
> super-majority, this doesn't have any effect.
> Winner Loser Margin
> A      SQ    16
> B      A     15
> B      SQ    12
> 
> Using Tideman, we 
> lock A->SQ 16
> lock B->A  15
> lock B->SQ 12
> 
> Order B > A > SQ
> 
> Even though B does not have a 13 margin directly against SQ, it does through
> a path.  B qualifies, and is therefore the winner.
> 
> ---
> Blake Cretney
> 

Rob Lanphier
robla at eskimo.com
http://www.eskimo.com/~robla