# [EM] Consensus Threshold Approval

Andy Jennings elections at jenningsstory.com
Mon Apr 8 10:12:04 PDT 2013

```Forest,

This is an interesting method.  It gives another good objective meaning for
numerical scores on a 0-100 scale.

The consensus threshold would be very useful in situations where compromise
is paramount.  For instance, I can pledge to support a new taxation scheme
only if 90% of the citizens support it.  (This is even more important in
situations where participation is voluntary, such as donations.)  If 90% of
the people say the same thing, it is probably a pretty good compromise.

On the other hand, I see some situations where it could be problematic.  If
90% of people give a candidate a score of 11 and the other 10% of people
give a score of 0, that candidate will have a consensus threshold approval
of 90%.  I can imagine those voters being surprised that such a large
coalition was built from people who liked the candidate so little.

As Jameson notes, CTA is the same as the chiastic median, but applied to
the thresholds instead of the scores.  However, the thresholds are 100
minus the scores, so it definitely changes the meaning of things.  The CTA
value probably has more natural real world meaning than the chiastic
median, but it brings the following concerns:

1. In terms of the visual representation, I consider the area to be the
best measure of support if everyone is honest.  Approval, median, and
chiastic approval, as you describe below, all measure something related to
the area under the curve.  Examining the other diagonal, like CTA, doesn't
seem to measure anything related to the area.  So it might not be a good
indicator of the total level of support.

2. CTA does not meet the unanimity criterion for aggregation functions,
which says that if all voters give the same score, then the societal score
should match.  Approval doesn't either, but median, score, and chiastic
approval do.

3. CTA also can be extremely sensitive to one voter's input.  One voter, by
lowering his score (raising his threshold) by one point, can scuttle the
whole coalition and cause the societal score to go from 100 to 0.

~ Andy

On Thu, Apr 4, 2013 at 4:44 PM, Forest Simmons <fsimmons at pcc.edu> wrote:

> For purposes of clarification, I would like to show how Approval, Bucklin,
> Range, Chiastic Approval, and Consensus Threshold Approval manifest
> themselves relative to each other visually.
>
>
>
> I assume versions of these methods that make use of range style ballots on
> a scale of zero to 100.  These methods also have in common that once the
> ballots are counted each candidate ends up with a score of some kind, and
> the candidate with the largest score is elected.
>
>
>
> So let’s concentrate on how each of these methods would assign a score to
> the same fixed candidate.
>
>
>
> All of these methods can be explained in terms of the graph of the
> function F given by
>
>
>
> p=F(r) is the percentage of the ballots that rate our candidate strictly
> greater than r.
>
>
>
> Each point (r, p) of this graph will lie somewhere in the 100 by 100
> square with corners at (0,0), (0,100), (100, 0) and (100, 100).
>
>
>
> Furthermore, the graph will descend from left to right in steps whose
> widths are whole numbers.
>
>
>
> The left endpoint of each step will be included but the right end point
> will not be included.
>
>
>
> Color this graph blue.  Now join the steps with vertical segments.  The
> interior points of the vertical segments are colored red, while the top end
> point of each red segment will be colored red, and the bottom point will be
> colored blue.
>
>
>
> Now the union of the red and blue separates the lower left corner from the
> upper right corner of the square.  Therefore the diagonal from (0, 0) to
> (100, 100) must cross the colored graph in either a red or blue point.  Since
> the red and blue are non-increasing while the diagonal is strictly
> increasing, there can be only one point of intersection.  The common
> value of the coordinates of this intersection point is the Chiastic
> Approval score.
>
>
>
> Now calculate the area of the region of the square that  lies to the
> lower left of the red/blue diagonal.  This area is the average rating of
> our candidate.  So the candidate whose lower left area is greatest is the
> Range winner.
>
>
>
> Now bisect our square horizontally with a straight line segment from (0,
> 50) to (100, 50).  The first coordinate (r) of the point of intersection
> (r, 50) of this line with the red determines the basic Bucklin score.  Ties
> are broken by various methods.
>
>
>
> Now bisect the square with a vertical segment from (50, 0) to (50, 100).  Assuming
> an approval cutoff of fifty, the second coordinate (p) of the intersection
> (50, p) of this segment with the blue is the approval score.
>
>
>
> Now consider the diagonal from the upper left corner (0, 100) to the lower
> right corner (100, 0).  If this diagonal does not intersect the blue,
> then the candidate’s Consensus Threshold Approval score is zero.  Otherwise
> it is the second coordinate of the highest (and therefore leftmost) blue
> point of intersection.
>
>
>
> In summary, we have bisected the 100 by 100 square vertically,
> horizontally, and diagonally.  The diagonal with positive slope leads us
> to the chiastic approval winner.  The other diagonal leads us to the
> consensus threshold approval winner.  The horizontal bisector leads us to
> the Bucklin winner.  The vertical bisector leads us to the Approval
> winner.  The area cut off by the colored graph determines the Range
> winner.
