1. Is it a Good Time to be a Mariners Fan? Ranking Baseball Teams Using Linear Algebra By Melissa Joy and Lauren Asher

2. 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…

3. MLB 2006 Regular Season (through April 17 th ) 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

4. How to find the ranking vector According to Charles Redmond, the vector yielding the ranking has this formula:

5. 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 rows represents the number of games played Sum of the rows =1 Making an Adjacency Matrix

6. -7 0 8 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 Finding an S vector

7. 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 Eigenvalues: = 1 = -1/3 = 1/3 Eigenvectors: 11111111 1 1 1 0 0 1 0 Normalized Eigenvectors: ½½½½½½½½ ½ -½ ½ -½ 1/√2 0 -1 /√2 0 1/√2 0 -1 /√2 Solving for Eigenvectors

8. A Linear Decomposition of S -7 0 8 -7 0 8 -7 0 8 -7 0 8 1/2 -1/2 1/2 -1/2 1/√2 0 -1/ √2 0 1/√2 0 -1/ √2 0 -5 -7/ √2 -9/ √2 -5/2 5/2 -5/2 5/2 -7/2 0 7/2 0 -9/2 0 9/2

9. 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:

10. The Final Ranking -7/2 0 7/2 0 -9/2 0 9/2 -5/2 5/2 -5/2 5/2 -1.75 0 1.75 0 -2.25 0 2.25 -.625.625 -.625.625 -2.375 -1.625 1.125 2.875

11. 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