Conservative and risky football strategies (and kurtosis)

Brian from Advanced NFL stats recently posited that some NFL teams (namely, the Washington Redskins under Jim Zorn) might have been throwing too few interceptions. This was because the lack of interceptions was a symptom of playing too conservatively, and therefore costing the Redskins games.

Implicit in Brian's thoughtful article are a couple of assumptions that I want to unpack, because radically different strategies might be appropriate depending on the level of football.

• The first assumption is that a lack of passing (or passing aggressively) costs the offense points. This is undoubtedly correct: on average, passes garner more yards per play than runs, and an equilibrium playcalling strategy will seek to maximize the returns for each play (whether in terms of yards, first downs, or points).
  
• The second assumption appears to be that maximizing yards and points is the optimal strategy for an offense. Hence, the lack of interceptions means that the team is leaving points on the board, thus costing it games. This is the assumption I want to address in slightly more detail.

Is it always "optimal" to set your strategy to maximize points scored?

In the NFL -- which is what Brian focuses on -- this is likely true and the assumption holds. NFL teams are almost all competitive with each other, and even the worst teams can beat the best in a given game. So any reduction in expected points is likely to hurt a team's chances of winning because they need to maximize that out to get wins.

But is that true in college? Or in high school? Think about when Florida plays the Citadel. The Gators have a massive talent advantage compared with the Bulldogs. As a result, what is the only way they can lose? You guessed it: by blowing it. They can really only lose if they go out and throw lots of interceptions, gamble on defense and give up unnecessary big plays, or just stink it up.

A fan or some uninitiated coach might see this as a lack of effort, but another view might be that Florida used an unnecessarily risky gameplan that cost them a victory. And since we know that they would win almost every time, what did they gain by being more aggressive? Even if they gained in expected points, this is something like the difference between a forty-point and sixty-point victory, which ought to be irrelevant. (I leave aside BCS calculation questions, which very well might make it worth it to increase the risk of loss to get a bigger chance of a blowout victory.)

The upshot then is that, for the storied programs with large talent advantages, there is seemingly more downside than upside to being very aggressive, either on offense or defense. While it might increase the risk of blowing the opponent out, it also increases the risk of stumbling.

The flipside: the underdog

It's a well-worn belief that underdogs -- i.e. the kind of severely outmatched opponent that cannot win without some good luck -- must employ some risky strategies to succeed. This has long been believed but now we have a reason, though it also teaches us that there is a price to this bargain. The underdog absolutely must take the riskier strategy, whether by throwing more and more aggressively, by onside kicking, or doing flea-flickers and trick plays. They have to get lucky. In the process, however, they also increase the chance that they will get blown out, possibly quite badly. But isn't that worth the price of a shot at winning? Florida might pick off the pass and run it back for a touchdown; they might sack the quarterback and make him fumble; they might blow up the double-reverse pass. If so, then things look grim. But what if they didn't? And if the team didn't do those things, how can it beat them by being conservative? By waiting for Florida to make mistakes?

Get technical

Let's take a quick step back and talk about what is happening from a probability standpoint. What does a more aggressive (and thus more risky) strategy do to our expected outcomes? Hopefully everyone is familiar with the bell-curve, which is a graphic way of depicting the range of possible outcomes based on the probability of their occurrence. The normal distribution is the most common, and it assumes that outcomes on the left and right are as likely as the average outcome. Here, let's assume this is the curve for an offense that can be expected to score around 28 points a game.

Now, let's say they decide to ramp it up. They want to score more points, but this is a riskier strategy, and therefore the range of outcomes will vary more wildly. Below is the new curve, which has moved to the right (to reflect the greater expected points) but is also flatter -- a measure of kurtosis -- which makes the "tails," or ends of the curve "fatter."

(Remember, the height of the curve is the probability of the event happening. Although with the moved curve the whole offense now is expected to score more points, it is now less bunched around the middle because the strategy employed is riskier and hence has more variance or variety.)

