You have to square p to have p in there at all, because r is defined as m/p. So r*p = (m/p)*p; it just cancels outthe division.
Kevin

Le mardi 28 mai 2019 à 21:44:02 UTC−5, robert bristow-johnson <rbj@audioimagination.com> a écrit : 

On Tue, May 28, 2019 7:32 pm, Kevin Venzke" stepjak@yahoo.fr wrote:

so weighting this with the square of participation is a good thing?  i need help understanding why.  of course as Y>X increases,the defeat strength should only go down, if it's about Margins.

but why is rp^2 better than rp?  i can see why r*p is better than just r.  but hyping up p might be too much of a good thing.
L8r   r b-j
 

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that ain't how i look at it. 
probabilistically, the bigger WV+LV is, we can expect a bigger WV-LV in magnitude.

the expectation value (or mean) of |WV-LV| increases as WV+LV does.
hmmm, i gotta think about this a little.  the probability a voter votes for the winning candidate is p=[WV/(WV+LV)] and the probability the voter votes for the losing candidate is q=[LV/(WV+LV)]   consider a
binomial distribution, with n, p and q=1-p,

the variance of WV is (WV+LV)[WV/(WV+LV)][LV/(WV+LV)] which is also the variance on LV.  the standard deviation of both are sqrt( WV*LV/(WV+LV) )  and the numerator increases as the square of the denominator.  so I would guess that |WV-LV| goes up with the sqrt(WV+LV).  if the
decisiveness (which Juho calls "r") remains the same, if you double the size of an election, the s.d. of the |WV-LV| margin is expected to increase by 41%.  but the means also go up.
so the standard deviation goes up with sqrt(WV+LV).  but i still expect the mean of |WV-LV|
to go up proportionately with participation if the mind of the electorate remains the same and the participation increases.
the more i keep reading this thread (that i guess that i started), the more convinced that i am that, considering simplicity as an asset to sell to both policy makers and
the public, i am convinced that the "Best Single-Winner Method" must be Ranked-Choice ballots (not FPTP, not Score, not Approval) and Condorcet-compliant (because the alternative to electing the CW is electing a candidate when explicitly more voters marked their ballots preferring a
different specific candidate) and i think that RP with simple Margins (WV-LV) is the most meaningful and simplest.

   WV-LV = [(WV-LV)/(WV+LV)]  x  (WV+LV)
the factor [(WV-LV)/(WV+LV)] (or "r") is the measure of the decisiveness of an election.  this is what we mean by "Brexit wins by 3.8% in 2016."
the factor WV+LV (or "p") is the measure
of how big the election is.  how many people are affected by it enough to weigh in on it.
the Defeat Strength is the product of the two.  Bigger elections count more.  And more decisive elections count more.  i dunno.
 
perhaps, to sell a Condorcet method,
rather than RP, we could sell IRV-BTR to make the IRV crowd happy.  we sorta lose the precinct summability (actually precincts can report pairwise defeat totals which can still be used to check up on the official central counting in the likely case there is no cycle).  then this Defeat
Strength discussion becomes moot again.
hmmmm.
r b-j

same"Margin is good, and if it means "I don't know but I trust other voters to express a valid opinion about this option", then RM is perfect. Just tell voters the chosen interpretation of blank tanks in advance so they can fill a sincere ballot...

WinningVotes is quite difficult to explain since it says that the strength of preference is discontinuous with fully ranked votes (51-49 is a strong victory but 49-51 is a heavy loss). I tried to write good explanations on how Winning Votes and Losing Votes (that is also discontinuous) treat
pairwise tiesand rankings, but the end results were not very intuitive, so I will not include any of that mess here :-).

thestrength of my vote" (the voter doesn't want to influence the strength of the final decision in any way), and Moderated Margins says "others shall decide, but I vote to make their final decision weaker" (lack of opinions should men that the outcome is weak). Relative Margins says
thatpreference 30-10 should be seen as stronger than 60-40. Margins says that 30-10 should be seen as equally strong. Moderated Margins says that 30-10 should be seen as weaker than 60-40 (maybe "less decisive" since so many voters didn't tell their opinion).

50%participation in the pairwise competition (when 50% of the preferences in the ballots are ties) should mean that the strength of the result should be only 50% of the strength it would otherwise be (Margins can be seen as the starting point here). While Relative Margins results can be seen to
beMargins results, where the Margins result will be divided by participation (percentage), in Moderated Margins the Margins result will be multiplied with participation. In that sense they are mirroring each others at the opposite sides of the Margins philosophy.

complicatedfor any regular election. These options try to capture the sincere opinion of the voter. Strategic implications ffs.

