It seems that one of the largest factors that could improve the accuracy of the beatpath ranking is to get home field advantage into the equation somehow.

I’m in Seattle, and our Seahawks have one of the more dramatic differences between home record and away record.

My first thought was to split the league’s thirty-two teams into sixty-four “virtual” teams, like “Seahawks_HOME” and “Seahawks_AWAY” for each team… then run the same analysis you’ve got now and see what pops out.

But you’d always have one team’s HOME version playing another team’s “AWAY” version, so the graph might be disjointed. I can’t quite visualize whether that would work or not.

Another (more labor-intensive) way to explore home-field advantage would be to split just one team into its home and away versions… so have “Seahawks_HOME” and “Seahawks_AWAY” in the graph with the other thirty-one (non-split-into-home-and-away) teams. That would give a useful graph, and useful rankings, with Seattle’s two versions each showing up in the graph… possibly with a beatpath from one to the other!

I don’t know how you could parlay that into a single useful graph that helps visualize all teams’ home field advantage into the beatpaths system…

For example, if INDY beats NE beats CINCY beats INDY, then the feedback arc set problem would recommend deleting (say) the INDY beats NE edge. But this would create a scenario where NE is superior to INDY and CINCY, not borne out by the data.

Actually, what you need here is a min-cost flow. This is a lot more mathematical than what you might like, but the basic idea (see the book by Ahuja, Magnanti and Orlin) is that you want to compute a flow from the “source” – in this case all undefeated teams) to the “sink” – all winless teams (if there are no undefeated or winless teams, that’s not a problem either). Moreover, you want the flow (think of pushing water along the edges) to have minimum cost (because pushing water is expensive). In that case, the min-cost flow will automatically exclude all cycles, essentially deleting them, and the resulting graph (where we only keep edges with flow on them) will be the graph you need.

So it isn’t exactly that the Patriots are 12th, even though that’s where I have them. It’s more that just based off of the existing wins and losses so far, there is more reason to rank those eleven teams ahead of the Patriots, than there is to rank the Patriots ahead of them. I’m not saying that the Patriots are the 12th-best team in the league, though. This is a tool – if someone is convinced the Patriots is the best team in the league, and then they come here and say, “oh wow, well hold on a second – it turns out they really haven’t beaten anyone of value yet…” then the beatpaths graph/ranking has served its purpose.

I’ll look up the minimum feedback arc set. It might not fit what I’ve wanted to do here – one thing is that I kind of have perversely enjoyed creating a system that relies on absolutely zero math. It makes intuitive sense to me that a team split should cancel itself out before a three-team beatloop, and so on.

And yes, it is early in the season. There is just something interesting in watching the graph change from round to round. Like it or not, teams get prematurely anointed or rejected from people not paying enough attention to short-term schedule difficulty. I think this point is actually a plus for beatpaths rather than a minus.

But what’s even more interesting is when the paradigm shifts. For instance, a seemingly good team can lose to a couple of seemingly crappy teams, making the seemingly good team seem bad. But then it turns out those bad teams are good. So the first good team is still good. We see that a lot – last year, New Orleans was seen as good here, long before the main media came around.

Interesting problem there with removal of beatloops. What you have is the classical graph theory out of combinatorial mathematics approach to determine winners of tournaments. It also turns out that there is a mathematical solution known as the minimum feedback arc set which can be used as a fair way to remove what you call beatloops. Just removing the small loops first won’t necessarily give you the correct answer for your rankings, while a minimum feedback arc calculation will.

I do have one problem with your rankings, and that is that if a team is undefeated it may still have a low ranking because it has not played anyone with significant wins. The Patriots are an example of this. Even though they could not possibly have a better beatpath than they currently have they end up with a low ranking. This is not college football where a team has some control over its schedule.

Any method that leads to a ranking of the Lions and 49′ers ahead of the Patriots is obviously a bit questionable.

#8, at various times in the development of the algorithm, I’ve thought of things like that, but the problem is how to determine the strength of the opponent you beat. They’re weaker since you beat them. If you go by what their ranking was before you beat them, then things start to not work as well.

#9, I came across mention of something called a Markovian matrix that sounds like that, and I heard it’s used to handicap race horses. But I haven’t found any good resources online on how to actually do it. Would probably need a question-answerer to volunteer to answer some questions for me on how to actually create and subtract and interpret the matrices. I took a matrix theory course in college but it’s been a while.