[EM] Election-Methods Digest, Vol 106, Issue 2

Forest Simmons fsimmons at pcc.edu
Thu Apr 4 16:44:39 PDT 2013

```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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