Well, it's a little late, but I did a test of the IFNOP (Ignore Fewest Number of Preferences) method using 4 examples from the Wikipedia article on the Schulze method. I picked Schulze because it is fairly well-behaved with fewer voting paradoxes than many other methods.
Example 1 (45 voters; 5 candidates): 5 ACBED 5 ADECB 8 BEDAC 3 CABED 7 CAEBD 2 CBADE 7 DCEBA 8 EBADC Putting the pairs in the appropriate boxes: (4,5) ED(8) CB(5) AC(8) BD(7) DE(2) BA(7) DC(8) (3,4) BE(8) EC(5) DA(8) EB(14) AD(10) (3,5) BD(8) EB(5) DC(8) ED(7) AE(2) EA(7) AC(8) (2,3) CB(5) DE(5) ED(8) AB(3) AE(7) BA(10) CE(7) (2,4) CE(5) DC(5) EA(8) AE(3) AB(7) BD(10) CB(7) (2,5) CD(5) DB(5) EC(8) AD(10) BE(2) CA(7) BC(8) (1,2) AC(5) AD(5) BE(8) CA(10) CB(2) DC(7) EB(8) (1,3) AB(5) AE(5) BD(8) CB(3) CE(7) CA(2) DE(7) EA(8) (1,4) AE(5) AC(5) BA(8) CE(3) CB(7) CD(2) DB(7) ED(8) (1,5) AD(5) AB(5) BC(8) CD(10) CE(2) DA(7) EC(8) The first number is the number of votes for the pair, the second is the number of votes for the reverse pair, and the number in parenthesis is the difference: 30 AD 15 (15) 25 BA 20 (5) 33 BD 12 (21) 27 CA 18 (9) 29 CB 16 (13) 24 CE 21 (3) 28 DC 17 (9) 23 EA 22 (1) 27 EB 18 (9) 31 ED 14 (17) Now, let's see how far we have to go (by ignoring votes) before each pair is broken. The order of comparison is (4,5)(3,4)(3,5)(2,3)(2,4)(2,5)(1,2)(1,3)(1,4)(1,5). Zeroes mean there are no corresponding pairs in that box. BD (21) 7+0+8+0+6 ED (17) 8+0+7+2 AD (15) 0+10+0+0+0+5 CB (13) 5+0+0+5+3 CA (9) 0+0+0+0+0+7+2 DC (9) 8+0+1* EB (9) 0+9 BA (5) 5 CE (3) 0+0+0+3 EA (1) 0+0+1 Pair winners are on the left of the arrow, pair losers on the right BCE-->A CE-->B D-->C ABE-->D C-->E DC is the first link broken -- at (3,5) -- leaving C the winner. Since C>E is now redundant, E is in second place. Since both C and E have been removed, B is in third place, which puts A in fourth place and D in fifth place. The completed order is: C>E>B>A>D. Just for comparison, Schulze selects E as the winner. Borda gives the order E>A>B>C>D, partly because it considers comparing a first and second place vote equivalent to comparing a fourth and fifth place vote. ************ Example 2 (30 voters; 4 candidates): 2 ACDB 3 ADCB 4 BACD 3 CBDA 3 CDBA 1 DACB 5 DBAC 4 DCBA Putting the pairs in the appropriate boxes: (3,4) DB(2) CB(4) CD(4) DA(3) BA(7) AC(5) (2,3) CD(2) DC(3) AC(5) BD(3) DB(3) BA(5) CB(4) (2,4) CB(2) DB(3) AD(4) BA(3) DA(3) AB(1) BC(5) CA(4) (1,2) AC(2) AD(3) BA(4) CB(3) CD(3) DA(1) DB(5) DC(4) (1,3) AD(2) AC(3) BC(4) CD(3) CB(3) DC(1) DA(5) DB(4) (1,4) AB(5) BD(4) CA(6) DB(1) DC(5) DA(4) The first number is the number of votes for the pair, the second is the number of votes for the reverse pair, and the number in parenthesis is the difference: 15 AC 10 (5) 19 BA 6 (13) 16 CB 9 (7) 16 DA 9 (7) 18 DB 7 (11) 13 DC 12 (1) D is the unique winner, while the three remaining candidates are in the circular tie C>B>A>C. The link A>C is the easiest one to break, resulting in the final order of D>C>B>A. For comparison, the Schulze method also gives D as the winner, and Borda gives the complete order D>C>B>A as well. ******************** Example 3 (30 voters; 5 candidates): 3 ABDEC 5 ADEBC 1 ADECB 2 BADEC 2 BDECA 4 CABDE 6 CBADE 2 DBECA 5 DECAB Putting the pairs in the