Possible Team Ranking Mehod

In a tournament of 6 teams, each team plays all the other teams. The results are summarised in the diagram. A arrow from a team to another team indicates that the first team beats the second.

network representing victories and defeats in a tournament

The table of results is
Winner\Loser A B C D E F
A 0 0 1 1 1 0
B 1 0 0 1 1 0
C 0 1 0 1 0 1
D 0 0 0 0 1 1
E 0 0 1 0 0 0
F 1 1 0 0 1 0
No team wins all their games and often a team loses to another team, but wins against another team that won against them. Teams A, B , and F all have three victories. One way of choosing a winner is this:
The matrix of the table is:
\[M= \left( \begin{array}{cccccc} 0 & 0 & 1 & 1 & 1 & 0 \\ 1 & 0 & 0 & 1 & 1 & 0 \\ 0 & 1 & 0 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 1 & 0\end{array} \right)\]

\[M^2= \left( \begin{array}{cccccc} 0 & 1 & 1 & 1 & 1 & 2 \\ 0 & 0 & 2 & 1 & 2 & 1 \\ 2 & 1 & 0 & 1 & 3 & 1 \\ 1 & 1 & 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 & 0 & 1 \\ 1 & 0 & 2 & 2 & 2 & 0\end{array} \right)\]
  represents the two step wins (team 1 loses to team 2, but beat team 3 who also beat team 2). We can use  
\[M+M^2= \left( \begin{array}{cccccc} 0 & 1 & 2 & 2 & 2 & 2 \\ 1 & 0 & 2 & 2 & 3 & 1 \\ 2 & 2 & 0 & 2 & 3 & 2 \\ 1 & 1 & 1 & 0 & 2 & 1 \\ 0 & 1 & 1 & 1 & 0 & 1 \\ 2 & 1 & 2 & 2 & 3 & 0\end{array} \right)\]
.
Adding up the rows gives 9 for team A, 9 for team B,11 for team C, 6 for team D, 4 for team E and 10 for team F. Team C wins, and the teams are ranked C, F, A=B, D,E.

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