This entry is written by guest blogger Rich Haglund

Look no further than the Bowl Championship Series, more specifically, the BCS rankings.

The BCS rankings, are determined by averaging the percent totals of the Harris Interactive Poll, the USA Today Coaches poll and a Computer Rankings Percentage (drawn from six computer polls).

If you're bored or looking for extra credit, a more detailed explanation is available here.

These six computer polls rely heavily on algebra to calculate the weekly BCS rankings.

1. Chris Hester and Jeff Anderson developed one of the six computer models. According to Hester, "The rankings were developed mostly with formulas we learned in high school algebra."

Hester and Andersion developed their ranking system while attending the University of Washington in 1993. The two were political science majors.

In 1998 the BCS asked to use the pair's formula in the BCS rankings.

Anderson declined to divulge how much money he gets for the use of his football rankings, but concedes the payout is mostly emotional.

Anderson told the Washington City paper, "The plan was to give more weight to strength of schedule than the existing ranking services did. "What we're trying to show with our rankings is who had the best season," he says. "We're not trying to show who we think would win if teams meet. That's not what our rankings are trying to do. We're trying to reward teams for their actual accomplishments."

2. Another computer ranking was created by Richard Billingsley, a business consultant who operates the College Football Research Center, He developed the "power rating" system, detailed here.

He tajkled with ESPN to discuss the "power rating" system: "I've been ranking College Football teams since 1970, developing the core of my formula at the age of 19. No, I'm not a math whiz, and yes, in times like these I wish I were. I look around the landscape of the BCS pollsters and see an MIT Math Graduate, a Rocket Scientist, a World Renowned Medical Researcher, A Mathematics Professor, and a Missile Tracking Expert and wonder how in this world I ever became a part of this esteemed group. Well, have you ever heard the old saying "find something you like to do, do it well, and eventually you'll be successful". I don't know how successful I am, but that old saying certainly held true for me."

"Many times I'm asked about the formula I use, and it's evolution through the years. Mathematically, it's not a complex formula, using only simple addition, subtraction, multiplication and division. What makes it unique I think are the "rules" that are used in the formula's evaluation process. These rules relate to specific scenarios that I've seen play themselves out over the course of my 40 some odd years of closely observing the sport."

3. Wesley Colley, a professor at the Center for Modeling, Simulation and Analysis at the University of Alabama-Huntsville, has a Ph.D. in astrophysical sciences from Princeton. He runs an interesting rankings site called "Bias free matrix ratings." Check it out here.

Colley currently conducts several research projects, including some for NASA and the Naval Air Warfare Center. Colley has also used his model for the football rankings to this year's presidential election.

4. Kenneth Massey, a mathematics professor at Carson-Newman College in Tennessee, has a system that he believes is "the most scientific and full-featured system available for analyzing the performance of members of a competitive league."

Massey's formula is complicated.

Each team's gametime performance is assumed to be normally distributed about a certain mean (its rating). The probability that team A would defeat team B is then determined from the cumulative distribution function (CDF) associated with a normal random variable.

Let p = Prob(A beats B) = F(rA,rB,hA,hB), where rA,hA and rB,hB are ratings and home advantages of teams A and B respectively. F is a function of rA,rB,hA,hB that is based on the CDF of a normal random variable.

All the game scores are translated to a scale from 0 to 1 by the GOF. Let g = GOF(pA,pB), where pA and pB are the points actually scored by teams A and B in a particular game.

A nonlinear function of the teams' ratings is formed by multiplying terms that look like:
p^g * (1-p)^(1-g)

Here ^ denotes an exponent. Also note that 0 <= p,g <= 1. By maximizing the resulting function, maximum liklihood estimates (MLE) are obtained for the ratings and home advantages. The optimization problem may be solved with standard techniques such as Newton's method.

Still confused? Check it out here.

In 1997, Massey published an undergraduate honors project applying statistical models to ratings of sports teams. He's also created a presentation showing how mathmetics has invaded competitive sports. View it here.

5. Peter Wolfe is a UCLA physician and infectious diseases professor. His rankings formula is also complex. It utilizes, His ranking system uses "a maximum likelihood estimate"

In it, each team i is assigned a rating value ??i that is used in predicting the expected result between it and its opponent j, with the likelihood of i beating j given by:
??i / (??i + ??j)

The probability P of all the results happening as they actually did is simply the product of multiplying together all the individual probabilities derived from each game. The rating values are chosen in such a way that the number P is as large as possible. This is often called a Bradley-Terry model, and is described in papers listed at Wilson's site (see Bradley and Terry 1952, Ford 1957, Elo 1986, Keener 1993).

Check out the whole things here.

6. Probably the most well-known ratings system was created by Jeff Sagarin, who earned a mathematics degree at MIT and an MBA from Indiana Univesrity.

He ranks a variety of sports and his rankings have been published in USA Today since 1985.

Sagarin system for the BCS:

In ELO-CHESS, only winning and losing matters; the score margin is of no consequence, which makes it very "politically correct". However it is less accurate in its predictions for upcoming games than is the PURE POINTS, in which the score margin is the only thing that matters. PURE POINTS is also known as PREDICTOR, BALLANTINE, RHEINGOLD, WHITE OWL and is the best single PREDICTOR of future games. The ELO-CHESS will be utilized by the Bowl Championship Series (BCS).

The overall RATING is a synthesis of the two diametrical opposites, ELO-CHESS and PURE POINTS (PREDICTOR).

Check out a detailed description of Sagarin's college football rankings system here.

Extra Credit
Here's the problem: Expected payouts for teams in the five BCS games in January 2009 will be approximately $18 million. Fox is paying $80 million per year to televise the BCS games through January 2010. ESPN recently agreed to pay $125 million each year to televise the BCS games from 2011 through 2014.

a = w + t

So, if w is the average annual payout from the BCS games and t is the cost to broadcast the games annually, then what is a, the amount of money generated in part by "formulas we learned in high school algebra"?

Of course there's some football involved in generating all this revenue. And an economist would be able to estimate the total amount of money being invested in the opportunity to win some of that $18 million and the national television exposure -- probably totaling over $1 billion.