Magic number (sports)

For other uses of the term, see Magic number (disambiguation).

In certain sports, a magic number is a number used to indicate how close a front-running team is to clinching a division title and/or a playoff spot. It represents the total of additional wins by the front-running team or additional losses (or any combination thereof) by the rival team after which it is mathematically impossible for the rival team to capture the title in the remaining games. This assumes that each game results in a win or a loss, but not a tie. Teams other than the front-running team have what is called an elimination number (or "tragic number") (often abbreviated E#). This number represents the number of wins by the leading team or losses by the trailing team which will eliminate the trailing team. The elimination number for the second place team is exactly the magic number for the leading team.

The magic number is calculated as G + 1 − WA − LB, where

For example, in Major League Baseball there are 162 games in a season. Suppose the top of the division standings late in the season are as follows:

TeamWinsLosses
A9658
B9362

Then the magic number for Team A to win the division is 162 + 1 − 96 − 62 = 5.

Any combination of wins by Team A and losses by Team B totaling to 5 makes it impossible for Team B to win the division title.

The "+1" in the formula serves the purpose of eliminating ties; without it, if the magic number were to decrease to zero and stay there, the two teams in question would wind up with identical records. If circumstances dictate that the front-running team would win the tiebreaker regardless of any future results, then the additional constant 1 can be eliminated. For example, the NBA uses complicated formulae for breaking ties, using several other statistics of merit besides overall win/loss record; however the first tiebreaker between two teams is their head-to-head record; if the frontrunning team has already clinched the better head-to-head record, then the +1 is unnecessary.

The magic number can also be calculated as WB + GRB - WA + 1, where

This second formula basically says: Assume Team B wins every remaining game. Calculate how many games team A needs to win to surpass team B's maximum total by 1. Using the example above and with the same 162-game season, team B has 7 games remaining.

The magic number for Team A to win the division is still "5": 93 + 7 − 96 + 1 = 5.

Team B can win as many as 100 games. If Team A wins 101, Team B is eliminated. The magic number would decrease with a Team A win and would also decrease with a Team B loss, as its maximum win total would decrease by one.

A variation of the above looks at the relation between the losses of the two teams. The magic number can be calculated as LA + GRA - LB + 1, where

This third formula basically says: Assume Team A loses every remaining game. Calculate how many games team B needs to lose to surpass team A's maximum total by 1. Using the example above and with the same 162-game season, team A has 8 games remaining.

The magic number for Team A to win the division is still "5": 58 + 8 − 62 + 1 = 5. As you can see, the magic number is the same whether calculating it based on potential wins of the leader or potential losses of the trailing team. Indeed, mathematical proofs will show that the three formulas presented here are mathematically equivalent.

Team A can lose as many as 66 games. If Team B loses 67, Team B is eliminated. Once again, the magic number would decrease with a Team A win and would also decrease with a Team B loss.

In some sports, ties are broken by an additional one-game playoff(s) between the teams involved. When a team gets to the point where its magic number is 1, it is said to have "clinched a tie" for the division or the wild card. However, if they end the season tied with another team, and only one is eligible for the playoffs, the extra playoff game will erase that "clinching" for the team that loses the playoff game.

By convention, the magic number typically is used to describe the first place team only, relative to the teams it leads. However, the same mathematical formulas could be applied to any team, teams that are tied for the lead, as well as teams that trail. In these cases, a team that is not in first place will depend on the leading team to lose some games so that it may catch up, so the magic number will be larger than the number of games remaining. Ultimately, for teams that are no longer in contention, their magic number would be larger than their remaining games + the remaining games for the first place team — which would be impossible to overcome.

Derivation

The formula for the magic number is derived straightforwardly as follows. As before, at some particular point in the season let Team A have WA wins and LA losses. Suppose that at some later time, Team A has wA additional wins and lA additional losses, and define similarly WB, LB, wB, lB for Team B. The total number of wins that Team B needs to make up is thus given by (WA + wA) − (WB + wB). Team A clinches when this number exceeds the number of games Team B has remaining, since at that point Team B cannot make up the deficit even if Team A fails to win any more games. If there are a total of G games in the season, then the number of games remaining for Team B is given by G − (WB + wB + LB + lB). Thus the condition for Team A to clinch is that (WA + wA) − (WB + wB) = 1 + G − (WB + wB + LB + lB). Canceling the common terms, we obtain wA + lB = G + 1 − WA − LB, which establishes the magic number formula.

Subtlety

Sometimes a team can appear to have a mathematical chance to win even though they have actually been eliminated already, due to scheduling. In this Major League Baseball scenario, there are three games remaining in the season. Teams A, B and C are assumed to be eligible only for the division championship; teams with better records in other divisions have already clinched the two available "wild card" spots:

TeamWinsLosses
A8772
B8772
C8574

If Team C were to win all three remaining games, it would finish at 88-74, and if both Teams A and B were to lose their three remaining games, they would finish at 87-75, which would make Team C the division winner. However, if Teams A and B are playing against each other in the final weekend (in a 3-game series), it would be impossible for both teams to lose the three remaining games. One of them will win at least two games and thereby clinch the division title with a record of either 90-72 or 89-73. The more direct consequence of this situation is that it is also not possible for Teams A and B to finish in a tie with each other, and Team C can't win the division.

One can say definitely whether a team has been eliminated by use of the algorithm for the maximum flow problem.[1]

Alternative Method

Another method can be used to determine the Elimination Number which uses only the Games Remaining (GR_L, GR_T) and Games Behind Leader (GBL) statistics, as follows: E= \frac{GR_L+GR_T}{2}-GBL+1,
where GR_L means Games Remaining for Leader (similarly, GR_T means Games Remaining for Trailer).


Refer back to the example presented above. The elimination number for Team B is once again "5": E= \frac{8+7}{2}-3.5+1.


It is necessary to use this method if the teams play different numbers of games in the full season, for instance due to cancellations or ties that will not be replayed. Note that this algorithm also is limited by the aforementioned subtleties.

See also

References

  1. ↑ Kleinberg, Jon; Tardos, Éva (2005). Algorithm Design. Addison-Wesley. ISBN 978-0321295354.

External links

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