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r b-j                         rbj@audioimagination.com

"Imagination is more important than knowledge."

I listed some of the properties of "good" preference functions. Should I read what you say so that f(x,y) should be non-increasing for all fixed values of x, and non-decreasing for all fixed values of y? If f(x,y) approaches plain p (or -p) (locally) somewhere in the x-y-plane, that would violate this rule. I guess the rule should cover both the losing side and the winning side. I note that the Moderated Margins preference function gets very flat close to (0,0), but of course stays within the described requirement there too. (Losing Votes and Relative Margins are totally flat at the x-axis and y-axis.)

Juho

I'll describe some possible viewpoints to Moderated Margins.

I agree with you that it is good if election methods are simple. I also agree that Margins is a simple and natural approach to measuring pairwise preferences. From this point of view Relative Margins and Moderated Margins are two modifications to the Margins approach.

I'll follow your Brexit example. The strength of result 52%-48% can be said to be a difference of 4%. Same with 4%-0%. Relative Margins philosophy says that 4%-0% is stronger than 52%-48%. Moderated Margins says that it is weaker. If you think about probabilities and uncertainties, you could say that 4%-0% should be weaker since 96% of the voters didn't give an opinion. It could well be that sufficient number of those voters could have the opposite opinion to make the other candidate win this pairwise comparison. In some ranked elections (e.g. when there are lots of candidates) it is quite possible that most voters will rank only some of the candidates. If this is the case, all those missing rankings are not intentional ties, but maybe more often an indication of lazy voters, or voters that didn't think those candidates are important enough to rank them. For people thinking this way, Moderated Margins could be an interesting option.

I'm not trying to say that Moderated Margins would be the best approach. I'm just saying that under certain assumptions, people might be interested in a pairwise comparison method that would consider weak participation in ranking two candidates to mean that the strength of that preference should be considered weak (when compared to basic Margins). If we want to have such a pairwise preference function, Moderated Margins can be said to be a very simple and natural such function.

Juho

Hi Juho, what I was seeing initially is that f(x,y) shouldn't increase when y increases, since that would be an increase in support for Y which presumably could make Y lose. It's a Mono-raise concern.
But you manage to put positive y in the definition without creating that issue. So that's interesting.
Regarding your specific wording, I hope I understand your meaning. I think f(x,y) is allowed to increase whenx is fixed and y decreases. It seems impossible to me to say that f(x,y) must not decrease when y is fixed,since f(x,y) will probably need to decrease when x decreases.
Kevin

Le mercredi 29 mai 2019 à 02:49:49 UTC−5, Juho Laatu <juho.laatu@gmail.com> a écrit : I listed some of the properties of "good" preference functions. Should I read what you say so that f(x,y) should be non-increasing for all fixed values of x, and non-decreasing for all fixed values of y? If f(x,y) approaches plain p (or -p) (locally) somewhere in the x-y-plane, that would violate this rule. I guess the rule should cover both the losing side and the winning side. I note that the Moderated Margins preference function gets very flat close to (0,0), but of course stays within the described requirement there too. (Losing Votes and Relative Margins are totally flat at the x-axis and y-axis.)

Juho

On 29 May 2019, at 05:32, Kevin Venzke stepjak@yahoo.fr wrote:
Hi Juho,
Moderated Margins has a good motivation. The math is interesting.

My first thought that the losing side, the Y>X votes, should not be able to increase the strength of the X>Y defeat. Otherwise they have incentive to remove their opposition to X. So participation is never used directly as defeat strength.
However, it seems that (x-y)*(x+y) simplifies to (x^2)-(y^2). So, as Y>X increases the defeat strength can only go down. That's a good trick.
Kevin

Le mardi 28 mai 2019 à 01:55:19 UTC−5, Juho Laatu <juho.laatu@gmail.com> a écrit :  

I like the approach of telling people clearly what their vote and not giving any preference between two candidates means. In Margins a tie can be said to mean "they are equally good", and in Relative Margins "I support the opinion of those voters that rank them". Winning Votes is quite difficult to explain since it says that the strength of preference is discontinuous with fully ranked votes (51-49 is a strong victory but 49-51 is a heavy loss). I tried to write good explanations on how Winning Votes and Losing Votes (that is also discontinuous) treat pairwise ties and rankings, but the end results were not very intuitive, so I will not include any of that mess here :-).

