<p class="MsoNormal">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.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">I assume versions of these methods that make use of range
style ballots on a scale of zero to 100.<span style>
</span>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.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">So let’s concentrate on how each of these methods would
assign a score to the same fixed candidate.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">All of these methods can be explained in terms of the graph
of the function F given by</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">p=F(r) is the percentage of the ballots that rate our
candidate strictly greater than r.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">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). </p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">Furthermore, the graph will descend from left to right in
steps whose widths are whole numbers. </p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">The left endpoint of each step will be included but the
right end point will not be included.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">Color this graph blue.<span style>
</span>Now join the steps with vertical segments.<span style> </span>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.<span style>
</span></p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">Now the union of the red and blue separates the lower left
corner from the upper right corner of the square.<span style> </span>Therefore the diagonal from (0, 0) to (100,
100) must cross the colored graph in either a red or blue point.<span style> </span>Since the red and blue are non-increasing
while the diagonal is strictly increasing, there can be only one point of
intersection.<span style> </span>The common value of the
coordinates of this intersection point is the Chiastic Approval score.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">Now calculate the area of the region of the square that<span style> </span>lies to the lower left of the red/blue
diagonal.<span style> </span>This area is the average
rating of our candidate.<span style> </span>So the
candidate whose lower left area is greatest is the Range winner.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">Now bisect our square horizontally with a straight line
segment from (0, 50) to (100, 50).<span style> </span>The first
coordinate (r) of the point of intersection (r, 50) of this line with the red determines
the basic Bucklin score.<span style> </span>Ties are broken
by various methods.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">Now bisect the square with a vertical segment from (50, 0)
to (50, 100).<span style> </span>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.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">Now consider the diagonal from the upper left corner (0,
100) to the lower right corner (100, 0).<span style>
</span>If this diagonal does not intersect the blue, then the candidate’s
Consensus Threshold Approval score is zero.<span style>
</span>Otherwise it is the second coordinate of the highest (and therefore
leftmost) blue point of intersection.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">In summary, we have bisected the 100 by 100 square
vertically, horizontally, and diagonally. <span style> </span>The diagonal with positive slope leads us to
the chiastic approval winner.<span style> </span>The other
diagonal leads us to the consensus threshold approval winner.<span style> </span>The horizontal bisector leads us to the Bucklin
winner.<span style> </span>The vertical bisector leads us
to the Approval winner.<span style> </span>The area cut off
by the colored graph determines the Range winner.</p>
<br><br><div class="gmail_quote">On Wed, Apr 3, 2013 at 6:18 PM, Jameson Quinn <span dir="ltr"><<a href="mailto:jameson.quinn@gmail.com" target="_blank">jameson.quinn@gmail.com</a>></span> wrote:<br><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
<div dir="ltr"><br><div class="gmail_extra"><br><br><div class="gmail_quote"><div><div class="h5">2013/4/3 Forest Simmons <span dir="ltr"><<a href="mailto:fsimmons@pcc.edu" target="_blank">fsimmons@pcc.edu</a>></span><br>
<blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex"><br><br><div class="gmail_quote"><div><div>On Wed, Apr 3, 2013 at 12:07 AM, Kristofer Munsterhjelm <span dir="ltr"><<a href="mailto:km_elmet@lavabit.com" target="_blank">km_elmet@lavabit.com</a>></span> wrote:<br>
<blockquote class="gmail_quote" style="margin:0pt 0pt 0pt 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex">
<div>On 04/03/2013 12:01 AM, Forest Simmons wrote:<br>
<blockquote class="gmail_quote" style="margin:0pt 0pt 0pt 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex">
Jobst has suggested that ballots be used to elicit voter's "consensus<br>
thresholds" for the various candidates.<br>
<br>
If your consensus threshold for candidate X is 80 percent, that means<br>
that you would be willing to support candidate X if more than 80 percent<br>
of the other voters were also willing to support candidate X, but would<br>
forbid your vote from counting towards the election of X if the total<br>
support for X would end up short of 80 percent.<br>
<br>
The higher the threshold that you give to X the more reluctant you are<br>
to join in a consensus, but as long as your threshold t for X is less<br>
than than 100 percent, a sufficiently large consensus (i.e. larger than<br>
t percent) would garner your support, as long as it it is the largest<br>
consensus that qualifies for your support.<br>
<br>
A threshold of zero signifies that you are willing to support X no<br>
matter how small the consensus, as long as no larger consensus qualifies<br>
for your support.<br>
<br>
I suggest that we use score ballots on a scale of 0 to 100 with the<br>
convention that the score and the threshold for a candidate are related<br>
by s+t=100.<br>
<br>
So given the score ballots, here's how the method is counted:<br>
<br>
For each candidate X let p(X) be the largest number p between 0 and 100<br>
such that p(X) ballots award a score strictly greater than 100-p to<br>
candidate X.<br>
<br>
The candidate X with the largest value of p(X) wins the election.<br>
</blockquote>
<br></div>
I think a similar method has been suggested before. I don't remember what it was called, but it had a very distinct name.<br>
<br>
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).<br>
<br>
I *think* it went like that, at least. Sorry that I don't remember the details!</blockquote></div></div><div><br>Good memory, that was Andy Jennings' Chiastic method. Graphically these two methods are based on different diagonals of the same rectangle.<br>
</div></div></blockquote><div><br></div></div></div><div>Different, how? It seems to me they're just the same, but with the numbers reversed. </div><div class="im"><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
<div class="gmail_quote"><div>
</div><div><blockquote class="gmail_quote" style="margin:0pt 0pt 0pt 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex"><div><br>
<br>
<blockquote class="gmail_quote" style="margin:0pt 0pt 0pt 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex">
If there are two or more candidates that share this maximum value of p,<br>
then choose from the tied set the candidate ranked the highest in the<br>
following order:<br>
<br>
Candidate X precedes candidate Y if X is scored above zero on more<br>
ballots than Y. If this doesn't break the tie, then X precedes Y if X<br>
is scored above one on more ballots than Y. If that still doesn't break<br>
the tie, then X precedes Y if X is scored above two on more ballots than<br>
Y, etc.<br>
<br>
In the unlikely event that the tie isn't broken before you get to 100,<br>
choose the winner from the remaining tied candidates by random ballot.<br>
</blockquote>
<br></div>
I imagine Random Pair would also work.<div><br>
<br>
<blockquote class="gmail_quote" style="margin:0pt 0pt 0pt 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex">
The psychological value of this method is that it appeals to our natural<br>
community spirit which includes a willingness to go along with the group<br>
consensus when the consensus is strong enough, as long as there is no<br>
hope for a better consensus, and as long as it isn't a candidate that we<br>
would rate at zero.<br>
</blockquote>
<br></div>
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.<br>
<br>
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.</blockquote>
</div></div></blockquote><div><br></div></div><div>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.</div>
<span class="HOEnZb"><font color="#888888">
<div><br></div><div>Jameson</div></font></span></div><br></div></div>
</blockquote></div><br>