[EM] XA's continuity property

[Andy Jennings]

Yes, using the graphical framework where XA is (the x-coordinate of) the
intersection of the diagonal and the distribution function, the median is
(the x-coordinate of) the intersection of y=50 and the distribution
function.  (Any other order statistic would be a horizontal line at a
different height.)

One voter changing their vote (or adding one voter or removing one voter)
moves the distribution function up or down slightly.  It's easy to see how
this can cause a large change in the intersection with a horizontal line.
But with a diagonal line, the intersection can move left or right no more
than the distance the distribution function was perturbed vertically.

~ Andy

On Tue, Nov 1, 2016 at 6:10 PM, Forest Simmons <fsimmons at pcc.edu> wrote:

>
>
> I'm afraid my original hurried example comparing the continuity of XA
> against the discontinuity of Bucklin was not quite accurate, let alone
> clear.  Now, while more relaxed, I think I can make this clear:
>
> Ignoring the other candidates, let's focus on the ballot ratings given to
> candidate X before and after a slight perturbation (perhaps caused by the
> process of a recount).
>
> Before:
>
> 50+epsilon percent of the ballots rate X at 100 percent, while the other
> 50 - epsilon percent of the ballots rate X at zero percent.
>
> After the recount 50-epsilon of the ballots rate X at 100 percent, while
> 50 + epsilon of the ballots rate X at zero.
>
> In other words the recount either corrected the count by 2 percent of
> epsilon, or further messed it up by 2 percent of epsilon (we don't know
> without a more careful recount).
>
> Before the recount Bucklin gives X a score of 100%.
> After the recount the Bucklin score for X is zero.
>
> A difference of 2epsilon percent causes a sudden drastic drop in the
> bucklin score.
>
> How about the before and after XA scores?
>
> Before the recount, the XA score for candidate X is 50 + epsilon percent
> because X was given at least that high of a rating on 50 + epsilon percent
> of the ballots.
>
> After the recount, the XA score was 50-epsilon percent, because X was
> given that high or higher rating on that percent of the ballots.
>
> No drastic sudden drop.  A change of 2 epsilon percent in the ballot count
> only occasions a change of the same magnitude in the XA score.
>
> That's the kind of continuity I meant to illustrate.
>
> Thanks,
>
> Forest
>
> .
>
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