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Monday, July 10, 2006

Run/Pass Balance and a Little Game Theory

Football is the most strategic of all sports. A big part of this is the unique feature that each game is 100-150 or so unique trials—the plays. This gives rise to the art and science of play-calling, “balance,” formations, sets, set-ups, counters, and whatever else that keeps us up at night thinking about this stuff. Stepping back for a second though, I wanted to simply look at the concept of balance and how we should best achieve it. Before that, however, I wanted to emphasize what I think are the most important offensive statistics.

Yards Per Carry and Average Yards Per Pass Attempt

I've long felt that the most important rushing stat, at least in terms of 1st and 2nd down performance, was yards per carry, and not total rushing yards or anything else. Quite simply, a running back who gets 120 yards on 15 carries plays in a better offense than a RB who gets 120 yards on 25 or 30 carries.

This same logic, however, applies to the passing game. More important to me than passing efficiency, or completion percentage (by itself), yards per completion, or any other statistic is Average Yards Per Pass play (including sacks). Bud Goode, legendary football statistician to the stars (Dick Vermeil, Bill Parcells, etc) has been harping on this stat for years.

The key is that it, in effect, combines completion percentage and yards per completion. The NFL QBs who have had the highest totals ever in a season are as diverse as Joe Montana (exceptionally high completion percentage) to more long-ball throwers. It penalizes the guy who inflates his completion percentage and the guy who points to his long-balls while ignoring how inefficient he is.

These two stats converge in the most important first and second down stats, which are average yards per play. The goal is to move forward to the other guy's goal-line, continually increasing your chances of scoring a TD. Further, you really want to do this on first and second down: Third down is a defense's down. The odds are in the defense's favor, and so are the strategies. Also, the teams with the best third down conversation rates are invariably the ones who have the shortest average distance to go on third down--further emphasizing that positive first and second down yards are the key.

So the goal is to find the mix of runs and passes that maximizes your teams' average gain per play.

[Note: This is not entirely true, as passes carry more risk. Turnovers make up the most important stats of all in terms of winning, and pass plays result in more turnovers than do run plays--both more fumbles and more interceptions. The answer, however, is not to ignore passing, but instead to require a "passing premium"--your passes should average more per play than your runs to counterbalance this risk.]

Offensive Identity and a Taste of Game Theory

How good you are in "absolute terms" at running or passing is a matter of talent, scheme, and reps. My argument here is that for the sake of "balance" it doesn't matter what you're better at, but, as Carolina offensive coordinator Dan Henning says, you pick a target mix and go for that, while adjusting to the defense.

This adjusting to the defense is where game theory comes in. The basic idea is that your offense and their defense have certain strengths and weaknesses, and, for the most part, everyone knows each other's weaknesses.

Imagine you are fortunate enough to have a future All-American guy at RB. He runs for a ton of yards as a junior, and now, a year later, you're ready to ride him to a state title. But everyone else knows about this guy now. They begin stacking the line. You've got this All-American at running back and you're averaging less per carry than you did three years earlier when you had three Academic All-Americans--and no football All-Americans--splitting time at RB. What's going on? What do you do?

You pass of course. You run bootlegs, you fake it to him, and you throw the ball. But how odd you say. You have the best running back you've had in 15 years, and you wind up running less? The answer is simply that everyone else knows you have this stud RB, so they commit so much effort and defensive scheme and structure to push your expected yards per rushing play down to a manageable number, your passing opportunities increase, even if you have less talent there than years past.

This same goes for great passing teams. (Think about all the spread offense teams that have used the defense's natural tendency to play pass against four wides to their running advantage.)

This little cat and mouse game is really an extension of the Nash Equilibrium from Game Theory (the subject of the movie A Beautiful Mind, about John Nash, the concept's namesake).


The idea is if you are a very good passing team you pass most of the time, then you run when it is favorable and see positive results without having had to practice it too much. Same goes vice-versa--we all know how dangerous play-action passes are from heavy run teams, especially say a veer option team.

Again, I don't think yards per rush and yards per passing attempt should be exactly equal--passes are riskier than running plays. Specifically, they more often result in lost yardage (sacks) and turn the ball over more often (both fumbles and interceptions). So you should expect your yards per pass attempt to be higher than yards per rushing attempt.

