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

Kevin Venzke stepjak at yahoo.fr
Tue May 28 19:32:33 PDT 2019

 Hi Juho,
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.

    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". 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%+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

    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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