[EM] Defeat strength, Winning Votes vs. Margins, what to do with equal-ranks on the ballot?
robert bristow-johnson
rbj at audioimagination.com
Tue May 28 23:36:28 PDT 2019
> 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 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
> Le mardi 28 mai 2019 à 21:44:02 UTC−5, robert bristow-johnson <rbj at audioimagination.com> a écrit :
> On Tue, May 28, 2019 7:32 pm, Kevin Venzke" <stepjak at yahoo.fr> wrote:
>
>> Moderated Margins has a good motivation. The math is interesting.
>>>Moderated Margins
>>> f(x,y) = (x-y)*(x+y)
>>> = m*p = r*p^2
>>
>> 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.
>
> 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 r*p^2 better than r*p? 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
>
>
>
>> Le mardi 28 mai 2019 à 01:55:19 UTC−5, Juho Laatu <juho.laatu at gmail.com> a écrit :
>>
>> > On 24 May 2019, at 02:24, Stéphane Rouillon <stephane.rouillon at sympatico.ca> wrote:
>>>
>>> All criterias (Winning Votes, Margins, Relative Margins) have advantages and are acceptable. The fine choice depends on the interpretation you told voters that would be made of blank ballots. If a blank rank means "all bad", WV is perfect. If it means "all the
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...
>>
>> 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".
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 :-).
>>
>> 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
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).
>>
>> 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
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.
>>
>> 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
complicatedfor 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%+2*x,50%-2*x) 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 = r*p
>> 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)
>> = m*p = r*p^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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> --
>
> r b-j rbj at audioimagination.com
>
> "Imagination is more important than knowledge."
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--
r b-j rbj at audioimagination.com
"Imagination is more important than knowledge."
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