I however want to discuss about another pairwise preference function. It could be called Moderated Margins. While Relative Margins says that ties mean that "others shall decide, and they shall use the strength of my vote too", Margins says "others shall decide, but without the strength of my vote" (the voter doesn't want to influence the strength of the final decision in any way), and Moderated Margins says "others shall decide, but I vote to make their final decision weaker" (lack of opinions should men that the outcome is weak). Relative Margins says that preference 30-10 should be seen as stronger than 60-40. Margins says that 30-10 should be seen as equally strong. Moderated Margins says that 30-10 should be seen as weaker than 60-40 (maybe "less decisive" since so many voters didn't tell their opinion).

In Moderated Margins a tie can be said to mean "I want them to be more equal" or "against any preference". I.e. this voter wants to flatten the final preferences, and make the final preference strength weaker. Mathematically Moderated Margins can be defined so that 50% participation in the pairwise competition (when 50% of the preferences in the ballots are ties) should mean that the strength of the result should be only 50% of the strength it would otherwise be (Margins can be seen as the starting point here). While Relative Margins results can be seen to be Margins results, where the Margins result will be divided by participation (percentage), in Moderated Margins the Margins result will be multiplied with participation. In that sense they are mirroring each others at the opposite sides of the Margins philosophy.

One could imagine also election methods where voters would be offered the option to cast different kind of ties. They could be e.g. relative, normal and moderated ties, or "others to decide", "I'm neutral", "make them equal". But probably that gets too complicated for any regular election. These options try to capture the sincere opinion of the voter. Strategic implications ffs.

I modelled the preference functions in OSX's Grapher (file available if someone is interested). It was a nice way to visualise and study these and other preference functions in 3D. I'll explain the nature of preference functions and their 3D modelling a bit more.

preference functions (f(x,y)) are defined in triangle (0%,0%), (100%,0%), (0%,100%) of the x-y-plane
    x and y coordinates refer to percentage of ballots that prefer A over B and B over A respectively
values of f are in range [-1, 1]
    positive value => A preferred over B
    negative value => B preferred over A
    0 => A and B are tied
    1 => A preferred over B with maximal strength
    -1 => B preferred over A with maximal strength
"tied" line from (0%,0%) to (50%,50%)
    all discussed preference functions (f) give the same value (0)
    f(x,x) = 0
"fully ranked" line from (100%,0%) to (0%,100%)
    all discussed preference functions (f), except Winning Votes, give the same result
    f(x,100%-x) = x
    at this line all ballots rank A over B or B over A
    values are linear in the sense that f(50%+2x,50%-2x) is always twice as strong preference as f(50%+x,50%-x)
    this linear approach is just a typical way to present the preference strengths (could be something else too)
the triangle can be divided in two smaller triangles
    (0%,0%), (100%,0%), (50%,50%)
    (0%,0%), (50%,50%), (0%,100%)
    all discussed preference functions are "symmetric" with respect to A and B
    i.e. the two smaller triangles have the same form
    they are rotated 180° in 3D around the tie line
    f(x,y) = - f(y,x)
    in one of the smaller triangles A always wins, and in the other one B always wins
    f(x,y)>0 when x>y
    f(x,y)<0 when x<y

p = participation
    values in range [0, 1]
    percentage of ballots that have ranked A over B or B over A
    p = x + y
m = margin
    values in range [-1, 1]
    m = x - y
r = ratio
    values in range [-1, 1]
    r = (x-y)/(x+y) = m/p

Margins
    f(x,y) = x-y
    = m = rp
Relative Margins
    f(x,y) = (x-y)/(x+y) (defined as 0 when (x+y)=0)
    = m/p = r
Moderated Margins
    f(x,y) = (x-y)(x+y)
    = mp = rp^2
Winning Votes
    f(x,y) = if x>y then x elseif x<y then -y else 0
Losing Votes
    f(x,y) = if x>y then 1-y elseif x<y then -1+x else 0

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Hi Robert,

I know that you feel it's adequate that there is a probabilistic relationship between (x-y) and (x+y). I think
the relationship to expect in reality is unclear. Consider that in real world experience major contests that
involve the strongest candidates and most of the voters are often close races, in which case (x-y) doesn't
predict (x+y) well at all.

