Game Theory Intro and 4th Down Decisions

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
PuntGo For It
Team APunt.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
PuntGo For It
Team APunt20, 20
24, 27
Go For It27, 2427, 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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6 Responses to “Game Theory Intro and 4th Down Decisions”

  1. JTapp says:

    I think it should be noted that in both matrices the Nash is Go for it, Go for it.

    While I think the tempo issue is the same in both football and basketball, I don't think it's been better understood in basketball than football.

    The fast-break Denver Nugget teams of old were the worst points-per-possession teams in NBA history, and had some of the worst records in history. All their uptempo style did was help seal their doom by giving the superior teams more possessions. They'd have been better off slowing it down.

    If you have an inferior team, you play a slow-down offense and increase the variance in outcomes by decreasing the # of possessions. For example, If Miami played Boston for 3:00, they would have better odds of winning than if they played them all 48 minutes.

    I'd say it's the same in football. Mike Martz' physically inferior teams score a lot of points by passing a lot, but all this strategy does is increase the chances of losing by increasing possessions in a game with a superior opponent.

    Inferior teams should run the ball more to try and milk clock and reduce possessions. Superior teams should exploit their advantage and play a higher tempo.

    I've seen many teams, particularly in college, that were said to "improve" after installing a pass-heavy spread offense. It allowed them to blow out their inferior opponents, but get creamed worse by superior opponents. It seems to me that once a coach picks a philosophy (ie: run-heavy or pass-heavy) he doesn't change it much from game-to-game depending on his opponent.

    I look forward to the next segment.

  2. Brian Burke says:

    I agree with your analysis on tempo. But I'd recommend a team play to its strength rather than run or pass too often. There are better ways to manage the clock that don't sacrifice optimum play choice.

    I think the 4th down strategy issue is similar in that it is heavily dependent on relative opponent strength. So far I've only discussed equally matched opponents because the go-for-it strategy also would increase variance in points.

    Even if the go-for-it strategy did not increase your average expected point advantage, it would still dramatically increase the range of scores. In other words, a team would often end up scoring 42 points but also end up giving up that many too.

    So if you're an underdog team, you'd want higher variance. It doesn't help you to stink just a little but do it consistently because you'd lose every week. You'd rather sometimes really really stink, and sometimes do very well. At least then you'd have good chance at winning.

    The go-for-it strategy might be better suited for underdogs than heavy favorites.

  3. Presh says:

    This is a great analysis. I'd like to offer one more aspect of the game--the coach's job security.

    Since coaches are expected to punt on fourth down, they don't look bad even if it is a bad strategy. If they elect to go for it, they play a much riskier strategy--they have to hope things work out, or else they lose their job.

    I liken this to the stag hunt game. Coaches make the sub-optimal choice because it is "safer."

  4. Derek says:

    I've been thinking about a similar problem to the high-tempo offense lately: is it better to have a good offense and bad defense or a bad offense and good defense? In other words, if I'm winning a lot of games by a field goal, am I better off winning/losing 13-10 or 38-35?

    My mind went to Pythagorean winning percentage, and hopefully this formula will be clear in plain text:

    lim{x->Inf} x^2 / (x^2 + (x-3)^2) = 1/2

    The 3 in (x-3)^2 is arbitrary, and you could use any point differential.

    Essentially, what it says is that the more points you score and allow, your Pythagorean winning percentage tends to .500. Given that the outcomes of close games (within a field goal) are essentially random, how do teams with good O/bad D and bad O/good D fare?

    Teams with good O/bad D are going to have a Pythagorean win % closer to .500, and with a bit of good luck, they can win one or two extra games by a field goal to reach a 9-7, 10-6 record and a playoff birth. Conversely, teams with bad O/good D are much more dependent on the swing of luck. Let's say you win a game 6-3. The Pythag win % for that game .800. Lose that game, however, and the Pythag win % is .200. The Pythag win % of a 38-35 game is .541, however.

    So teams with bad O/good D are much more susceptible to swings of luck in close games, and I think it helps to explain why teams with very good offenses tend to make the playoffs more often than teams with very good defenses.

  5. Brian Burke says:


    I think I understand what you're saying now. The (x-3) term is "winning by 3." I agree that teams with bad O/good D will appear stronger than they probably are in Pythag win %.

    I think that may also explain the old adage that the fastest way to improve a team is to improve its defense. If the bad O/good D team miraculously improves (but it's really just due to lots of luck) then it would reinforce that belief.

  6. Anonymous says:


    The dilema of which is better a good O/bad D or a bad O/good D is an interesting question. But using Pythagorean winning percentage to find the answer is flawed.

    Pythagorean winning percentage is not routed in theory, rather it is an empirical device known to be a good estimate of wins based on points scored vs. points allowed.

    The exponent used in Pythagorean winning percentage is subjective to the total points scored. You used x^2 but, in higher scoring sports such as basketball x^16.5 is commonly used.

    When x^2 was used in the low scoring game the expected win percentage was 80%. If x^16.5 is used in the high scoring game the expected win percentage is 80%.

    Some other analysis must be used to answer your question.

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