[EM] Clock Methods

[Gervase Lam]

> Date: Tue, 1 Feb 2005 14:23:08 -0800 (PST)
> From: Forest Simmons <simmonfo at up.edu>
> Subject: Re: [EM] Clock Methods

> Thanks for taking time to explore.  And nice text graphics for the
> clock!

I think I spent a bit too long calculating how to draw the text graphics...

> It turns out that as long as you allow only strict rankings, the center
> of gravity of the distribution will fall in the the four hour (i.e. 120
> degree) sector of the clock face centered on the Borda winner, so that's
> why I didn't suggest using the center of gravity approach.

> >                 A
> >                 .
> >       A>C>B .       .[4] A>B>C
> >
> >   Not(B) .              . Not(C)
> >
> >C>A>B [2].                . B>A>C
> >
> >        C .              . B
> >
> >       C>B>A .       .[3] B>C>A
> >                 .
> >
> >               Not(A)

[C>A>B was written at the 7 o'clock position.  I have now corrected it to 
C>B>A.]

I don't think the centre of mass is on B.

The total of all the votes is 2 + 3 + 4 = 9.

Draw a vertical and horizontal lines so that they are perpendicular to 
each other and cross each other at the centre of the circle.  With respect 
to these lines, the co-ordinates of C>A>B is obviously (-1, 0).

The A>B>C and B>C>A voters are respectively 60 degrees above and below the 
horizontal line.  Therefore, their x co-ordinates are respectively cos +60 
and cos -60 (i.e. 1/2 and 1/2).  Also, their y co-ordinates are 
respectively sin +60 and sin -60 (i.e. sqrt(3)/2 and -sqrt(3)/2).

The x co-ordinates of each of the rankings is weighted by the number of 
votes cast for each ranking.  This is then used to work out the weighted 
mean x co-ordinate of the clock or spinning-top.

((2 * -1) + (4 * 1/2) + (3 * 1/2)) / 9 = 3/18

The weighted mean y co-ordinate of the spinning-top is worked out in the 
same way except that the y co-ordinates of the rankings are used instead 
of the x co-ordinates.

((2 * 0) + (4 * sqrt(3)/2) + (3 * -sqrt(3)/2)) / 9 = sqrt(3)/18

Therefore, the co-ordinates of the centre of mass is (3/18, sqrt(3)/18)).

Now to work out the angle of the line from the centre of the circle to the 
centre of mass:

Tan [Angle] = (sqrt(3)/18) / (3/18) = sqrt(3)/3 = 1/sqrt(3).

Therefore, the angle of the line is 30 degrees above the horizontal line.  
This means the line points at Not(C).

This is not nice in this case because the voters are equally split between 
candidates A and B.  There is nothing between them.

Thanks,
Gervase.