2-color 1-sided strategy (2C1SS) and voting systems robust against it

[Warren D Smith]

2C1SS is:
there are 2 colors of candidates, "red" & "blue," and probably also
some more (who are neither red nor blue).
Some voters, 1-sidedly, exaggerate their red scores toward MAX
and their blue scores toward MIN.
By 1-sided I mean, there is no analogous counterattack by the
pro-blue anti-red forces.

It has recently been brought to my attention by Jameson Quinn that
STAR voting and JQ's own "321 voting" system exhibit good
robustness against 1-sided exaggerating voters.
This can be regarded as an advantage for those systems, as well
as for the 2-round "range + top-2 runoff" (R+R) system, by comparison to
plain score voting (which is comparatively vulnerable to 1-sided
strategy).

However, 2C1SS is a more general 1-sided strategy scenario than
Jameson Quinn's notion of 1-sided strategy which had
allowed, essentially, just a single red and single blue candidate.
And against 2C1SS, both STAR and 321 and R+R all suck.

Can we devise voting methods robust against 2C1SS?
Yes, we can.

First, I will discuss a few variants of STAR, and explain that some
of them can be made robust versus 2C1SS.

Original STAR (vulnerable to 2C1SS):
1. ballots are score-style
2. find the two candidates A & B having the highest score-sums
(or highest mean scores)
3. the pairwise-victor among {A,B} wins, i.e. if there
are more A>B ballots than B>A, then A wins.

The following variant, whose revolto-acronym could be "STORM"
(score then rival-max), is however, robust versus 2C1SS:
1. ballots are score-style
2. find the candidate W with greatest mean score
3. among all rivals R of W which defeat W pairwise,
elect the one with greatest mean score. (But elect W if R is the empty set.)

It might, however, be that STORM is worse than STAR in other respects
(I have no idea). But it is possible to devise a method along these lines
which I can guarantee will perform, overall, superior to (or
at least no worse than) all three among STORM, STAR, and plain-score voting.
The trick to accomplish that is what I call "the method of tunable parameters."

Tunable score-STAR-storm (TSSS) method:
1. ballots are score-style
2. find the candidate W with greatest mean score
3. find the candidate S with the 2nd-greatest mean score
3. among all rivals R of W such that
meanscore(R)>=T1*meanscore(S) and
meanscore(R)>=T2*meanscore(W) and
which defeat W pairwise,
elect the one with greatest mean score. (But elect W if R is the empty set.)

Here T1 and T2 are real constants with 0<=T1<=1 and 0<=T2<=1.
The point I am driving at is: If we choose (T1,T2) OPTIMALLY
then TSSS will be at least as good as the best among
STORM, STAR, and plain score voting.
Because: if T1=T2=1 then TSSS=plain score.
If T1=1 and T2=0 then TSSS=STAR.
If T1=0 and T2=0 then TSSS=STORM.
I presume we only are interested in the triangular region
0<=T2<=T1<=1 or parameter-space.
To choose (T1,T2) optimally just do a lot of numerical experiments using
Jameson's VSE tester to find the parameter combination yielding best
overall results.

Now I must say I find STAR, and especially 321 voting (whose definition,
by the way, seems to vary depending on the date Jameson talks about it)
rather ugly. They violate a lot of properties like monotonicity,
favorite betrayal, and (for 321) cloneproofness. They are kludgy. TSSS also
is subject to the same complaint.
The question is can we get a non-ugly method that is robust versus 2C1SS.

Well, a method was proposed long ago which is simple, natural, has
pretty good properties, and designed to be robust versus 2C1SS:
"trimmed-mean range voting":

1. Ballots are score style.
2. For each candidate C, compute its "trimmed mean" by
discarding the top fraction F and bottom fraction F of C's scores,
then taking mean of the 1-2F fraction that remain.
3. candidate with greatest trimmed-mean is elected.

Here F is a tunable parameter. If F=0 then this is plain score voting.
If F=0.4999 then this is median-based score voting. Choose the
value of F with 0<=F<1/2 that yields the best overall performance in
VSE testing.

