# [EM] Candidate-Space Method

MIKE OSSIPOFF nkklrp at hotmail.com
Sat Dec 14 19:19:40 PST 2002

```As sometimes happens, I don't remember the exact subject title of
the posting I'm replying to, so I made up my own descriptive subject line.

Forest described a method that infers voters' issue-space positions
from how they rate the candidates, and, estimates candidates' positions
from those of the people who consider them favorite. Since the method
doesn't have a name, I'll just call it "Candidate-Space" (CS), short
for Candidate Ratings Space. I've run across the name "attribute
space" for that space, but if a Nader preferrer votes also for Gore,
strategically, it seems unfair to say that he and Nader have a Gore-like
attribute. :-|

I'm not saying that CS is the best name, only that I'm using it because
the method as yet isn't named.

>From preliminary experience with a few examples, CS, in its
Approval+favorite version, and ranking version, looks pretty good.

I don't know what Hamming distance or L1 norm are, but I used
Pythagorean distance and city-block rectangular distance. With the
Approval+favorite version, there's no difference between those two.

I don't often have time to get to a library, so if someone knows what
Hamming distance or the L1 norm is, will you post the definition?

I'm interested in voting systems because of their practical importance,
and so I'm interested in the voting systems that can be publicly adopted
soonest. So I don't usually take part in discussion about more complicated
methods that would only be proposed after we've achieved
more modest single-winner reform. But I make an acception, at least a
little, with CS, because, for one thing, its way of estimating issue-space
positions makes it helpful for judging whether Pythagorean distance or
city-block rectangular distance is better for simulations;
and also because it's interesting.

Of course, when society has already adopted and been using a more
modest reform like Approval, -1,0,1, or other CR, or maybe Condorcet(wv),
all the social improvement that single-winner reform can bring
had become democratically involved enough to consider more complicated
methods like CS, or the other excellent but complicated methods that
have been discussed here, then we'd certainly already have a much better
and more nearly ideal society by that time. Then, further refining
the voting system would still be an interesting & worthwhile pursuit,
but society wouldn't urgently need it the way it does now.

Of course when I say that society urgently needs voting reform
(and a number of other reforms, all of which would be advanced by
voting reform), I don't mean that _we_ need it. If a person _needs_
social improvement, then s/he is setting himself/herself up for
unhappiness. Of course we like to do what we can though.

But that's getting off the subject. I tried CS with my standard
IRV badexample in which A, B, & C have 40, 25, & 35 voters. In that
example, the Approval+favorite CS behaves much like regular
Approval+Favorite, except that when the C voters vote also for B,
they not only make B win, but they make C come in 2nd.

When the numbers are 22, 23, & 55, and the C voters mistakenly vote
for B even though they have a majority, they don't thereby give the
election away to B, as they would in ordinary Approval. (That's ignoring the
majority favorite provision, which would elect C
automatically in ordinary Approval+favorite or CS Approval+favorite).

Also, I wanted to find out if the ranking version of CS would elect
the CW in the 40,25,35 example. The rankings were:

40: ABC
25: BAC
35: CBA

With city-block rectangular distances,
A is elected, though B is the CW. With Pythagorean distance, B is
elected. I checked truncation & order-reveresal, with Pythagorean
distance.

If the A voters truncate, giving minimum rating to B & C, that
doesn't steal the election from B in that example. If they order-reverse,
that will elect A. But the B voters can prevent that result
by merely upranking C to 2nd place, with A. That's an remarkably &
unusually mild defensive strategy. I don't know if that would always
work, or, if not, under what conditions it would work; or if
offensive truncation always fails.

I don't know what criteria CS meets. Condorcet(wv) was intentionally
chosen to minimize the strategy requirements on a majority, and
so it would be a little surprising if another method met the
majority defensive strategy criteria or FBC. But who knows--for all
I know now, maybe CS does.

