As I anxiously await the start of the big Jets vs Colts AFC championship game, I am reminded of a few cool computational/math things related to football that I thought about over break. First of all, I found two pretty cool websites which examine football from a bit more of an academic perspective. First, there is www.advancednflstats.com which has a bunch of articles about game theory in football and some other statistics stuff. He devotes a lot of analysis to the decision between running and passing in the NFL, has his own metric for ranking teams, and generally discusses other stuff going in the meta-game of football. There is also the fifth down which examines all aspects of football but generally focuses on the media side of things. Both are worth a read during the mid-week but will probably fall by the way side after the superbowl.
What I really want to discuss is a type of football gambling tournament that my father participated in. I’m not sure if this style of pool has a particular name, but allow me to describe the rules.
# 1 – All players put K dollars into the pool. With N players, there will be NK dollars for the winner(s) at the end.
# 2 – Each week (there are 17 weeks in the NFL season), each player chooses one team to win (or one team to lose). For instance, the weekend schedule might be A vs B and C vs D. I will choose team A to win. If team A wins, I move onto the next week. If team A loses, I am out of the tournament. As opposed to normal football gambling, team A does not have to beat the point spread or anything like that, they just to have to win.
# 3- There is a stipulation to how you choose teams to win. A player cannot choose a team to win more than once and a player cannot choose a team to lose more than twice. Thus if A is playing B, I cannot choose A if I had previously chosen A to win in an earlier week. Also, even I hadn’t chosen A to win before, but I chose against B twice (by choosing both C and D to beat B), then I cannot choose A to win (because doing so would cause me to choose against B for the 3rd time). I like to call this rule the Detroit rule.
#4 – At the end of the regular season, all the players left split the prize pool evenly. If all remaining players are eliminated in a given weekend, that weekend is not counted and they try again next weekend (this could theoretically happen indefiniely, but it doesnt). Players are permitted to split portions of the prize pool before the season is over if all remaining players agree to such.
Rule #3 is what makes this game interesting. When the problem was first described to me, it screamed of a maximization problem (you want to maximize the probability of you being alive after week 17) that maybe had some substructure to it (maybe a knapsack-like thing). For now, lets assume that we have probabilities of team A beating team B for all pairs (A,B) and that they don’t change throughout the season (which is not reasonable but it gives us a starting point). Let represent our choice of team for week
. Note that
where
or else we would violate rule #3.