Is it a Good Time to be a Mariners
Fan?
Ranking Baseball Teams Using
Linear Algebra
By Melissa Joy and Lauren Asher
How are sports teams usually ranked?
Winning Percentage system: The team with the highest
percentage of wins is ranked first.
Problems:
•If all the teams do not play all the other teams
then your winning percentage depends on how
good the teams you play are.
•Possibility of ties
Solution: Linear Algebra…
MLB 2006 Regular Season
(through April 17th)
Los Angeles Angels (A)
Seattle Mariners (B)
Oakland Athletics (C)
Texas Rangers (D)
A vs. B:
W 5-4
L 8-10
L 4-6
A vs. D:
W 5-2
W 5-4
L 3-11
B vs. C:
W 6-2
L 0-5
L 0-3
C vs. D:
L 3-6
W 5-4
L 3-5
Sum of Points Scored in the 3 games
A vs. B: 17-20
A vs. D: 13-17
B vs. C: 6-10
C vs. D: 11-15
How to find the ranking vector
According to Charles Redmond, the vector yielding the ranking
has this formula:
Making an Adjacency Matrix
Sum of rows represents the number of games played
1/3 1/3 0 1/3
1/3 1/3 1/3 0
0
1/3 1/3 1/3
1/3 0 1/3 1/3
Sum of the rows =1
Finding an S vector
Sum of Points Scored in the 3 games
A vs. B: 17-20
A vs. D: 13-17
B vs. C: 6-10
C vs. D: 11-15
A: -3 + -4 = -7
B: 3 + -4 = -1
C: 4 + -4 = 0
D: 4 + 4 = 8
-7
-1
0
8
Solving for Eigenvectors
=1
1/3 1/3 0 1/3
1/3 1/3 1/3 0
0
1/3 1/3 1/3
1/3 0 1/3 1/3
Eigenvectors:
1
1
1
1
Eigenvalues:
= -1/3
= 1/3
1
-1
1
-1
1
0
-1
0
0
1
0
-1
Normalized Eigenvectors:
½
½
½
½
½
-½
½
-½
1/√2
0
-1 /√2
0
0
1/√2
0
-1 /√2
A Linear Decomposition of S
-7
-1
0
8
1/2
1/2
1/2
1/2
-7
-1
0
8
1/2
-1/2
1/2
-1/2
0
-5
-5/2
5/2
-5/2
5/2
-7/2
0
7/2
0
-7
-1
0
8
1/√2
0
-1/ √2
0
-7
-1
0
8
0
1/√2
0
-1/ √2
0
-9/2
0
9/2
-7/ √2
-9/ √2
Plugging S into the Limit
The limit can be expanded
into the decomposed form
of S
The eigenvalues are
substituted in for M/3
The limit becomes:
The Final Ranking
-7/2
0
7/2
0
0
-9/2
0
9/2
-5/2
5/2
-5/2
5/2
-2.375
-1.625
1.125
2.875
-1.75
0
1.75
0
0
-2.25
0
2.25
-.625
.625
-.625
.625
And the winner is…
-2.375
-1.625
1.125
2.875
1. Texas Rangers (D)
2. Oakland Athletics (C)
3. Seattle Mariners (B)
4. Los Angeles Angels (A)
This ranking is based on points.
It is a better early season predictor because:
• Measures skill rather than simply wins and losses
• Eliminates ties