To reiterate the earlier points of how this can be counter-intuitive, look at Urban Meyer at the Florida Gators (stats below). Let's say next year, with a year in his system, the passing game stays the same but the running game improves by a full yard per play. Now, what happens? Well, first Meyer will run the ball more--less risky, same reward. But then the defense will see this and begin to step up to stop the run, and drive the average yards per run back DOWN. Yet, the defense will be weaker to the pass. The result?

Counterintuitively, the passing game yards per attempt could go up and Meyer should then actually pass more. Surprised? Just think about it: If the D had to do more to stop the run, the pass gets more attractive, so Florida starts getting maybe 6.7 or 7+ per play every time they throw it, so of course they are going to throw it, even if in absolute terms it was the run game, not the pass game, that improved. Regardless, the improvement in the run game should affect the entire offense's production, which is what is important.

The lesson? If your passing game is suddenly working better, it might not be because you are suddenly Bill Walsh. It might be because you've got a stud running back everyone wants to stop.

The point of this is that you can hang your hat on one thing, but you might be leaving production on the table by not running or passing enough.


I just pulled some basic stats off of for major college teams to give some examples. I don't mean this as a criticism of these teams since my stats include some downs like 3rd down that may inflate or skew the stats, and college football includes sacks as a running play. To counteract that I added the QB's rushing numbers to the passing stats (except for Vince Young). This may be problemmatic for Chris Leak at Florida, since Urban Meyer uses a system where the QB runs the ball, but Leak was not particularly good at this and did not run near as much as Meyer's previous quarterbacks.

I fairly randomly selected these teams, though I did want to highlight teams of interest and on different ends of the spectrum.

Texas Tech

Pass-happy Mike Leach at Texas Tech attempted 697 passes for 4857 yards, averaging 6.97 yards per pass attempt. (I also recognize how many of these are shovels and the like but I'm just being simplistic.)

They ran the ball 172 times for 1040 yards, or 6.05 per rushing attempt.

So we compare 6.97 per pass to 6.05 per rush. Putting the two together the average yards per play is 6.77. We can see you can make an argument that they should have passed MORE, since that would have raised their average yards per play, but a passing premium of about a yard seems about consistent with most other teams.

The result? Tech, for all its crazy stuff, is pretty balanced.

Florida Gators

Next I looked at the Florida Gators. They got 2801 yards on 490 passing attemts (5.72 average) and 1680 yards rushing on 350 attempts (4.71). Together, the total yards per play was 5.33. Again we see about a yard of "passing premium" indicating that Urban was pretty balanced but that his team was not as productive, on a per play and total basis, as Tech.

Minnesota Gophers

Let's look at the Gophers: They ran it 586 times for 3247 yards (5.54), and threw it 347 times for 2690 yards (7.75).

That's the biggest passing premium we've seen, over 2.25 yards. Unless Minnesota is extremely risk averse, it appears that the Gophers should have passed more than they did. This result makes sense with what we said above: Minnesota had one of the best backs in the country, Maroney, and another guy who got 1000 yards. Their QB, some guy named Cupito, I didn't even remember. But defenses and defensive coordinators know the same thing. They were all geared to stop Maroney and the Gophers zone run game.

Should they have gone pass happy? No, of course not. Yet, imagine if they had thrown 30-50 more passes instead of runs (only 2-4 more per game). With more passing, the yards per pass attempt would have gone down, but I don't imagine it would have gone down to less than 6 yards like the rushing average. Also, yards per rush would have probably gone up as well. Thus, Minnesota likely would have been more productive to the point of 3 or more points in several games. In the Big 10, that is the difference between winning and losing.

The fact is that Minnesota's strength was definitely running the ball, but everyone else knew it too: Minnesota could have seen some easy success in the passing game and helped out their offense in total by throwing a bit more.

[Note: My numbers are rough so I'm not really trying to criticize Minnesota per se, just use them as an example.]

Southern Cal

These numbers are less helpful for the truly dominant teams (and less important, being smart about things matters less when you've got all the best talent). Nevertheless, let's look at the teams in the National Title Game.

USC threw 523 passes for 4193 yards = 7.88 yards per pass attempt
and 474 rushes for 3344 yards = 7.06 per rush attempt

This indicates that USC, no surprise, was very balanced and efficient in its playcalling. Maybe they should have run a bit more since that "passing premium" was kind of low, but USC is also a very efficient passing team and they do not turn the ball over very much, so they can have a smaller passing premium and get away with it.