Personally I would say that (x-y) just reveals the lowest possible value for (x+y).

Your arguments about balancing decisiveness and size/salience are unique I think. I think it's more 
common to argue that salience is irrelevant or imaginary. One issue in particular is that margins is 
supposed to incentivize voters to cast strict rankings. If this works, then every contest will have about the
same salience anyway.

Kevin

I agree.

Yes, when x decreases, f(x,y) may decrease. Maybe the difference in formulating our equations is that I referred to monotonicity assuming that the changing parameter would increase (not decrease).

Also the fact that I used negative values when modelling the preference functions (when y>x, and candidate B is considered better) make my equations a bit different when compared to the usual EM approach. I'll include a screen capture of the preference functions to make it easier for all to visualize the functions (https://www.dropbox.com/s/5flw6dza9lko24a/3D%20preference%20functions%20for%20EM.tiff?dl=0). In the picture the red lines are the tie line (from (0%, 0%) to (50%, 50%)), and the fully ranked line (from (100%, 0%) to (0%, 100%)). The blue lines show the Relative Margins and Moderated Margins functions. The Margins function is a flat plane between these two functions. The plane is defined by the two red lines. (I didn't include it in the picture to keep it clearer.)

Juho

---------------------------- Original Message ----------------------------

Subject: Re: [EM] Defeat strength, Winning Votes vs. Margins, what to do with equal-ranks on the ballot?

From: "Kevin Venzke" stepjak@yahoo.fr

Date: Wed, May 29, 2019 9:54 pm

To: "EM" election-methods@lists.electorama.com

Hi,

it seems to me that one would expect x-y to scale with x+y.  if the decisiveness (which is the name that i give the percent margin, (x-y)/(x+y) ) is about the nature of the individuals voting, not about their quantity) is unchanged with size, then x-y is expected to get bigger as x+y
does.  if the decisiveness is the equal between two paired runoffs in RP or Schulze, which decision is more important to satisfy?  the one involving more voters weighing in or the pairing with fewer?  

and [(x-y)/(x+y)] * (x+y)^2 = x^2 -y^2 is interesting and it on the way to Winning Votes (it's the L^2 distance norm and as p gets higher, the L^p norm more and more emphasizes the Winning Votes, making the Losing Votes less relevant).
But my interest is in the most simplest, yet adequate
multi-candidate, single-winner method that can be sold to lawmakers and the public.
IRV and FPTP is inadequate.  The minimum fundamental adequacy is, in my opinion:

1. One-person, one-vote.  Whether you prefer your candidate with much greater fervor than my preference for my candidate must not matter.  Our votes must count equally.
2. Majority preference prevails.  If more voters mark their ballots that Candidate A is preferred over
   Candidate B than the number of voters marking their ballots to the contrary, then Candidate B is not elected.
   Also important to me is:
3. Decisive on Election Day.  The election should be resolved, without additional voting, the moment all of the ballots marked are known (by the
   central counting computer).  Delayed runoffs are decided by half as many people that participated in the original election.
4. Precinct summability for a check on any nefarious election meddling at the central counting computer.

No, i think that, if statistics are generated from actual elections you'll find (assuming x>y) that x-y will correlate with x+y.  The expectation value of x-y will increase with x+y .  I think
viewing it as a binary probability distribution where the probability that a random voter votes for candidate X is x/(x+y) and the probability this randomly chosen voter votes for candidate Y is y/(x+y).  In this group I am leaving out the voters that preferred neither X nor Y. 

Well, thank you, I think.  :-)
It just seemed to me to be a natural or serendipitous reason that simple margin, x-y, is a good and simple measure of defeat strength of an election to compare to
the defeat strength of other elections.
I just think that Margins should be more mainline than WV or LV, which seem a little fringey to me.

Given the same decisiveness of the two elections, why
should a smaller election count more than a larger election.  there are people disappointed with the result of any election.  the idea of majority rule is to reduce the number of disappointed persons with franchise.  assuming all voters have equal franchise, the net number of people
being disappointed are the Winning votes minus Losing Votes.  If we want to minimize that, we emphasize the results of elections with larger margins over those of smaller margins.

yes.  i would expect that. 
regards,
robert

sqrt(WV+LV).  >if the decisiveness (which Juho calls "r") remains the same, if you double the size of an election, the s.d. of the |WV-LV| margin is expected to >increase by 41%.  but the means also go up.

not Score, not >Approval) and Condorcet-compliant (because the alternative to electing the CW is electing a candidate when explicitly more voters marked their >ballots preferring a different specific candidate) and i think that RP with simple Margins (WV-LV) is the most meaningful and
simplest.

case there is no >cycle).  then this Defeat Strength discussion becomes moot again.

