Last fall Jobst analyzed the 2004 team-to-team stats of the American League as an example of using River.
I decided to revisit the example since I hadn't followed his work the first time, and found that his River analysis had an error. In the example, scores were listed as percentages of head-to-head matches won. Jobst suggested that I try to find something like Approval to use to see how DMC would work. I started using row sums. If I were using winning games instead of percentages, the row sum would be total games won, exactly what is used to rank teams currently. The row sum of wv is equivalent to the Borda score. It then occurred to me that dividing the row sum by N-1 is the average number of winning votes. When applied to winning percentages, this gives the average percentage of games won, a perfectly reasonable way to rank baseball teams, if not candidates. So here is that example, reposted. Because the average percentage is hard to tell apart, I put brackets around it. Best viewed with fixed-width font (e.g. Courier) and a wide screen: Original matrix (row averages on diagonal): Tms a b c d e f g h i j k l m n o a ( Bal) [50] 53 33 50 100 67 33 44 26 0 78 58 71 58 28 b ( Bos) 47 [60] 67 43 86 67 56 33 58 89 56 74 44 74 50 c ( CWS) 67 33 [51] 53 42 68 44 47 43 22 78 67 67 43 44 d ( Cle) 50 57 47 [50] 47 58 56 37 33 67 56 50 11 71 56 e ( Det) 0 14 58 53 [42] 42 22 37 57 44 56 50 44 67 50 f ( KC) 33 33 32 42 58 [33] 0 37 17 22 29 33 44 50 33 g ( LAA) 67 44 56 44 78 100 [60] 56 56 53 65 86 47 44 39 h ( Min) 56 67 53 63 63 63 44 [55] 33 29 56 44 71 67 61 i ( NYY) 74 42 57 67 43 83 44 67 [62] 78 67 79 56 63 56 j ( Oak) 100 11 78 33 56 78 47 71 22 [58] 58 78 55 67 56 k ( Sea) 22 44 22 44 44 71 35 44 33 42 [39] 29 37 22 50 l ( TB) 42 26 33 50 50 67 14 56 21 22 71 [43] 22 50 83 m ( Tex) 29 56 33 89 56 56 53 29 44 45 63 78 [54] 78 56 n ( Tor) 42 26 57 29 33 50 56 33 37 33 78 50 22 [42] 44 o (Intr) 72 50 56 44 50 67 61 39 44 44 50 17 44 56 [50] Grid reordered in descending order of row average: Tms i b g j h m c a d o l e n k f i ( NYY) [62] 42 44 78 67 56 57 74 67 56 79 43 63 67 83 b ( Bos) 58 [60] 56 89 33 44 67 47 43 50 74 86 74 56 67 g ( LAA) 56 44 [60] 53 56 47 56 67 44 39 86 78 44 65 100 j ( Oak) 22 11 47 [58] 71 55 78 100 33 56 78 56 67 58 78 h ( Min) 33 67 44 29 [55] 71 53 56 63 61 44 63 67 56 63 m ( Tex) 44 56 53 45 29 [54] 33 29 89 56 78 56 78 63 56 c ( CWS) 43 33 44 22 47 67 [51] 67 53 44 67 42 43 78 68 a ( Bal) 26 53 33 0 44 71 33 [50] 50 28 58 100 58 78 67 d ( Cle) 33 57 56 67 37 11 47 50 [50] 56 50 47 71 56 58 o (Intr) 44 50 61 44 39 44 56 72 44 [50] 17 50 56 50 67 l ( TB) 21 26 14 22 56 22 33 42 50 83 [43] 50 50 71 67 e ( Det) 57 14 22 44 37 44 58 0 53 50 50 [42] 67 56 42 n ( Tor) 37 26 56 33 33 22 57 42 29 44 50 33 [42] 78 50 k ( Sea) 33 44 35 42 44 37 22 22 44 50 29 44 22 [39] 71 f ( KC) 17 33 0 22 37 44 32 33 42 33 33 58 50 29 [33] "Intr" is inter-league play. I didn't count Intr victories when doing River. Percentages >50 are winning, 50 is a tie. After reordering, the winner is quickly seen to be Boston, which agrees with River. I found that the 2004 National League grid was similar -- St. Louis was the winner with both DMC-AvgPct and River. I find it interesting that both league winners were predicted by Condorcet -- possibly one could use this in betting pools ;-). But the DMC method is much faster to find by hand than River. One could of course use Borda/Row-average-seeded DMC for elections as well. That would be equivalent to Pairwise Sorted Borda. And no extra approval cutoff would be required. But using Borda score as the seed ranking would overly encourage strategic burying and eliminate the ability to adjust the approval cutoff without changing ranking. Ted -- araucaria dot araucana at gmail dot com ---- Election-methods mailing list - see http://electorama.com/em for list info
