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Yet Another Mathblog

This is the fourth of a series of expository posts on matrix-theoretic sports ranking methods. This post discusses the Elo rating.

This system was originally developed by Arpad Elo (Elo (1903-1992) was a physics professor at Marquette University in Milwaukee and a chess master, eight-time winner of the Wisconsin State Chess Championships.) Originally, it was developed for rating chess players in the 1950s and 1960s. Now it is used for table tennis, basketball, and other sports.

We use the following version of his rating system.

As above, assume all the $n$ teams play each other (ties allowed)
and let $latex r_i$ denote the rating of Team $latex i$, $latex i=1,2,dots,n$.

Let $latex A=(A_{ij})$ denote an $ntimes n$ matrix of score results:

$latex
A_{ij}=
left{
begin{array}{rr}
-1,& {rm if team } i {rm lost to team } j,
+1,& {rm if team } i {rm beat team } j,
0…

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Yet Another Mathblog

This is the third of a series of expository posts on matrix-theoretic sports ranking methods. This post discusses the random walker ranking.

We follow the presentation in the paper by Govan and Meyer (Ranking National Football League teams using Google’s PageRank). The table of “score differentials” based on the table in a previous post is:

$latex

begin{tabular}{c|cccccc}
verb+xy+ & Army & Bucknell & Holy Cross & Lafayette & Lehigh & Navy
hline
Army & 0 & 0 & 1 & 0 & 0 &…
Bucknell & 2 & 0 & 0 & 2 & 3 &…
Holy Cross & 0 & 3 & 0 & 4 & 14 &…
Lafayette & 10 & 0 & 0 & 0 & 0 &…
Lehigh & 2 & 0 & 0 & 11 & 0 &…
Navy & 11 & 14 & 8 & 22 & 6 &…
end{tabular}

$
This leads…

View original post 400 more words

Yet Another Mathblog

This is the second of a series of expository posts on matrix-theoretic sports ranking methods. This post discusses Keener’s method (see J.P. Keener, The Perron-Frobenius theorem and the ranking of football, SIAM Review 35 (1993)80-93 for details).

See the first post in the series for a discussion of the data we’re using to explain this method. We recall the table of results:

XY Army Bucknell Holy Cross Lafayette Lehigh Navy
Army x 14-16 14-13 14-24 10-12 8-19
Bucknell 16-14 x 27-30 18-16 23-20 10-22
Holy Cross 13-14 30-27 x 19-15 17-13 9-16
Lafayette 24-14 16-18 15-19 x 12-23 17-39
Lehigh 12-10 20-23 13-17 23-12 x 12-18
Navy 19-8 22-10 16-9 39-17 18-12 x

sm261_baseball-ranking-application_teams-digraph Win-loss digraph of the Patriot league mens baseball from 2015

Suppose T teams play each other. Let $latex A=(a_{ij})_{1leq i,jleq T}$ be a non-negative square matrix determined by the results of their games, called the preference matrix

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