appropriate boxes: (4,5) EC(5) BC(5) CB(1) CA(4) DE(10) AB(5) (3,4) DE(5) EB(5) EC(5) BD(4) AD(6) CA(5) (3,5) DC(5) EC(5) EB(1) EA(4) BE(4) AE(6) CB(5) (2,3) BD(3) DE(8) AD(2) AB(4) BA(6) BE(2) EC(5) (2,4) BE(3) DB(5) DC(3) AE(2) AD(4) BD(6) BC(2) EA(5) (2,5) BC(3) DC(5) DB(1) AC(2) DA(2) AE(4) BE(6) BA(2) EB(5) (1,2) AB(3) AD(6) BA(2) BD(2) CA(4) CB(6) DB(2) DE(5) (1,3) AD(3) AE(6) BD(2) BE(2) CB(4) CA(6) DE(2) DC(5) (1,4) AE(3) AB(5) AC(1) BE(2) BC(2) CD(6) DC(2) DA(5) (1,5) AC(8) AB(1) BC(2) BA(2) CE(10) DA(2) DB(5) The first number is the number of votes for the pair, the second is the number of votes for the reverse pair, and the number in parenthesis is the difference: 30 DE 0 (30) 21 AD 9 (12) 21 AE 9 (12) 20 DC 10 (10) 20 EC 10 (10) 19 BE 11 (8) 19 CA 11 (8) 18 AB 12 (6) 17 BD 13 (4) 16 CB 14 (2) DE (30) 10+5+0+8+0+0+5+2 AD (12) 0+6+0+2+4 AE (12) 0+0+6+0+2+4 DC (10) 0+0+5+0+0+5 EC (10) 5+5 BE (8) 0+0+4+2+2 CA (8) 4+4 AB (6) 5+0+0+1 BD (4) 0+4 CB (2) 1+0+1 C-->A AC-->B DE-->C AB-->D ABD-->E The link that breaks first is CA, at (3,4), making A the winner. BD breaks next, so D is second place. EC breaks next, so E is in third place. This leaves C>B, so the final order is: A>D>E>C>B For comparison, the Schulze method gives B as the winner, or exactly the opposite. Using Borda, we get D>A>B>C>E *********** Example 4 (9 voters; 4 candidates): 3 ABCD 2 DABC 2 DBCA 2 CBDA Putting the pairs in the appropriate boxes: (3,4) CD(3) BC(2) CA(2) DA(2) (2,3) BC(5) AB(2) BD(2) (2,4) BD(3) AC(2) BA(4) (1,2) AB(3) DA(2) DB(2) CB(2) (1,3) AC(3) DB(2) DC(2) CD(2) (1,4) AD(3) DC(2) DA(2) CA(2) The first number is the number of votes for the pair, the second is the number of votes for the reverse pair, and the number in parenthesis is the difference: 7 BC 2 (5) 2+3 6 DA 3 (3) 2+0+0+1 5 AB 4 (1) 0+1 5 AC 4 (1) 0+0+1 5 BD 4 (1) 0+1 5 CD 4 (1) 1 D-->A A-->B AB-->C BC-->D The link A>B is the easiest one to break, taking one point, at (2,3). This leaves B as the overall winner. D is next, with the link C>D broken at (3,4) combined with the winner B. A resolves to third place, and C to 4th place. The completed order is: B>D>A>C For comparison, the Schulze method gives a tie win to B and D, and Borda gives the order B>D>A>C, which is the same as for IFNOP. Some thoughts: The Schulze and IFNOP methods can give give the same winners (in example 2 and a tie win in 4), different winners (example 1), and completely opposite winners (e.g. example 3). IFNOP places a higher priority on top choices than on bottom choices, with the justification being that while top choices are generally what the voter honestly prefers (unless compromising), bottom choices are a mix of most hated, most unknown, and most strategic votes. Looking at the votes, the winners given by IFNOP seem reasonable -- even in the example where it gave the exact opposite of result from Schulze, the candidate it chose defeated the Schulze winner. If anyone can come up with a good voting paradox, or an obvious winner that loses (or obvious loser that wins), please let me know. It's more fun seeing where voting systems fail than it is trying all the examples that work well. Michael Rouse [EMAIL PROTECTED] ---- election-methods mailing list - see http://electorama.com/em for list info
