People in 49 of the 50 United States, as well as the District of Columbia and Puerto Rico, seem to be grousing about the "unfairness" of the BCS system this week. Fortunately for everyone, my multiple graduate level courses in probability & statistics at Michigan taught me how to design a perfectly fair college football season. It's really easy! First, you make an infinite number of clones of all the players, and then have each set of clones play a 120-team round robin. Once everyone's done, we figure out which team had the largest average number of wins, and there ya go, there's your national champion! If two teams end up with the exact same average number of wins (an event that can occur, but with probability zero - that's the kind of silliness you deal with in graduate courses in probability), we can whip up two new batches of clones and have them play a championship game.
Of course this can't happen. If I could produce an infinite army of clones, they'd be divided between securing my benevolent dictatorship and playing far deadlier games than football for my amusement. But this is how probability works. Axiomatic probability theory is based on the understanding that if we play the same game multiple times, we'll end up with different outcomes, so we should figure out what would happen if we played the game an infinite number of times. The fact that this is the basic approach to the subject is why explaining probability and statistics to the Paul Finebaums and Michael Weinrebs of the world is a fool's errand. Don't even bother trying to explain it to people who can't even count to 85.
In theoretical computer science, there is a formal definition of fairness. Roughly, it says that if you visit a state x infinitely often, and an action a can occur from the state x, then a will occur infinitely often. In real life, we don't get to work with infinite amounts of time and space, so we can't design a system that's formally fair. Since we only get 12-14 games to work with, we'll have to make some compromises to be reasonably fair, like Tony Hoare suggested back in 1978.
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| Tony Hoare. Sounds like he should be a wide receiver for the Steelers. Looks like the old British dude he is. |