Anyway, about the distance measures: Simplicity recommends the
city-block distance. Instead of assuming the two quantities,
policy difference, and policy difference disutility proportional
to the square of policy difference, it's simpler to just treat them
as the same, as in city-block distance. But the fact that, at least
in that one example that I checked, Pythagorean distance is the
distance measure that found the CW to be closest to the voters overall,
that suggests that Pythagorean distance is better.

All those things, the strategy properties and criteria complicances,
the better voter-closeness accuracy of Pythagorean distance, those
things would have to be determined in a more general way than just
looking at one example. But CS seems to have especially mild strategy
requirements. The fact that uprating a candidate lowers your candidate's
distance to his voters seems to act as a built-in buffer against
strategy harshness. But it could possibly also lead to a nonmonotonic
example. Of course CS has a lot more merit to outweigh nonmonotonicity
than IRV does.

Does anyone have any opinion on Pythagorean distance vs
city-block rectangular distances, for spatial simulations of SU?
Or what about Hamming distance for that purpose (As I said, I
don't know what Hamming distance is)?

Mike Ossipoff

Here's the crucial idea that allows us to avoid the survey: a voter's
position in issue space is reflected in the voter's choice of approved
candidates.

The fewer disagreements two voters have about which candidates are
acceptable, the closer they are to each other in issue space (on average).

Suppose we use Majority Choice ballots, and no candidate is a favorite of
more than fifty percent of the voters. Then (without violating the
secrecy of any voter's ballot) we can ascertain the approximate position
of each candidate by averaging the ballots that list him/her as a
favorite.

Here we are assuming that the most avid supporters of candidate X are
relatively near candidate X in issue space, so their average location is
a reliable estimate of his/her location.

Another reason for estimating the candidate's position rather than taking
the candidate's self estimate, is that the "candidate" might not be a
person; voting methods are used in other contexts as well.

Once we have an estimate of the position of each candidate, the distance
separating each voter from each candidate can be estimated also.

I suggest using the Hamming distance, which is just the L1 norm of the
difference in position vectors.

For example, if there are ten candidates and my (ordered) list of
acceptable candidates is C1, C4, C5, and C9, then my position vector is

[1,0,0,1,1,0,0,0,1,0].

Candidate C7 might have an estimated position of

[.1, .2, .3, .1, .2, .5, 1.0, .6, .4, .8] .

Then my Hamming distance is the sum

.9 + .8 + .7 + .9 + .8 + .5 + 1 + .4 + .6 + .8 .

Note that your Hamming distance is always at least one unit from any
candidate that you do not list as acceptable.

Now how do we figure the winner?

For each candidate X we calculate the sum of the distances from X to each
voter, and then divide by the number of voters. This number E may be
interpreted as the expected distance from X to a randomly chosen voter.

The candidate with the smallest value of E wins the election.

Any other candidate is more distant (on average) from a randomly chosen
voter.

If there are two seats to be filled, we consider all pairs of candidates,
and average the distances from the voters to the nearest member of the
pair.

In other words, if the pair is {X,Y} and voter Z is closer to X than to Y,
then the distance from X to Z is the number that this ballot contributes
to the sum.

Obviously this method can be extended to multiwinner elections with
several seats to be filled, with no more computational effort than PAV.

In fact, the access to the shape of the issue space that this method
affords us can drastically reduce the number of combinations of candidates
that need to be considered.

For example, if the issue space is essentially one dimensional, and if two
candidates are both to the left of the median voter, then there is no use
in considering them as a candidate pair.

By the way, it is obvious how to adapt this method to CR style ballots,
and, since it is easy to convert rankings to ratings, the method can be
adapted to ranked ballots just as easily.

However, I don't think that the extra refinement of ratings or rankings is
really necessary, or that there is any statistical chance that they would
change the outcome in any election with more than one hundred voters.

The Majority Choice ballots are just the right compromise in simplicity
and expressivity for the day when we can go beyond Approval ballots.

In the mean time, Approval Ballots can be adapted to the method by the
simple device of conflating the two categories of Favorite and Acceptable
with Approval, i.e. approved candidates are considered as both Favorite
and Acceptable.

Forest

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