the mathematics of rare matchups in the March Madness tournament

This year’s Final Four is set in the NCAA men’s basketball tournament, with Duke, the University of North Carolina (UNC), Kansas, and Villanova facing off this weekend. This is the first time Duke and UNC will play in the tournament. At first blush this is hard to believe when considering how often these two teams have played in the tournament (a combined total of 334 games!). It’s easier to believe when considering the mathematics used to create the bracket.

I once blogged about the constraints required to seed the 68 teams in the tournament and build the bracket. The NCAA’s website indicates the same rules are still in use.

First, the 68 teams are selected, sorted, and seeded. This is a long process. Then, the 68 teams are assigned to one of the four regions to create the bracket. There are many rules for this last step. Here is the rule that explains why Duke and UNC haven’t played in the tournament before:

“Each of the first four teams selected from a conference shall be placed in different regions if they are seeded on the first four lines.”

Duke and UNC are almost always in the first four teams of their conference, the Atlantic Coast Conference. They typically play each other twice during the regular season and sometimes a third time in the ACC conference tournament. Duke and UNC played each other twice this season. According to the NCAA constraints for constructing a bracket, Duke and UNC are not allowed to meet in the tournament before the Final Four. This is when they are meeting in the 2022 tournament. Mathematical constraints secretly guide the tournament.

Fun fact: it is not always possible to create a feasible bracket that conforms to all of the rules.

There are several other constraints for constructing a bracket. Infeasibility can happen in real applications of mathematical optimization. Mathematical constraints do not make nuanced exceptions to the rules the way human decision makers do, so infeasible problem instances must be addressed with humans.

The selection committee addresses the problem of infeasibility by moving a team’s seed up or down by one and sometimes two. This seems like a small change, but it can drastically change a team’s path to the Final Four. The good news is that about a decade ago the rules were tweaked to change teams’ seeds less often, in a victory for the tournament and also for mathematics.

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