In this installment I would like to summarize (by example) the "how to" of the current Condorcet Flavored Proportional Representation (CFPR) method that takes into account the constructive criticism of Adam Tarr:
Here's information from one typical ballot: C1>C2=C3>C4>C5=C6=C7>C8=C9>C10>C11>C12=C13=C14=C15 This information might have come from Grade Ballot information: Grade Candidates A C1 B C2,C3 C C4 D C5,C6,C7 E C8,C9 F C11 G C12 H C13,C14,C15 Now suppose that the two subsets being compared head-to-head are S1={C2,C5,C6,C7,C14,C15} and S2={C1,C3,C8,C9,C12,C13}. To score these subsets relative to this ballot, first we make a copy of the ballot restricted to the members of the union of S1 and S2: C1>C2=C3>C5=C6=C7>C8=C9>C12>C13=C14=C15 We see that these candidates are divided into six equivalence classes. The respective subsets S1 and S2 have the following respective numbers in each of the six classes: X1=(0,1,3,0,0,2) and X2=(1,1,0,2,1,1) . Now we take the difference of these two ordered six-tuples and let Y1 and Y2 be the respective positive and negative parts (with negative signs removed): X1-X2=(-1,0,3,-2,-1,1) , Y1=(0,0,3,0,0,1), and Y2=(1,0,0,2,1,0). Next we find the median positions in Y1 and Y2 respectively. The higher and lower ranking medians M and m are in positions 3 and 4, respectively. Now we calculate k1 and k2, the number of candidates in S1 and S2, respectively, ranked higher than M (while counting 1/2 for each candidate with rank M): k1=2.5, and k2=2 . Next, we calculate j1 and j2, the number of candidates in S1 and S2, respectively, ranked lower than m (counting 1/2 for each candidate on the boundary): j1=2, and j2=3 . Now the sums k1+j2=5.5 and k2+j1=4 Since k1+j2 is greater than k2+j1, this ballot supports S1 over S2. Since k1+j2 is an odd multiple of .5, the margin is calculated as follows: -ln(4)+1/.5+1/1/5+1/2.5+1/3/5+1/4.5+1/5.5 - (1+1/+1/3+1/4) which is approximately equal to .286793663 . This last value is awarded to S1 in the head-to-head comparison of S1 to S2, while S2 gets zero in this head-to-head comparison. That's it. Forest ---- For more information about this list (subscribe, unsubscribe, FAQ, etc), please see http://www.eskimo.com/~robla/em