For a while I've been a proponent of more aggressive decision making on 4th down in the NFL. On this site and others, and in academic research such as the Romer paper, it's been shown that a team is usually better off going for it on 4th down, as long as it's not buried deep within its own territory or facing a very long 'to go.'There are two possibilities here. Either most research on the subject is wrong, or NFL coaches are timid and more concerned with short term job security than winning. I've previously suggested the real answer may lie in decision theory, namely that the uncertainty surrounding such tactical decisions lead coaches to choose the option that promises the best worst case scenario rather than the option that provides the best chance of winning in the long run.
Although I still stand by that analysis, it may be only part of the story. Any tactical decision in football has to take into account the opponent. And this is exactly what game theory can do.
There are two kinds of games in game theory: zero-sum and non-zero-sum games. Zero-sum games are typically those in which the winner takes all. Whatever is good for me is equally bad for my opponent. In contrast, non-zero-sum games allow for "win-win" or "lose-lose" scenarios.
Football defines the zero-sum game at every level. At the end of 60 minutes, one team earns 100% of the W and the other eats the whole L. Even ties are zero-sum. Only in the most extremely bizarre scenario would a football game be non-zero sum, such as when a tie would qualify both opponents for the playoffs. And within each game, every single play represents a zero-sum "sub-game." Every yard gained by the offense is a yard given up by the defense. Football screams zero-sum.
But just as winning is a zero-sum proposition, scoring is not. A team could adopt a strategy that allows itself to score more often, but also allows its opponents to score slightly more too, hoping to gain the greater advantage. Mike Martz's Rams may have been an example of such a team. Their relative disregard for turnovers led to aggressive high-scoring outcomes, but it also allowed opponents to score more frequently themselves. The question is, on balance, does the strategy favor the more aggressive team itself or its opponents?
Consider a football game between two equally matched teams, Team A and Team B. Each team has the choice of two strategies: either the conventional punt strategy or an aggressive go-for-it on 4th down strategy. Assuming the research on 4th down strategies is correct, the possible outcomes of the game are listed in the table below. In terms of the probability of winning, when both teams employ the conventional punt strategy they each have a .5 probability of winning. Similarly, if both teams employ the go-for-it strategy they would also have equal chances to win. But if one were to adopt the go-for-it strategy and the other did not, the aggressive team would enjoy a .6 to .4 advantage in its probability of winning. (The table can be read [outcome for Team A, outcome for Team B].
| Team B | |||
| Punt | Go For It | ||
| Team A | Punt | .5, .5 | .4, .6 |
| Go For It | .6, .4 | .5, .5 | |
If the research is correct, and coaches were purely rational, they would adopt the more aggressive strategy on 4th down. But they don't, and part of the reason my be related to how coaches think of outcomes.
Coaches might sometimes confuse points for the ultimate outcome of interest--winning. There is a paradox at work. A strategy that ultimately allows an opponent to score more points may actually be superior in terms of winning. But because in every other respect football is zero-sum, a coach would naturally be averse to any strategy (more aggressive than the conventional status quo) that allows his opponent to score more, even if it theoretically improved his team's chances to win. After all, if scoring is good for my opponent, how could it be good for me?
Here is the same football game from the table above, but this time the outcome is described in terms of points scored instead of win probability. As you can see, the game now appears to be non-zero-sum.
| Team B | |||
| Punt | Go For It | ||
| Team A | Punt | 20, 20 | 24, 27 |
| Go For It | 27, 24 | 27, 27 | |
The average score in the NFL is about 20 points, so equally matched teams playing the conventional punting strategy could expect to average 20 points per game. The go-for-it strategy, according to the research, would allow the aggressive team to score more often--say 27 points per game. But it would also allow an opponent to score slightly more often--say 24 points per game.
If a coach is thinking in purely zero-sum terms, he would not be able to reconcile the non-zero-sum [27, 24] advantage with everything else he knows about the football.
Basketball has accepted this concept for years. NBA teams like the Phoenix Suns or Denver Nuggets employ fast-paced tempos that allow for more total possessions by both themselves and opponents. Consequently, they both score and allow more points per game. The idea is that their team strengths give them an advantage when playing at a quick pace. I think it might be particularly difficult to accept in football because of the zero-sum nature of yards gained and lost on every play.
It's all in how you define your utility. Outcomes in game theory must adhere to strict rules such as linearity and transitivity to be valid. This is why previous research on run/pass balance has missed the mark. I'll illustrate exactly how and why in a forthcoming article. Plus, I'll propose a valid measure of utility in football.
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