Weighted Powers Ranking Method Introduction The Weighted Powers Ranking Method is a method for ranking sports teams utilizing both number of teams, and strength of the schedule (i.e. how good are the teams they beat, and how bad are the teams that they lost to). To do this, a matrix is created based on the real schedule that each team plays. This matrix is then multiplied by itself to achieve the second level wins. If Team A has a win over Team B and Team B has beaten Team C, then Team A has a second level win over Team C. We can then create a matrix of all second level wins. The new matrix is then scaled and added to the original matrix. The process continues on for as many levels as the user chooses. The ratings for each team are then simply the sum of the row corresponding to that team. The Process Consider the n n matrix A where n is the number of teams and a ij is 1 if team i beat team j, or zero otherwise. This is the first matrix described above. Note that the sum of each row is equal to the number of games that team i won. Now consider A 2 ; this is the second matrix as described above, or the second level wins. We can sum k matrices, in order to consider k level wins. Their sum at this point can be described as k R k = A i. Now, we wish to include a scale on each A i, otherwise, every A i is weighted equally, meaning second, third, or even fifteenth level games that never existed in reality have just as much influence on a rating as the original games that were actually played. For our purposes, we will use 1 as our constant, where n is the number of n i 1
rows in the matrix A. Our new sum becomes R k = k 1 n i Ai. Now that we have created a matrix R k whose row sums represent the ratings, we must get those sums. We will use the vector e (an n 1 column vector) in order to get the rating vector r k. In order to do this, we calculate r k = R k e so that every row i in r k corresponds to team i s ranking. An Example In order to fully understand the process of calculating the Weighted Powers Ranking Method, let us consider an example of five teams, Team A, Team B, Team C, Team D, and Team E. The following table represents their wins and losses, and what the scores were. Teams Team A Team B Team C Team D Team E Team A - Win 16-12 Loss 41-57 Win 13-12 Win 27-21 Team B Loss 12-16 - Loss 30-38 Win 36-28 Loss 35-10 Team C Win 57-41 Win 38-30 - Loss 18-29 Loss 14-37 Team D Loss 12-13 Loss 28-36 Win 29-18 - Loss 0-39 Team E Loss 21-27 Win 35-10 Win 37-14 Win 39-0 - 2
From this table, we see that A = Team A Team B Team C Team D Team E Team A 0 1 0 1 1 Team B 0 0 0 1 0 Team C 1 1 0 0 0 Team 0 0 1 0 0 Team E 0 1 1 1 0 and if we consider the ratings using only first level wins, k = 1, then 1 1 R 1 = 5 i Ai = 1 5 A and r 1 = A 0.6 B 0.2 C 0.4 0.2 E 0.6 and so teams A and E are tied for first, Team C is in third place, and teams B and D are tied for fourth place. When we consider the case k = 6 (and we are looking at all the wins through the fifth level wins), we find that 6 1 R 6 = 5 i Ai and r 6 = A 3.25 B 1.05 C 2.20 1.10 E 3.20 3
and there is a clear distinction in each rating. Now we can rank the teams: A is first, E is second, C is third, D is fourth, and B is fifth. Advantages and Disadvantages Clearly, as seen in the above example, the Weighted Powers method is effective in separating ties in rankings by using multiple level wins. It uses wins and loses to take into account strength of schedule. The user of the method can also choose a scale that is appropriate for the situation. If they deem the unplayed games to be more or less important, they can scale to fit this opinion. Unfortunately, there is no way to currently account for ties, as they are added into the matrix A as 0s. Along a similar vein, the scores of each game are not taken into account the point differential is ignored, so a close loss does not help a team, nor does a large win. Potential Improvements In order to accommodate the lack of ties, the user can modify A such that a ij is 1 if team i beat team j, 1 if team i tied with team j, or 0 if team i lost to team 2 j. This would keep A being a non-negative matrix, yet would still distinguish between the three possibilities. A different adjustment, which would account for the point differential would be to replace all the values of 1 with the point differential of the game it represents. If we apply this to our example, we see that A = A B C D E A 0 4 0 1 6 B 0 0 0 8 0 C 16 8 0 0 0 0 0 11 0 0 E 0 25 23 39 0 4
and subsequently (using the same weights as before) r 6 = A 42.5 B 13.3 C 40.1 25.4 E 153.8 which changes the ratings from what we previously had. Now we have the order of E, A, C, D, then B, the only difference being that E is now in first place instead of A. This is a rather major change in terms of the rankings. A comparison with other rating methods would help to solidify which might be better in this instance. Conclusion As seen throughout this paper, the Weighted Powers Ranking Method is effective at making clear splits when ties occur, whether or not they occur in the highest or lowest places. It can be altered to fit the needs of the set being ranked by using different weights, and can be more severely altered in order to accommodate ties, or contain the point differentials of the games. It is also a fairly simple ranking method to implement, since it is based in a simple summation of matrices. 5