Friday, February 6, 2009

football meets economics

Slate had an interesting article recently about proposed changes to an overtime rule in the NFL, specifically, how it is decided which team gets the first possession. This paper on the subject makes the point more formally, for those who, like me, prefer their stories with a side of math. The analysis leaves out some considerations, however, that I think should change the conclusion.

Currently, a coin flip determines who gets the first possession, and a kick-off by the other team determines the initial field position. As overtime is decided by sudden death, the team with first possession has an advantage: they win in overtime in the last ten years more than 60% of the time. (Interesting, the historical average is lower. But better offensive drives have pushed up the advantage.) This coin-flip determination is thus nonideal, although there is no ex ante advantage for either team, because the probability of winning is randomly decided.

Two solutions to this unfairness are proposed. One is the simple fair-division mechanism familiar to any rival siblings: one chooses a split, and the other chooses which side to take. That is, one team would choose an initial field position, and the other team would choose whether to play or defend from that position. The "fair" position would depend on the teams involved of course, and it would take a couple seasons for teams to work out the best strategy, but the average fair position is estimated to be 15-20 yards. Also, note that any improvement in offense will never push the fair position as far back as the 0 yard line, since the probability of giving up a safety enforces this lower bound. (The paper makes this assumption explicitly, but for some reason does not point out this reason for its realism.)

The imperfection in this mechanism arises from imperfect information: The choosing team knows its strengths better than the dividing team, so the dividing team is at a disadvantage. A different mechanism, based on auctions, does better in this respect. In this rule, each team "bids" on field position, and the one willing to take possession closer to their own goal gets the ball at some compromise between the bids of the two teams. This is a perfectly symmetric rule, and thus favored by the economists and by the guys who wrote to the NFL to try to get the rule changed.

My first objection to this conclusion is that a little bit of unfairness in the divide-and-choose scheme is ok. It's true that if a coin toss determines which team divides and which chooses, this introduces an arbitrary element into the game. But, instead of introducing additional arbitraryness, the asymmetry could be used to counteract an arbitrary advantage that already occurred: the coin flip at the beginning of the game to assign first possession. The winner of that coin flip had a small advantage, and thus if the game ends up tied, they played slightly worse. They should thus be forced to divide the field and allow the other team to choose possession in overtime. Both advantages are small, so cancelling them out in this way should be a nearly perfect resolution.

My second objection is to the pragmatic details of the implementation of an auction rule. There are several ways it could be done, as listed by the guy, Quanbeck, who tried to get the rules changed. First is a live dutch auction, then an ascending live auction, and last a sealed bid auction. The last way is just boring. The first two ways would of course cause the coaches to plan ahead what yard they would seize or concede possession at, but in a live auction format, they would also try to read each other's body language for subtle clues and adjust their strategy on the fly. This puts too much visible responsibility on the coach, when it's better for the game if the players are the ones responsible for their fate and the coaches stay on the sidelines. Additionally, can you picture a non-trivial pause in a super-tense hard-fought violent game for a geeky field-position auction exercise? Quanbeck can, but he's an electrical engineer =) A divide-and-choose rule would take no longer than the coin toss, and make a lot more sense to the masses.


bodhi said...

fascinating! I had no idea you had such interest in football :) Many people I know complain incessantly about how stupid overtime is in the NFL these days, because even though the probability of victory for the coin-toss-winning team is over about 60%, as you state, it seems in real life to be much higher. Almost so much so that "oh, they won the toss, thus they will win the game, and that is stupid", which is what people I know say. A bit of mental combat and knowing your strengths as in the divide-and-choose could be cool, but you still have to deal with the entertainment aspect of it all - will it be fun to watch on tv?

and by the way - hi! :)

Vera said...

hi bodhi! yeah i do love football although i mostly think this is a convenient motivation for economics, which I like more =) yeah i know what you mean about the entertainment aspect. i figure even if it is boring, at least it's really fast, and probably not more boring than the coin toss.

somebody said...