[EM] XA (Andy Jennings)

Tue Nov 1 17:54:39 PDT 2016

I don't think you understand my point. Say candidate A has 60 votes at 0, 5
votes at 35, and 35 votes at 100; while candidate B has 63 at 0, 5 at 40,
and 32 at 100. B wins with 37. Now add 5 to the middle scores of both;
nothing fundamental changes about how any voter views either candidate, but
now A wins with 40.

My system of EXA (or percentile chiastic approval, PXA) would build ratings
scales automatically, so that an electorate that liked numbers around 80
could get the same results as one that liked numbers around 20.

2016-11-01 20:32 GMT-04:00 Forest Simmons <fsimmons at pcc.edu>:

> I guess I was assuming normalized ballots all along, in other words you
> would have at least one candidate that you approved unconditionally and at
> least one that you would not approve no matter what the other voters did,
> at least in an election setting.
>
> On the other hand if you are just contributing to a five star rating guide
> ...
>
> On Tue, Nov 1, 2016 at 2:28 PM, Jameson Quinn <jameson.quinn at gmail.com>
> wrote:
>
>> One issue I have with XA is that it makes numerical votes inherently
>> meaningful; it is entirely possible to change the election outcome by
>> adding or subtracting a constant from all ballots. I'm wondering if this is
>> fixable.
>>
>> What if you transformed all ratings into percentiles first? Let's call
>> this system empirical chiastic approval, EXA. So if you had something like
>> (candidates W-Z and grades A-F)
>>
>> 3: WA XB YC ZF
>> 2: WF XC YD ZA
>>
>> ... that would be an empirical grade distribution of 5As 3Bs 5Cs 2Ds 5Fs,
>> or in percentiles A75 B60 C35 D25 F0. So the EXA score is W60, X60, Y35,
>> Z40. (This example gave a tie because I deliberately made it so there would
>> be round numbers. In general, a tie would be highly unlikely. Clearly, in
>> order to minimize strategy, this tie should be broken in favor of W, the
>> candidate who had the most excess goodwill on their weakest
>> positive-influence ballot.)
>>
>> I think there are probably strategies involving manipulating the
>> empirical distribution (non-semi-honestly), but I doubt that anyone would
>> have the fine-grained info necessary to pull such a strategy off.
>>
>> If you condition on a given percentile distribution, it satisfies the
>> same kind of criteria that XA does, including "individual non-strategy".
>>
>> Interestingly, though not very usefully, this is a voting system which
>> would work just fine with allowing a negative infinity to positive infinity
>> ballot scale. (it would work even better than Bucklin in that sense.)
>>
>> 2016-11-01 16:54 GMT-04:00 Forest Simmons <fsimmons at pcc.edu>:
>>
>>> It is more likely that two candidates will have the same median score
>>> (an MJ tie situation) than having the same XA score.
>>>
>>> Part of the reason is that the XA scores depend continuously on the
>>> distribution of ratings, while the median can be a discontinuous function
>>> of the distribution.
>>>
>>> From another point of view, the graph of y = x is more likely to be
>>> perpendicular to the graph of the distribution function F(x) = Probability
>>> that on a random ballot candidate X will have a rating of at least x.  An
>>> orthogonal intersection minimizes error due to random perturbations.
>>>
>>> The graph of F stair steps down from some point on the y axis between
>>> (0, 0) and (0, 1) to some point on the vertical segment connecting (1, 0)
>>> to (1, 1).  If the distribution is uniform, then the graph of F is the
>>> diagonal line segment connecting (0, 1) to (1, 0), perpendicular to the
>>> line y = x.
>>>
>>> The median point (used in MJ and other Bucklin variants) is the
>>> intersection of the graph of F with the vertical line given by x = 1/2,
>>> cutting the square with diagonal corners at (0, 0) and (1, 1) in half.
>>>
>>> The midrange Approval value is the intersection of the graph of F with
>>> the horizontal line y = 1/2.
>>>
>>> The XA value is the intersection of the graph of F and line y = x, which
>>> bisects the right angle formed by  x = 1/2 and y = 1/2 at the intersection
>>> (1/2, 1/2).
>>>
>>> So XA can be thought of as a method half way between midrange Approval
>>> and score based Bucklin.
>>>
>>> More later ...
>>>
>>> Forest
>>>
>>>
>>>
>>>
>>>> From: Andy Jennings <elections at jenningsstory.com>
>>>> To: Michael Ossipoff <email9648742 at gmail.com>
>>>> Cc: "election-methods at electorama.com"
>>>>         <election-methods at electorama.com>
>>>> Subject: Re: [EM] XA
>>>>
>>>> On Mon, Oct 31, 2016 at 7:13 PM, Michael Ossipoff <
>>>> email9648742 at gmail.com>
>>>> wrote:
>>>>
>>>> > What makes XA do that more effectively than MJ? What's the main
>>>> advantage
>>>> > that distinguishes how XA does that from how MJ does it, or the
>>>> results,
>>>> > from the voters' strategic standpoint?
>>>>
>>>>
>>>> Michael,
>>>>
>>>> As Rob said, the median is not terribly robust if the distribution of
>>>> votes
>>>> is two-peaked:
>>>> http://www.rangevoting.org/MedianVrange.html#twopeak
>>>> And I'm afraid many of our contentious political elections are
>>>> two-peaked,
>>>> at least in the current environment.
>>>>
>>>> With MJ, I like the fact that if the medians for all candidates will
>>>> fall
>>>> between B and D, then I can use the range outside that for honest
>>>> expression.  Yet in the back of my head, I know that if everyone tries
>>>> to
>>>> "use the range outside that for honest expression", then the medians
>>>> won't
>>>> be in that range anymore and it seems like a slippery slope to everyone
>>>> using only the two extreme grades.
>>>>
>>>> XA solves this problem by making the more extreme grades more difficult
>>>> to
>>>> achieve.  As Rob said, in the case where everyone grades at the
>>>> extremes,
>>>> the XA will match the mean.
>>>>
>>>> On the other hand, I admit that:
>>>> 1) with the median, 50% would have to give the top grade for a
>>>> candidate to
>>>> receive that grade.  And 50% would have to give the bottom grade for a
>>>> candidate to receive that grade.  I consider both of these very
>>>> unlikely.
>>>> 2) MJ is not just "the median", it has a tie-breaking scheme which
>>>> mitigates this somewhat.
>>>>
>>>> ~ Andy
>>>>
>>>>
>>>
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>>>
>>>
>>
>
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