Major League Baseball

RIOT HOME

1. What is the magic number and how is it computed?

2. The so-called magic number is usually defined as the smallest number such that any combination of wins by the first-place team and losses by the second-place team totaling the magic number guarantees that the first-place team will win the division.

The magic number can be computed using the following numbers (see [1] for alternative formulae):

• w1 - the number of games the team currently in first place has won so far
• w2 - the number of games the team currently in second place has won so far
• g2 -  the number of games the team currently in second place has left to play

Now, suppose that the first-place team wins x more of its remaining games and the second-place team loses y more of its remaining games (i.e., it wins g2-y games).  We will now derive a formula for the magic number, x + y.

The team currently in first place will finish with w1+ x wins and the team currently in second place will finish with w2 + g2 - y wins.  The team currently in first place will finish ahead of the team currently in second place as long as w1 + x > w2 + g2- y.  The magic number is the smallest number  x + y  such that x + y > w2 + g2 - w1.

Since we are dealing with integers (whole numbers), the magic number is w2  + g2 - w1+ 1.

Here is an example using the National League East standings as of 9 am, EST, Sunday September 8 1996:

## National League East

 Clinch Avoid Elim Team W L GB Games Left 1st Play 1st Play Atlanta 86 55 - 21 13 9 0 0 Montreal 78 63 8 21 21 17 8 0 Florida 69 74 18 19 * * 17 9 New York 62 80 24 1/2 20 * * Elim 16 Philadelphia 58 85 29 19 * * Elim Elim

The first-place team, Atlanta, has 86 wins and the second-place team, Montreal, has 78 wins and 21 games left to play.  So, w1= 86, w2= 78 and g2= 21. Thus, Atlanta's magic number is 78 + 21 - 86 + 1 = 14.

This means that any combination of wins by Atlanta and losses by Montreal totaling 14 ensures that Atlanta will win the National League East.  For example, if Atlanta wins 14 more games, they will finish with at least 100 wins.  The best Montreal can do is 78 + 21 = 99 wins.  Thus, Atlanta would finish ahead of Montreal.  Likewise, if Montreal were to lose 14 games, they would have 7 games left to play and could finish with at most 78 + 7 = 85 wins.  Since Atlanta already has 86 wins, they would finish ahead of Montreal in this scenario as well.

Finally, suppose Atlanta wins 4 games and Montreal loses 10 - a combination adding up to the magic number, 14.  In this scenario, Atlanta would have 90 wins and Montreal would have 78 wins with 11 games left to play.  This means that Montreal could finish with at most 78 + 11 = 89 wins and could not catch up with Atlanta.

Notice that RIOT lists Atlanta's first-place clinch number as 13 - one less than the magic number.  In this case, the difference is that first-place clinch number includes ties for first place while the magic number does not.   The first-place clinch number is often just one less than the magic number, but sometimes there is a larger difference.  For example, consider another example from September 8 1996:

## National League West

 Clinch Avoid Elim Team W L GB Games Left 1st Play 1st Play Los Angeles 78 63 - 21 17 17 4 1 San Diego 78 65 1 19 17 17 4 0 Colorado 71 71 7 1/2 20 * * 11 7 San Francisco 59 81 18 1/2 22 * * Elim 19

Here, Los Angeles' magic number is 78 + 19 - 78 + 1 = 20, but the first-place clinch number is only 17.  What you can't see from the standings is that Los Angeles and San Diego will play each other 7 more times before the end of season.  Thus, if L.A. wins 17 more games, then at least 3 of them will be against San Diego.  Notice that 17 wins for L.A. plus 3 losses for San Diego adds up to the magic number, 20.  This example illustrates how the magic number does not always tell the whole story.

Another drawback with the magic number is that it really only applies to a pair of teams.  For instance, if San Diego loses 20 of its remaining games in the example above it does not necessarily mean that Los Angeles will win the division.  It just means that L.A. will finish ahead of San Diego.  Since Colorado has not yet been eliminated, it is still possible for the Rockies to win the division.   The first-place clinch number, however, is a guarantee; no matter what else happens, the Dodgers will at least clinch a tie for first place if they win 17 more games.

3. How are the wild-card teams determined?
4. Each league (American and National) sends five teams to the postseason: the three division winners and two wild card teams. The wild card teams are the two teams in the league with the best records among all teams that are not division winners. In princple the wild card teams are the teams with the 4th and 5th best records in the league, and are the two best second-place teams. However, it's possible that the wild card teams come from the same division, and it's possible that a wild card team could have a better record than the division winner of another division.

5. What happens when there are ties for postseason spots?
6. This can get quite complicated; see Wikipedia or the official Major League Baseball web site for details.

7. If RIOT says that my team's first-place elimination number is zero, does that mean my team has clinched first place?
Not necessarily. In the National League East example above, the first-place elimination numbers for Atlanta and Montreal are 0 and 8, respectively. Notice that Atlanta's record is 86 and 55, and that Montreal's is 78 and 63. Since there more than 20 games left to play, Montreal still has a chance to win the division. So, Atlanta hasn't clinched anything yet. Having a first-place elimination number of 0 simply means that it's possible for them to win the division without winning another game. In other words there is at least one scenario in which no team in the National League East wins more than 86 games. Based on the current standings and schedule of remaining games, Atlanta needs to win at least 13 more games to guarantee a first-place finish. Meanwhile if Montreal doesn't win at least 8 more games, they can't possibly catch up to Atlanta. So, they need to win at least 8 more games to avoid being eliminated from first place. This is why their first-place elimination number is 8.

References

[1]  M. T. Battista "Mathematics in Baseball." Mathematics Teacher. 86:4. 336-342. 1993

RIOT HOME