Rankings are important for many different applications in a wide variety of fields; in essence, any time there are multiple options to choose from, it is generally useful to have a systematic way of deciding which choice is the best. Whether considering colleges, national economies, or web pages, people value well-designed ranking systems that help them to make informed decisions. Ranking systems are especially important within the world of sports, where team rankings can make or break a season or a career.
A natural way to compare two athletic teams exists already; they play against each other, and their relative order is determined by the game’s outcome. However, given the variety of complicated factors that can decide who wins or loses, these one-to-one comparisons can’t reliably be extended to give a complete ordering of a set of many teams -- that is, beating one team doesn’t in turn guarantee superiority over every team that they have already beaten. Some method is needed to take the basic information provided by individual games and consolidate it into a measure of overall strength.
One way of doing this would be simply to compare the win-loss records of teams, but this oversimplifies the situation. For example, suppose Team A and Team B are evenly matched in skill, but Team A faces stronger opponents. Team A will win fewer games than Team B, and this model would incorrectly rank Team B ahead of Team A.
Alternatively, we can rank n teams by constructing an n-by-n matrix, initially filled with zeroes. In this matrix, there is a column and a row to represent each team, and the matrix entries represent each possible match-up between two teams. For each completed game, some value is added to the matrix entry in the row of the winning team and the column of the losing team (depending on the type of model, this value could simply be a 1 to represent a win, a point margin of victory, or some other weighted measure of the win). At this point, the sum of each row represents the corresponding team’s “primary wins”, or the games that they themselves won. By itself, this is very similar to a ranking strategy based solely off of win-loss records. However, once the matrix is set up in this way, several possibilities for deeper analysis become available:
Squaring the matrix will form a new matrix in which the row sums now represent “secondary wins”, or the games that a team’s defeated opponents have won. This allows the model to reward teams for victories over very strong opponents. This process can be continued to include third, fourth, and nth level wins if desired -- but each level becomes increasingly irrelevant to the problem at hand.
The original matrix can be normalized so that each column total is the same; this is another way to account for strength of schedule, because it devalues victories over very unsuccessful teams.
We can also consider an arbitrary ranking vector, which is refined by multiplying it with the matrix to produce a new ranking vector. Over many cycles of refinement, this will converge to the dominant eigenvector of our matrix - so in order to determine a final ranking we need only to find the dominant eigenvector.
The proposed project will attempt to accurately rank 2013 SEC football teams using some combination of the approaches discussed above, based on the statistics available from the season so far.