--

r b-j                         rbj@audioimagination.com

"Imagination is more important than knowledge."

 
I just thought of a good reason for why equal-ranked votes marked on an RCV ballot should not count for either candidate, even if simple Margins is used as Defeat Strength which, unlike Winning Votes, would be unaffected by whether equal ranks are counted for both or for neither
candidate.
This is about selling an RCV method to legislators and the public for use in real governmental elections.  I've always been leaning toward Ranked-Pairs using Margins and even with Margins, equal ranks should count for neither candidate because of the possibility of the total of
Winning Votes + Losing Votes exceeding the number of voters or the number of ballots issued.  If a large proportion of the electorate had no opinion regarding TweedleDee vs. TweedleDum, but rank them above Satan (who is left unranked and tied for last place with anyone else unranked), it might
seem odd in the TweedleDee vs. TweedleDum pairing if the sum of WV+LV exceeded the number of people voting.
I dunno, in multiwinner FPTP elections, the number of votes cast (assuming each voter can cast as many votes as there are seats) can be as high as the number of seats times the
number of voters.  But it would look weird if, in a single-winner RCV race if the total number of votes in any pair runoff exceeds the number of voters.
So I think (given the 2 to 4 principles below and given the need for simplicity and transparency in tallying) that Tideman Ranked Pairs
using Margins is the best to sell to the public and policy makers. Any unranked candidate is tied for last place on a ballot.  Equal ranking allowed on the ballot, but counts for neither candidate equally ranked..  This doesn't mean that Schulze (Margins) is not better, but it's more
complicated and makes no difference for a Smith Set of 3 or less.  And I don't think there will ever be a Condorcet-compliant RCV election that will ever have more than 3 in the Smith Set.  I think having a Condorcet cycle at all will be extremely rare in the context of a governmental
election.
It's possible that IRV-BTR, which is Condorcet compliant, might be easier to sell to policy makers that have already been supporting RCV using IRV.  But there are also IRV haters (that probably don't want any kinda RCV and will always conflate RCV with IRV), so I am not sure if
any association with IRV, that IRV-BTR is a better sell than Ranked Pairs.  The precinct summability can still be used to verify the Condocet Winner, but if there is no CW, the tallying of the election (and the decision regarding the winner) is done only at the central location as is the case
with IRV.
just my rambling.
r b-j

---------------------------- Original Message ----------------------------

Subject: Re: [EM] Defeat strength, Winning Votes vs. Margins, what to do with equal-ranks on the ballot?

From: "robert bristow-johnson" rbj@audioimagination.com

Date: Sun, June 2, 2019 12:10 am

To: "EM" election-methods@lists.electorama.com

people

--

r b-j                         rbj@audioimagination.com

"Imagination is more important than knowledge."

Hi Robert,

Le dimanche 2 juin 2019 à 02:10:13 UTC−5, robert bristow-johnson rbj@audioimagination.com a écrit : 

And: 

Most candidates can't actually win, in a reasonable method, so their contests shouldn't be affecting the result (so asto minimize spoilers). So even if you are correct that (x-y) scales with (x+y) for most candidates, the situation wherewe need it to work is where (x+y) is high; whether it's true with unviable candidates shouldn't matter.

In the real world with two candidates there is an incentive to compete over the median voter, because the candidatewho can do that has the majority (or close). If the two candidates are nominated and campaign effectively then (x-y)should be small.

Possibly RCV in some form could change this dynamic, although that doesn't seem like it would be a good thing(i.e. higher margins between the most competitive candidates).

I like the idea of measuring disappointment, but I think it has to be done per voter and based on what results themethod can actually offer. When X beats Y with some margin or vote count, we don't know much about what woulddisappoint X>Y voters or what their goals are. If Y is neither a voter's favorite nor least favorite, whether they wantY to be defeated depends on what else is on the table. If the method can only realistically elect Y or Z, thendisappointment from the resolution of X vs. Y will depend a lot on the opinions on Y vs. Z.
Kevin