[EM] Consensus Threshold Approval

[Andy Jennings]

>
> 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.


I should correct myself.  CTA is not exactly the same as the chiastic
median of the thresholds.  CTA calculates the largest number, x, between 0
and 100 such that x percent of people gave a threshold of x or below.
 Chiastic median applied to the thresholds would calculate the largest
number, x, between 0 and 100 such that x percent of people gave a threshold
of x or above.

But the three concerns I mentioned are still valid.

~ Andy


On Mon, Apr 8, 2013 at 10:12 AM, Andy Jennings
<elections at jenningsstory.com>wrote:

> 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
>>>>>> for your support.
>>>>>>
>>>>>> 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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>
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