For example, the Alabama Crimson Tide play the Ohio State Buckeyes on New Years Day in the first round of the College Football Playoffs. These two teams have not played each other this season, so there are no direct paths connecting them. In addition, they do not have any common opponents, so there are no paths of length two connecting them either. But some of their opponents have played each other, so there are paths of length three available:
- Alabama beat LSU by 7, LSU beat Wisconsin by 4, Wisconsin lost to Ohio State by 59. That's a +48 advantage to Ohio State ( = -7 - 4 + 59).
- Alabama beat West Virginia by 10, West Virginia beat Maryland by 3, Maryland lost to Ohio State by 28. A +15 advantage to Ohio State ( = -10 - 3 + 28).
- Alabama beat Missouri by 29, Missouri lost to Indiana by 4, Indiana lost to Ohio State by 15. A +10 advantage to Alabama (= +29 - 4 - 15).
If we average the net differentials over these three paths, we get an average advantage of 17.7 points in favor of the Buckeyes. But we don't have to stop at three-paths. There are actually 43 distinct paths of length four connecting Alabama and Ohio State. I won't list them out here, but you can view them all on the new Bowl Games page. The Buckeyes have an average advantage of 5.5 points when summed over these 43 paths.
The number of paths grows exponentially as we increase the length, which is why the tables stop at paths of length four. But with the help of some simple matrix manipulations, I can still summarize the average advantage for each path length. The "Chart" tab summarizes the results out to paths of length of fifty. For the Alabama-Ohio State matchup, you can see that the advantage flips to the Crimson Tide at around path length six, and eventually converges to a value of 7.7 points in favor of Alabama. That final value is more in line with the 9 points that Alabama is currently favored by in the point spread.
This feature is still in its entertainment phase at this point (please, no wagering). I'm still working out if there is any predictive value to this approach. But the general concept of "summing over paths" reminds me of a concept from my physics studies: Richard Feynman's path integral formulation of quantum mechanics. Without getting too into the weeds, the basic idea is to relate the probability of an event, say a particle going from point A to point B, to all the (infinite) ways that particle can travel between those two points. Each "path" connecting point A and point B has an associated probability amplitude. The total probability amplitude of going from point A to point B is just the sum of the probability amplitudes for each of those distinct paths - much in the same way that we summed over all distinct paths connecting Alabama and Ohio State.
The similarity may end there, but it would be fun to check whether there are common mathematical threads underlying both approaches - for certain values of "fun".