[EM] XA's continuity property

[Forest Simmons]

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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