Now there was something that bothered me about trimmed-mean.
The Jews hate Hitler. With trimmed mean, we are saying "Sorry Jews,
but your opinion of Hitler is an outlier so we are going to discard
it. Have a nice day."
This seemed undemocratic to me. However, I suppose that the Jamesonian view
would be, this undemocracy is forgivable if we choose the OPTIMUM value of F,
because we'll get the best overall performance and make the "unhappy Jews"
scenario rare enough that the price we pay for it is worth the
benefits we obtain.

Now it also is possible to define fancier variants still.
Sort the scores for a given candidate into increasing order.
Say there are N+1 scores S[0], S[1], S[2],..., S[N]. Now compute
{SUM(0<=i<=N)of W(i/N) * S[i]} / {SUM(0<=i<=N)of W(i/N)}
where W(x) is a unimodal positive-valued FUNCTION of x with 0<=x<=1.
This is C's "W-robustified mean."
Elect the candidate with the greatest W-robustified mean.

Choose the function W(x) which yields the best overall performance
on VSE testing. One simple 1-parameter family of W-candidates include
W(x) = [x*(1-x)]^P
where P>=0 is a tunable parameter. If P=0 then we get plain score voting.
If P goes to infinity we get median-based score voting.



--
Warren D. Smith
http://RangeVoting.org <-- add your endorsement (by clicking
"endorse" as 1st step)

[parker friedland]

There is one thing that bothers me about trimmed mean voting and similar voting methods that list the scores from highest to lowest and then give the scores in the middle of the list a higher weight. Those types of weighting methods usually fail some disable properties.

Desirable property number 1 - No change criterion: Suppose that a rating method takes in individual ratings of a candidate to compute a final rating for that candidate. If you then add more individual
ratings that are equal to the candidate's final rating, the candidate's final rating should not change.

Here is an example of trimmed mean voting violating this property when F = 0.25:

# of voters     Candidate A

7                    5 stars
12                  0 stars

Five 5 star ratings and a Five 0 star rating are then removed for being too extreme, and then the remaining ratings are the fallowing:

2                    5 stars
8                    0 stars

Trimmed mean voting then assigns A a rating of 1 star because among the remaining ratings, the average is 1 star.

However, if you then add ten 1 star ratings:

# of voters     Candidate A

6                    5 stars
4                    1 star
14                  0 stars

Non-extreme ratings:

# of voters     Candidate A

4                    1 star
8                    0 stars

Now, A's rating has gone down from 1 star to 0.333 stars even though all that was added was 1 stars.

Violating this property also implies a violation of a second desirable property:

Desirable property number 2 - Rating participation criterion: If giving a candidate a rating changes the rating the voting method assigns that rating, the new rating the voting method assigns that candidate should always be closer to that voter's rating of the candidate then the previous rating that voting method assigned to that candidate.

Median based voting and mean based voting pass these two criterions, and both of these properties appear to be desirable properties, so perhaps a voting method that is a compromise between mean based voting and versions of median based voting should have both of these two properties. Here is one such voting method with a tun-able parameter K that when K = 1, you get median based voting, and when K = 2, you get mean based voting:

Each candidate gets the rating R that best minimizes the total cost produced by each voter's cost function for that candidate, where each voter's cost function is defined as:
Voter cost = |S - R| ^ K
Total cost = ∑ voter cost
Where S is the score that voter gave that candidate.

When K approaches 0, the value of R that best minimizes the total cost is a candidate's mode score.
When K equals 1, the value of R that best minimizes the total cost is a candidate's median score.
When K equals 2, the value of R that best minimizes the total cost is a candidate's mean score.
When K approaches infinity, the value of R that best minimizes the total cost is a candidate's mid-range (max score + their min score divided by 2).

So you can adjust K to get almost every possible statistical measure of center in a data set. But the values of K that we would be interested in would be values of K that are between 1 and 2.

