[EM] More Condorcet Flavored PR examples

[Forest Simmons]

Scoring two candidate subsets relative to a ranked ballot can be
organized into a tableau as follows:

rank  X1   X2   (X1-X2)   Y1   Y2
---------------------------------
 1    0    0       0      0    0
 2    0    1      -1      0    1
 3    2    1       1      1    0
 4    0    0       0      0    0
 5    1    0       1      1    0
 6    0    0       0      0    0
 7    3    2       1      1    0
 8    1    3      -2      0    2
 9    4    4       0      0    0


The columns X1 and X2 tell us how many candidates of each rank (according
to the ballot in question) are found in each of the candidate subsets.
These two columns contain all of the information necessary for scoring the
two candidate subsets relative to the ballot in question.

Columns Y1 and Y2 are the positive parts of X1-X2 and X2-X1, respectively.

The median ranks for Y1 and Y2 are 5 and 8, respectively.

The greater and lesser of these are M=8, and m=5, respectively.

Columns X1 and X2 have k1=2.5 and k2=2 candidates, respectively, ranked
above M=5, and j1=4.5 and j2=5.5, respectively, ranked below m=8.

Since k1+j2=8 is greater than k2+j1=6.5, this ballot favors the first
candidate subset over the second.

One might be tempted to use the difference 8-6.5=1.5 as the margin of
preference, but this approach would not lead to Proportional
Representation.

For PR results the margin is calculated as follows:

         The integral from x=0 to x=1 of (x^6.5 - x^8)/(1-x),

which turns out to be approximately  .194  .


If the above integral expression is too daunting, you can approximate it
by a difference of natural logarithms:

  ln(1+2*8)-ln(1+2*6.5) which is also .194 to three decimal places.


In any case, this marginal value is awarded to the first candidate subset,
and the other subset receives a contribution of zero from this ballot in
this head-to-head comparison.

Before going on to an example of a complete election with many ballots, we
note the following facts:

(1)  If X1 has a higher median rank than X2, then Y1 will have a higher
median than Y2, and the first candidate subset will be preferred.

(2)  If tied ranks are not allowed, then the X1 and X2 columns will have
only zeroes and ones as entries. In that case columns Y1 and Y2 represent
precisely the set differences between the two candidate subsets.

(3)  The ranks in the first column could be replaced by grades A through I
without any change in the results.  In other words, this method can be
used with grade ballots as easily as with ranked ballots.


To be continued ...

----
For more information about this list (subscribe, unsubscribe, FAQ, etc), 
please see http://www.eskimo.com/~robla/em