Abstract

Mathematical ranking systems, such as those used in college football's Bowl Championship Series (BCS), can be classified in two broad categories. Predictive methods seek to forecast outcomes of future games, while retrodictive rankings aim to most closely match the results of contests already played. Ideally, a retrodictive method would order teams such that each team is ranked ahead of all teams it defeated, and behind all the teams to which it lost. However, this is generally impossible at the end of the season, as any ranking will “violate” the results of some games, by having the loser ranked above the winner. For any given set of game results, there is a minimum possible number of violations, and we call a ranking that induces the minimal number a perfect ranking; computing such rankings is an NP-complete problem. Jay Coleman, an operations research professor at the University of North Florida, developed a model called MinV to identify perfect rankings. For the 2008 season, each of the six computer ranking systems used in the BCS had at least 80% more violations than MinV. However, perfect rankings are not unique, raising the question of which perfect ranking is the best ranking. If all perfect rankings agreed on the top teams, this might not be a major concern, but in the 2008 season, there were five teams that could have been #1 in a perfect postseason ranking. Because of clustered scheduling, it is possible to move groups of teams up or down, and sometimes even whole conferences, while maintaining a perfect ranking.