What does this tell us? It really just reaffirms what we'd already guess (and assumes that we know what strategies are both riskier and more rewarding, which is an assumption but generally involves passing more). Our offense now: (a) averages more points, (b) has an increased chance of scoring in the forties and blowing out the opponent than before (represented by the shaded green area), but (c) has an increased chance of blowing it and scoring fewer points than our more conservative -- and less variant -- strategy from before. Hence, you might maximize your points but you might actually increase your chance of losing in the process.

Now, remember I'm making assumptions about the nature of the curve. There's also a probability phenomenon known as skewness, which might mean that the improved strategy actually will rarely ever incur a bad game and all the variance will be good.

But the reason I took this mathematical approach to this is that this is really the lesson of the financial crisis as applied to these Wall Street gurus, imported to football: you can "improve" your strategy, you can increase your expected gain, you can increase your chance of blowout wins, but in the process you might be sowing the seeds of your own unlikely, but catastrophic demise. Sort of Black Swans for football.

Spurrier and keeping it close

So in the NFL, where teams are almost all competitive (save, maybe the Detroit Lions), it's likely the best strategy to simply maximize expected points and to go from there. But in other levels, with talent disparities of all sorts, it is trickier, as we have seen.

In the 1990s, Steve Spurrier's Florida Gators were undoubtedly some of the most talented teams of the decade. They were also some of the most aggressive. As a result, they absolutely destroyed some teams. Of course there were the seventy-point blowouts of Kentucky, but what about when they scored more than sixty against Phil Fulmer's Tennessee Volunteers? Yet, Spurrier never once went undefeated with the Gators: his teams always seemed to drop a game or two that maybe they shouldn't have. And those losses almost always had the same profile -- too many interceptions, couldn't run the ball at all, and too many big plays given up on defense. I can't believe I'm inclined to say this, but maybe Spurrier should have been more conservative? He might not have won as many games by sixty or seventy, but maybe they would have gone undefeated and won more than one title?

On the flipside, almost every week of the season I see teams go to Southern Cal, LSU, or Ohio State, and pretty much give up all hope of winning in the name of "keeping it close and winning it in the fourth quarter." As outlined above, this might be the worst strategy against such teams. They have little chance of winning on the merits, so what they need to do is flatten the tails and increase the chance for a shocker: take risks, and hope their coin flips go in their favor. Maybe they won't. Maybe they get blown out. But not taking those chances is a surefire way to set their low chance of winning in stone.

Yet, much like with David Romer's paper where he observed that NFL coaches probably don't go for it on fourth down enough, there are external and likely irrelevant reasons that deter coaches from employing a true "risky-underdog" strategy: the risk that the coach will get fired. I am advocating here that underdogs go for it and increase the calculated risk they take on. (Keep in mind that you can go overboard on this. Chucking the ball forty yards downfield every play, while risky, would not increase your scoring or even chance of winning because you'd become predictible and downright silly. It's about calculated risk.)

But there are real costs -- at least for the coach -- of getting blown out. And make no mistake, the bargain for a greater chance of winning includes the greater chance of getting thrashed. Maybe this should be irrelevant -- a win is a win and a loss is a loss. But a blowout loss has collateral effects, even if they are purely psychological and emotional. You can lose recruits, you can lose donations, and you can lose your job. Look at Mike Shanahan with the Broncos. He was on the hotseat, but he lost his job primarily because Denver got blown out in their final game. I don't necessarily think that was because his team took on increased risk, but people do not tolerate ugly defeats, rational or not.

Similarly, there might be real gains for an underdog to just "keep it close" with a big boy without ever having a real chance of winning. People discount moral victories, but if such and such team can "keep it close" with USC, then they get all kinds of accolades and possibly even confidence going into the following weeks. But if they employed the risky-underdog strategy, then they might gain a slight marginal increase for a victory, with a steeper increase in the chance of getting buzzsawed right off the field (remember skewness).

So, from the perspective of being purely focused on winning football games, I think the implications of the risky/conservative strategy dynamic in the context of teams with wide talent disparities has some pretty dramatic implications. But in the real world, there's lots of other factors, including the felt need by the coach to protect his own skin. Yet, he might be costing his team a chance at victory.