Rethinking Magic Numbers For Sports Leagues

Working out the "magic" or elimination number (the number of games a team has to win to avoid elimination) is a classic problem taught to computer science students, said Dan Gusfield, chair of computer science at UC Davis.

Sports fans usually make a simple calculation of the number of games their team has left to play, compared to the win-loss difference with their nearest rival, to come up with the number of wins needed to top the league. This calculation is too simple, said Gusfield. It does not take into account that if the other team loses a game to a third team, the third team gains points.

"You have to look at all the teams and all the possible future games, and work out a scenario for the team you're interested in," Gusfield said. UC Berkeley computer scientists run a Web site that calculates these numbers and provides examples: http://riot.ieor.berkeley.edu/~baseball/.

Two years ago, a Cornell University researcher showed that the answers for any team can be tied together, so the elimination number for all the teams can be found at the same time. At any specific time in the season, the number of games won plus the number to play must be higher than a threshold number, or the team will be eliminated.

Gusfield and colleague Charles Martel have shown that this can be extended to sports leagues such as hockey and European soccer, which award points for wins, losses and draws. The same type of threshold number can also be used to find which teams still have a chance to make the playoffs as a wildcard team, Martel said.

"The phenomenon is universal, not tied to any scoring system," Gusfield said. But whether you can actually compute the threshold number is a different matter: under some scoring systems, it might be too difficult to work out, he said.

The results are published in the January 2002 issue of Algorithmica.

More information: UC Berkeley's "magic numbers" site is at http://riot.ieor.berkeley.edu/~baseball/.