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Tuesday, September 21, 2004

The Beauty of the Magic Number

Baseball has a large number of mathematical nuggets but since my childhood I have always liked the simplicity of the magic number. In a division race, the magic number is the minimum number of wins by the first place team and number of losses of the second place team to guarantee the first place team wins the division.

Let's do an example. As I write this the New York Yankees have 94 wins, the Boston Red Sox have 60 losses. The easiest way to compute the magic number comes from working backwards from the definition. There are 162 games in a season so the Yankees magic number is 162+1-(94+60) = 9. Any combination of nine Yankees wins and Red Sox losses and the Yankees wins the American League East. The "+1" comes from the fact that in a tie the Yankees would still need to win a one-game playoff to win the division.

What can the magic number teach us about complexity? Consider the RIOT Baseball Project at Berkeley. Not satisfied with the magic number, the project computes the First Place Clinch Number as the "Number of additional games, if won, guarantees a first-place finish." To compute this number one has to look not only at the current standings but the schedule of remaining games between the teams.

My main issue of the clinch number relates to complexity. Not only is it more complicated to compute; to update the clinch number after a game sometimes requires recomputing the number from scratch. The magic number has a simple update function counting down like a rocket launch. Yankees win the magic number drops by one. Red Sox lose the magic number drops by one. If the Yankees beat the Red Sox, both events happen so the magic number drops by two. And once the magic number hits zero you pop the champagne. That's the beauty of the magic number.

7 comments:

  1. The link to the "First Place Clinch Number" was great! I am familiar with SABRmetrics, "The Hidden Game of Baseball", Bill James's work, and STATS but had never come across RIOT before.

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  2. If the Yankees have 6 games left (3 against the Sox and 3 against Minnesota), and the Sox have 7 games left (3 against Yankees and 4 against Tampa Bay), and the Yankees are currently 4.5 games ahead of the Sox and 4 in the loss column, then how can the Magic Number be 6? For example, let's say the Yankees lose 3 to Red Sox this weekend. The Magic # stays the same, because no Yankee wins and no Sox losses, and the Yanks now lead by 1.5 games and 1 in the loss column. Now the two teams WIN all of their remaining games. Magic number is now 3 (3 more Yankee losses), but they have WON THE DIVISION AND THE SEASON IS OVER. Therefore the Magic Number must currently be less than six. Please let me know where my logic is faulty. thanks.

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  3. Correction: 3 more Yankee WINS in last 3 games reduce the magic number by 3.

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  4. You didn't account for the Boston-Baltimore and New York-Toronto series in early October that end the season.

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  5. Say the Sox sweep - Yanks lead is down to 1.5. The Yanks then have 6 games left. If they win all 6, they reach the magic #. The Red Sox will have 7 games left. If the Yanks lose 1, their lead is down to 1 game. With their remaining 5 losses and the Red Sox's 7 wins, they'd be tied. Ergo, the magic # is still magic.

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  6. You're right that you have to add 1 to the Yankees magic number, but not because of a one-game playoff. In the case of a tie in teams' records, with the winning team taking the division and the losing team taking the WC, the decisive factor would be the series record. Sox have beaten the Yankees 11 games to 8, so in the (now very unlikely) event of a tie, the Sox get the division and the Yankees get the Wild Card.

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  7. yankee magic number 20!

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