Why magic numbers are imprecise

By Adam Rubin

For the New York Mets, who trail the wild card-leading Atlanta Braves by 12 games with 12 games left, the math is pretty simple: One more Atlanta win or Mets loss and the Amazin’s officially are eliminated.

Sometimes, however, it’s over before it’s officially over. That was actually true with the Mets in the NL East race.

After Friday’s action, the Mets still appeared mathematically able to catch the Phillies. That was true because the Mets still had 14 games left and trailed in the division by fewer games than that number.

As it turned out, the Mets already had been eliminated. If the Phillies were to have lost the remainder of their games, that would have meant six wins for the Atlanta Braves, who have two series remaining against Philadelphia.

That’s where a group of Cal-Berkeley mathematicians came in 14 years ago.

“Linear programming formulations” designed by the group compute every possible outcome of each contender’s remaining schedule and determine when a team is eliminated from wild-card or first-place contention before the magic number officially hits zero. The window typically lasts about three days.

“Mathematicians have known about this window for a long time, and the earliest academic record of that goes back to 1966,” said Eli Olinick, an associate professor in the Department of Engineering Management, Information and Systems at Southern Methodist University. “In fact, it's an example in a widely used textbook on what are called network flow models.”

Olinick as well as fellow Ph.D. student Al Erera and Cal-Berkeley faculty members Ilan Adlerand Dorit Hochbaum developed a web site that would crunch the numbers and figure out when a Major League Baseball team was eliminated before the team’s tragic number had officially reached zero.

“The [text]book came out in '93. And, as you know, that was the last year before realignment and wild-card teams,” Olinick recalled. “I was a teaching assistant for a class that was using the book in '96 and the professor happened to assign the example as a homework problem. Another student and I told him about divisional play and got him interested in trying to work out the math when wild-card teams are involved. This was when the World Wide Web was really starting to take off, and so another one of our professors, who also teaches that class, suggested setting up the website.”

How complicated is the program?

“The process of trying to apply that kind of logic can get pretty complicated,” Olinick said. “There's a good example on the website involving what at the time was the entire AL East. And that's just for elimination from first place. Elimination from the wild card can be even more complicated.

“The linear programming formulations are whole systems of equations that describe all the possible end-of-season scenarios for each team's won-lost record based on their current records and remaining schedule of games. The formulation to determine the first-place elimination number for the Mets has 15 equations involving 21 variables. The formulation to determine the wild-card elimination number for the Mets has over 130 equations and more than 250 variables. Each equation by itself is no more complicated than the formula for computing the traditional magic number, and is less complicated than some of the sabermetric formulas (e.g., runs created). However, the process of solving the whole system of equations is very complicated and involves advanced mathematics.”

Of course, this type of mathematics has more practical applications than solving baseball elimination issues.

“There are lots of practical uses,” Olinick said. “The airlines use linear programming (LP) and integer linear programming (IP) models to optimize plane routing, crew scheduling and fleet planning. LP and IP models are used a lot in logistics. Companies like FedEx and UPS use them for routing packages and delivery vehicles. There are even medical applications such as planning radiation therapy. … They are used extensively in planning and managing telecommunications networks. In fact, the same linear program that is used to determine the maximum amount of traffic that can be sent from a source node to a destination node in a telecom network can be used to determine whether or not a team has been eliminated from first place in its division.”