Sports Ranking
General Information: Colley’s method of ranking, which was used in the BSC ranking prior to the change in the system. Colley’s method gives us a rating for each team, which we can use to find a ranking for the teams. Helpful Link.
In the 2013 season, the Big Ten football games in Table below occurred with W representing the
winner. The question is how to rank these teams based on these games.
Michigan State W – Indiana Michigan State W – Purdue
Michigan State W – Illinois Michigan State W – Iowa
Indiana W – Penn State Penn State W – Michigan
Iowa W – Penn State Penn State W – Michigan
Michigan W – Minnesota Michigan W – Indiana
Minnesota W – Northwestern Minnesota W – Wisconsin
Minnesota W – Nebraska Nebreska W – Purdue
Nebraska W – Illinois Ohio State W – Wisconsin
Ohio State W – Penn State Ohio State W – Iowa
Ohio State W – Northwestern Wisconsin W – Illinois
Wisconsin W – Northwestern Wisconsin W – Purdue
Table 1: 2013 Big Ten Results
1. Method 1: Directed graph
General Information: Colley’s method of ranking, which was used in the BSC ranking prior to the change in the system. Colley’s method gives us a rating for each team, which we can use to find a ranking for the teams. Helpful Link.
In the 2013 season, the Big Ten football games in Table below occurred with W representing the
winner. The question is how to rank these teams based on these games.
Michigan State W – Indiana Michigan State W – Purdue
Michigan State W – Illinois Michigan State W – Iowa
Indiana W – Penn State Penn State W – Michigan
Iowa W – Penn State Penn State W – Michigan
Michigan W – Minnesota Michigan W – Indiana
Minnesota W – Northwestern Minnesota W – Wisconsin
Minnesota W – Nebraska Nebreska W – Purdue
Nebraska W – Illinois Ohio State W – Wisconsin
Ohio State W – Penn State Ohio State W – Iowa
Ohio State W – Northwestern Wisconsin W – Illinois
Wisconsin W – Northwestern Wisconsin W – Purdue
Table 1: 2013 Big Ten Results
1. Method 1: Directed graph
a. Create the preference matrix, A, for the Big Ten games played. The entries
ai,j = wi,j /ni, where wi,j is the number of times team i beats team j and ni is the number of games played by team i.
ai,j = wi,j /ni, where wi,j is the number of times team i beats team j and ni is the number of games played by team i.
b. Determine the ranking vector r such that Ar = λr, where λ is the eigenvalue of the largest magnitude. In this ranking, the strength of a team is proportional to its score.
c. Write out the rating from highest to lowest.
2. Method 2: Colley’s Method
a. Create the Colley matrix, C, where Ci,j =
2 + ti, i = j
−ni,j , i 6 = j
, where ti is the total number
of games played by team i, ni,j is the number of times team i faced team j
2 + ti, i = j
−ni,j , i 6 = j
, where ti is the total number
of games played by team i, ni,j is the number of times team i faced team j
b. Create a vector b where bi = 1 + wi − li
2 where wi is the total number of wins accumulated
by team i and l − i is the total number of losses accumulated by team i.
by team i and l − i is the total number of losses accumulated by team i.
c. Solve for the rating vector (r1, r2, …, rn)T from the system of equation Cr = b.
d. Compare the ranking of the team obtained using the Method 1 and the Method 2.