>
>
> On Wed, Apr 3, 2013 at 6:18 PM, Jameson Quinn <jameson.quinn at gmail.com>wrote:
>
>>
>>
>>
>> 2013/4/3 Forest Simmons <fsimmons at pcc.edu>
>>
>>>
>>>
>>> On Wed, Apr 3, 2013 at 12:07 AM, Kristofer Munsterhjelm <
>>> km_elmet at lavabit.com> wrote:
>>>
>>>> On 04/03/2013 12:01 AM, Forest Simmons wrote:
>>>>
>>>>> Jobst has suggested that ballots be used to elicit voter's "consensus
>>>>> thresholds" for the various candidates.
>>>>>
>>>>> If your consensus threshold for candidate X is 80 percent, that means
>>>>> that you would be willing to support candidate X if more than 80
>>>>> percent
>>>>> of the other voters were also willing to support candidate X, but would
>>>>> forbid your vote from counting towards the election of X if the total
>>>>> support for X would end up short of 80 percent.
>>>>>
>>>>> The higher the threshold that you give to X the more reluctant you are
>>>>> to join in a consensus, but as long as your threshold t for X is less
>>>>> than than 100 percent, a sufficiently large consensus (i.e. larger than
>>>>> t percent) would garner your support, as long as it it is the largest
>>>>> consensus that qualifies for your support.
>>>>>
>>>>> A threshold of zero signifies that you are willing to support X no
>>>>> matter how small the consensus, as long as no larger consensus
>>>>> qualifies
>>>>>
>>>>> I suggest that we use score ballots on a scale of 0 to 100 with the
>>>>> convention that the score and the threshold for a candidate are related
>>>>> by  s+t=100.
>>>>>
>>>>> So given the score ballots, here's how the method is counted:
>>>>>
>>>>> For each candidate X let p(X) be the largest number p between 0 and 100
>>>>> such that p(X) ballots award a score strictly greater than 100-p to
>>>>> candidate X.
>>>>>
>>>>> The candidate X with the largest value of p(X) wins the election.
>>>>>
>>>>
>>>> I think a similar method has been suggested before. I don't remember
>>>> what it was called, but it had a very distinct name.
>>>>
>>>> It went: for each candidate x, let f(x) be the highest number so that
>>>> at least f(x)% rate the candidate above f(x).
>>>>
>>>> I *think* it went like that, at least. Sorry that I don't remember the
>>>> details!
>>>
>>>
>>> Good memory, that was Andy Jennings' Chiastic method.  Graphically these
>>> two methods are based on different diagonals of the same rectangle.
>>>
>>
>> Different, how? It seems to me they're just the same, but with the
>> numbers reversed.
>>
>>>
>>>>
>>>>  If there are two or more candidates that share this maximum value of p,
>>>>> then choose from the tied set the candidate ranked the highest in the
>>>>> following order:
>>>>>
>>>>> Candidate X precedes candidate Y if X is scored above zero on more
>>>>> ballots than Y.  If this doesn't break the tie, then X precedes Y if X
>>>>> is scored above one on more ballots than Y.  If that still doesn't
>>>>> break
>>>>> the tie, then X precedes Y if X is scored above two on more ballots
>>>>> than
>>>>> Y, etc.
>>>>>
>>>>> In the unlikely event that the tie isn't broken before you get to 100,
>>>>> choose the winner from the remaining tied candidates by random ballot.
>>>>>
>>>>
>>>> I imagine Random Pair would also work.
>>>>
>>>>
>>>>  The psychological value of this method is that it appeals to our
>>>>> natural
>>>>> community spirit which includes a willingness to go along with the
>>>>> group
>>>>> consensus when the consensus is strong enough, as long as there is no
>>>>> hope for a better consensus, and as long as it isn't a candidate that
>>>>> we
>>>>> would rate at zero.
>>>>>
>>>>
>>>> That's an interesting point. I don't think that factor has been
>>>> considered much in mechanism design in general. Condorcet, say, is usually
>>>> advocated on the basis that it provides good results and resists enough
>>>> strategy, and then one adds the reasoning "it looks like a tournament, so
>>>> should be familiar" afterwards.
>>>>
>>>> Perhaps there's some value in making methods that appeal to the right
>>>> sentiment, even if one has to trade off "objective" qualities (like BR,
>>>> strategy resistance or criterion compliance) to get there. The trouble is
>>>> that we can't quantify this, nor how much of sentiment-appeal makes up for
>>>> deficiencies elsewhere, at least not without performing costly experiments.
>>>
>>>
>> I'm currently doing such "costly experiments" on Amazon MTurk (with money
>> from Harvard). I'm evaluating Approval, Borda, Condorcet (3-candidate, so
>> the differences between the most common varieties doesn't matter), GMJ,
>> IRV, Plurality, Score, and SODA (with honest-declaring and
>> mutually-rational-assigning AI candidates), with an 18-voter, 3-candidate
>> scenario in factions of 8, 4, and 6 (with utilities for each voter of 0-3,
>> summing to 12, 16, and 11). I'll let the list know as results are available.
>>
>> Jameson
>>
>>
>
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