[EM] Motivation for Chiastic Approval (XA)

[Andy Jennings]

Michael asked what motivated XA.

When I came up with XA, I believe I was thinking about how voters might
manipulate a score voting election if they had a target score for each
candidate and were trying to make each candidate's mean score match their
target.  I was thinking of an ongoing election where the voters can see the
mean scores and change their votes at any time.  Whenever the candidate's
mean fell below their target, they would change their vote to MAX and
whenever it rose above their target they would change their vote to MIN.
Score voting is not even required.  Approval voting would be good enough!

If they could do this continuously (or hire a vote butler or write a
computer program to do it for them), would there be an equilibrium point
for a given candidate?  Yes, it would be the chiastic median (XM), the
largest number, x, where at least x% of the people wanted a score of x or
higher.  (For simplicity, we assume MIN=0 and MAX=100.)

It's like a declared-strategy variant of score voting.  Once we know that
we can just ask voters for their 0-100 score of each candidate and then
calculate each candidate's XM.

After that, I figured out that any voter who gave a candidate a higher
score than their XM, if given the opportunity to change their vote, could
do nothing to raise the XM.  Likewise, a voter who gave a candidate a score
lower than the XM could have done nothing to lower their XM.  So voters
should be honest, there's no reason to be dishonest when submitting your
scores!  Whether the candidate ends up with an XM higher or lower than the
score you submitted, your vote is, as much as possible, pulling their XM
towards your score.  In essence, it turns your vote into an approval vote,
approving everyone who society gives a lower score than you would like, and
disapproving everyone who society gives a higher score than you would
like.  So in this sense, the method can be called Chiastic Approval (XA).

Of course, this all assumes that voters only care about making the
candidate's final scores match their personal evaluations.  It ignores that
they would strategize to affect the winner.  You can construct profiles
where voters give honest scores and later discover that there were two
frontrunners and they gave scores to both that were higher than their XMs.
Those votes are pulling up on the XMs of both frontrunners, thus their
votes are not really helping distinguish the frontrunners, which they might
want.  (This is related to the chicken dilemma.)  You can construct
profiles where the voters would change their votes and that would change
the winner.  But whatever it's faults here, XA must be better than plain
approval, no?

In any case, I note that the more strongly a voter feels about the
difference between two candidates, the bigger difference he will put
between their scores, and it becomes much less likely that his scores will
be on the same side of both their XMs.

I also note that this method (XA) is appropriate for a society trying to
decide one number, say an approval rating for the president.

Lastly, I note that Rob LeGrand discovered this same system and studied it
in his thesis.  Later, I discovered it independently and studied it in
mine.  I have more to say about it if you want, but these are the basics.

~ Andy
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