[EM] XA (Andy Jennings)

[Andy Jennings]

On Tue, Nov 1, 2016 at 1:54 PM, Forest Simmons <fsimmons at pcc.edu> wrote:

> It is more likely that two candidates will have the same median score (an
> MJ tie situation) than having the same XA score.
>

I know you reformulated your continuity argument, which I agree with.

But as for this point, I'm not sure that XA is less likely to have ties.

Chiastic medians will be more densely clustered around 50 than will normal
medians.  (For any given distribution function, compare the intersection
with the diagonal to the intersection with the vertical midpoint.
Horizontally, the former will be between 50 and the latter.)

For a discrete grading scale, say integers from 0 to 100, this would
increase the probability for collisions.  On the other hand, the diagonal
might intersect with a horizontal segment of the distribution function and
could give a chiastic median which is not in the discrete grading scale,
but corresponds to a percentage of the electorate.  (If exactly 75.43% of
the electorate gave a grade of 76 or above, then the chiastic median is
75.43.)  I guess this may make collisions less common.  I'm not sure
exactly how these two forces would balance out.  It seems like you're
definitely most likely to see collisions on the integers.

For a continuous grading scale, say all real numbers from 0 to 100, any two
grades will collide with probability 0, so the medians will almost never
collide.  Whereas the chiastic medians have some probability of colliding
if they are returning discrete percentages of the electorate.  (The
horizontal segments of the distribution function are at discrete heights,
so if the diagonal hits one of them, then there could be collisions.)

So perhaps it depends on the grading scale and the presumed distribution of
votes...

~ Andy
-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://lists.electorama.com/pipermail/election-methods-electorama.com/attachments/20161104/4da37326/attachment-0001.htm>