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

robert bristow-johnson rbj at audioimagination.com
Sun Jun 2 00:10:00 PDT 2019

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

From: "Kevin Venzke" <stepjak at yahoo.fr>

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

To: "EM" <election-methods at lists.electorama.com>

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> Hi Robert,
Hi,
> 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.

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.

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

> Your arguments about balancing decisiveness and size/salience are unique I think.
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.
> I think it's more common to argue that salience is irrelevant or imaginary.
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.
> 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.
yes.  i would expect that.
regards,
robert
>

>

>

>>Le mercredi 29 mai 2019 à 01:36:41 UTC−5, robert bristow-johnson <rbj at audioimagination.com> a écrit :

>>

>>> 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 out the division.

>>>

>>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

>>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

>

>

--

r b-j                         rbj at audioimagination.com

"Imagination is more important than knowledge."

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