[parker friedland]

I messed up this example:

> Desirable property number 1 - No change criterion: Suppose that a rating method takes in individual ratings of a candidate to compute a final rating for that candidate. If you then add more individual
> ratings that are equal to the candidate's final rating, the candidate's final rating should not change.
>
> Here is an example of trimmed mean voting violating this property when F = 0.25:
>
> # of voters     Candidate A
>
> 7                    5 stars
> 12                  0 stars
>
> Five 5 star ratings and a Five 0 star rating are then removed for being too extreme, and then the remaining ratings are the fallowing:
>
> 2                    5 stars
> 8                    0 stars
>
> Trimmed mean voting then assigns A a rating of 1 star because among the remaining ratings, the average is 1 star.
>
> However, if you then add ten 1 star ratings:
>
> # of voters     Candidate A
>
> 6                    5 stars
> 4                    1 star
> 14                  0 stars
>
> Non-extreme ratings:
>
> # of voters     Candidate A
>
> 4                    1 star
> 8                    0 stars
>
> Now, A's rating has gone down from 1 star to 0.333 stars even though all that was added was 1 stars.
>
> Violating this property also implies a violation of a second desirable property:
>
> Desirable property number 2 - Rating participation criterion: If giving a candidate a rating changes the rating the voting method assigns that rating, the new rating the voting method assigns that candidate
> should always be closer to that voter's rating of the candidate then the previous rating that voting method assigned to that candidate.

Corrected version:

Desirable property number 1 - No change criterion: Suppose that a rating method takes in individual ratings of a candidate to compute a final rating for that candidate. If you then add more individual
ratings that are equal to the candidate's final rating, the candidate's final rating should not change.

Here is an example of trimmed mean voting violating this property when F = 0.25:

# of voters     Candidate A

7                    5 stars

13                  0 stars

Five 5 star ratings and a Five 0 star rating are then removed for being too extreme, and then the remaining ratings are the fallowing:

2                    5 stars
8                    0 stars

Trimmed mean voting then assigns A a rating of 1 star because among the remaining ratings, the average is 1 star.

However, if you then add eight 1 star ratings:

# of voters     Candidate A

7                    5 stars
8                    1 star
13                  0 stars

Non-extreme ratings:

# of voters     Candidate A

8                    1 star
6                    0 stars

Now, A's rating has gone down from 1 star to 0.57142857 stars even though all that was added was 1 star ratings.

Violating this property also implies a violation of a second desirable property:

Desirable property number 2 - Rating participation criterion: If giving a candidate a rating changes the rating the voting method assigns that rating, the new rating the voting method assigns that candidate should always be closer to that voter's rating of the candidate then the previous rating that voting method assigned to that candidate.

[Warren D Smith]

seems like it is even worse than you said:
with trimmed mean, we can adjoin some new voters providing scores S,
where S is BELOW some candidate's trimmed mean M, and that will cause
M to RISE.

Not very nice.

[NoIRV]

On Monday, April 23, 2018 at 7:47:49 PM UTC-4, Warren D. Smith (CRV cofounder, http://RangeVoting.org) wrote:
> seems like it is even worse than you said:
> with trimmed mean, we can adjoin some new voters providing scores S,
> where S is BELOW some candidate's trimmed mean M, and that will cause
> M to RISE.

How?

[Warren D Smith]

--Ok, let's see.
If the score-set for a candidate is (allowed scores range from 0 to 9,
trim off the top 25% and bottom 25% of the scores, then take mean)

{0,0,0,0, 1,5,5,5, 5,5,5,5, 9,9,9,9}

then the trimmed set of scores is

{1,5,5,5,5,5,5,5}

with trimmed mean = 4.5.
Now adjoin 4,4,4,4, which, note, all lie below 4.5. Now the scores are

{0,0,0,0,1, 4,4,4,4,5, 5,5,5,5,5, 5,9,9,9,9}

which is trimmed to

{4,4,4,4,5,5,5,5,5,5}

with trimmed mean = 4.6,
which, note, rose.