A totally fictitious league

Four fictitious teams play each other in an unnamed sport. We’ll call the sport FB and we’ll denote the teams:

1. OSU
2. MSU
3. PSU
4. UM

After playing each other twice each during the regular season their result matrix can be written:

They then play a four team tournament with the following results:

fake_tourney.png970x298 10 KB

Problem 1:

Write down the new, normalized result matrix that arises after the tournament.

Problem 2:

A numerical computation finds the eigenvector

Interpret this in the context of the Perron Frobenius theorem.

After the four team tournament we get a new matrix as follows,

\begin{bmatrix} 0 & 3 & 2 & 2 \\ 0 & 0 & 1 & 1 \\ 0 & 1 & 0 & 0 \\ 1 & 1 & 3 & 0 \\ \end{bmatrix}

The eigenvector given is a dominant eigenvector since all of its entries are real and nonnegative. This eigenvector corresponds to the rankings of the teams. This means if we were to rank the teams we would have

1. OSU
2. UM
3. MSU
4. PSU

I feel like we forgot to divide each row through by the number of games that each team played. Thus, new matrix should be:

This is particularly important after the tournament because some teams have played more games than others!