Who's really #1?

Science News, Dec 18, 1993 by Ivars Peterson

It happens every fall. Fierce arguments erupt over which college football team is the best in the country, As the season progresses, this frenzy of head scratching and navel gazing mounts until the climactic bowl games on New Year's Day finally settle the issue.

Or do they?

Even as the mud on the fields of battle begins to congeal, select sportswriters and coaches vote to rank the top 25 U.S. college teams and award the unofficial national championship. But in most years, when the final polls appear, grumbling echoes throughout the land. Allegations of bias, unfairness, cronyism, and petty politicking taint the result.

This season, those cries of anguish and disgust sounded early, In an epic confrontation on Nov. 13, Notre Dame defeated Florida State, 31 to 24, to claim the top spot in the national polls. The Associated Press media poll dropped Florida State from first to second place, while the CNN/USA Today coaches poll put undefeated Nebraska second and Florida State third.

The media voting reflected a fanatic desire to see a rematch between Notre Dame and Florida State in the season-ending Fiesta Bowl, and that could occur only if Florida State retained its second-place standing. But such obvious maneuvering was enough to trigger outbursts of outrage from a number of commentators.

"Voters are no longer rating the best teams. They are creating their idea of the best game," puffed Malcolm Moran of the New York Times.

"Credibility will always be a problem in a business of self-interest and mutual distrust," he concluded. "But a computerized power rating... would restore a sense of fairness."

It was a direct appeal for a clean, mathematical answer to a messy, human situation.

Similar thoughts had occurred to mathematician James P. Keener of the University of Utah in Salt Lake City after the 1984 college football season. That was the year sportswriters and coaches voted Utah rival Brigham Young University (BYU) the national title on the strength of its undefeated season. But these victories had come against generally undistinguished opponents.

Did this BYU team, despite its record, deserve to be number one? The voters in the polls had said yes.

Perturbed by this result, Keener set out to see whether a mathematical scheme, which automatically takes into account the strength of a team's opponents, would provide a more satisfactory answer. "My aim was to get some kind of fair, mathematically based ranking system that removes the subjectivity," Keener says.

Ranking places teams in an order from first to last, while rating assigns numerical values to their relative strengths. Once you have ratings, you can easily generate rankings. Keener turned to a ranking system suggested a few years earlier by Joseph B. Keller, an applied mathematician at Stanford University,

Keller had devised his ranking strategy while serving a term on the board of trustees of the Society for Industrial and Applied Mathematics (SIAM). He had become interested in ascertaining how well SIAM's published journals stacked up against others in the same field.

He decided he might rank SIAM journals by counting how many times they were cited by other publications. But this wasn't good enough. There had to be a way of adjusting the scores so that being cited in a prestigious, influential journal counted for more than being cited in some "garbage" journal, Keller noted at the time.

This sounds like a chicken-and-egg situation. It's hard to see how one can determine which journals are more influential without knowing how they rank, and ranks can't be determined without knowing how influential the journals are.

Remarkably, a relatively obscure mathematical result known as the Perron-Frobenius theorem furnishes a recipe for calculating just such a ranking. "The theorem shows that this circular reasoning actually has a solution," Keener says. "It also hints at how to arrive at that solution." Using this recipe and given the right data, a computer can establish the rankings in a matter of seconds.

Keller never got a chance to apply his scheme to journals. He did try it on baseball teams, however, using the number of times one team beat another team as the entry in the table of values at the heart of his computation of rankings.

In the 1984 National League baseball season, the Eastern Division had proved stronger than the Western Division, but teams within a division play more games against each other than against teams in the other division. Keller's scheme provided a way of ranking all 12 teams while taking into account the fact that the six teams in the stronger division faced tougher opponents more often and deserved greater credit for their victories.

College football, however, involves many more teams, which individually play far fewer games than professional baseball teams. The issue comes down to what value should go into the space reserved for the outcome of a matchup between each pair of teams in the table used for computing rankings.

In the simplest possible scheme, one can assign a single point for a win, half a point for a draw, and zero for a loss and calculate rankings on this basis. But unlike baseball teams, football teams rarely play each other more than once during a season. Moreover, all the credit goes just to the winner, whether the score is close or lopsided.