However, the stat that jumped at me was 1740/200 = 8.7. As in 8.7 yards per rushing attempt, as in Reggie Bush's yards per rushing attempt. As in, handing the ball off to Reggie Bush had a greater expected gain than did throwing the football, which is just unheard of. This implies that USC should have handed it to him more. Now there are other issues, like durability, and Reggie's receiving prowess, but that is such a substantial number you will not see anything like that.

Texas Longhorns

Texas's stats were interesting too.

336 passes for 3083 yards = 9.18 yards per pass attempt
605 rushes for 3574 yards = 5.9 per rush attempt

That's a huge discrepency--that dwarfs Minnesota's number earlier. Texas' numbers may be skewed because it was on the good-end of a lot of blowouts and probably ran the ball much more in the second half. Nevertheless, coupled with the fact that Vince Young was the nation's passing efficiency leader, this implies that Texas probably held Vince's hand to much and should have let him throw more (or he should have stayed in the pocket and thrown more). Especially since as a runner Young averaged nearly 7 yards a carry, better than all but one of Texas' running backs. This exceptionally high passing yards per attempt number is probably correlated with Vince's running ability--the D had to take men out of position to spy him on pass plays.


To reiterate, my stats here are a bit on the simple side but the point is not the stats, it's the thinking: Typically a fan or coach looks at numbers like 9 yards per pass and 6 yards per rush and says "well, you don't run it as well you throw it." I think the right response, though, is "you ran it too much" or "you didn't throw the ball enough." That's a very different approach. It makes perfect sense though. It's recognizing that you're coaching against a smart person on the other side who knows where your strengths are, and then exploiting that to your advantage.

I remember someone asking Hal Mumme when he was at Kentucky about how his teams' yards per carry had dropped around a yard or so from the season before. The reporter was incredulous and turned red faced at Mumme's response: Mumme told him that he saw the same thing, and that to fix it he would throw the ball more. The reporter cut him off and essentially called him an idiot, mentioning that everyone knows you run better by simply running more (wear them down!). I'm pretty sure Mumme's point was that he coached a passing team, and if his yards per carry was going down, at least one reason was that the defense was spending too much time on the run and that he, as playcaller, was not taking advantage of passing game weaknesses defenses were leaving open.


Anonymous said...

Chris, as always I love the analyis and have always thought about balance in this way.

I do have one slight issue/correction in your analysis.

If a teams running game improved from one year to the next they would not in equilibrium reduce the number or running plays. They would run more and pass less, but both their average yards per pass and run would increase.

Lets say one year you ran 25 times and passed 25 times and averaged 6.0 and 7.0 yards respectively 1 yd pass premium.

The next year you have a very strong running game but you are still the same at passing. ie if you ran the ball 25 times this year you would average 7 yds per carry and if you passed 25 times you would still average 7 yds per carry.

Now lets assume that every additional time that you run the ball beyond 25 your yds per attempt goes down 0.10 and every carry less than 25 your yds per goes up a similar amount. Let also assume that this same relationship holds for passing.

To get back to the 1 yd passing premium you assuming that the number of plays is constant you would need to run the ball 5 more times and pass 5 less. In this case You would averge 6.5 yds running the ball and 7.5 passing and run 30 times and pass 20. The key is you allways will run more.

Intuitively this is why Texas Tech passes more and Minnesota runs more but they both have have pass and run averages close to each other. The reason is that TT has is better overall at passing were they both to runn the same number of passes.

This same logic carries over from one season to the next.

This may all be obvious to you, but I thought you seemed to be saying that a team thats running game improved would counter intuitively run less. I appoligize if I misunderstood.

Anyway I love the article and the site. Keep it coming!

Anonymous said...

Hey great article Chris. I was wondering if you've looked at how these premiums would compare to NFL teams?

Chris said...

Brad: Thanks for the kind words. I agree with your stance, and to some extent I wrote about it in the way I did for rhetorical shock value to get people thinking. That being said, I do think it is possible for a team to get better at running and wind up running less, both from a game theory and a practical standpoint.

Most defenses cannot scale their focus continuously, and instead shift discretely. For example, imagine a defense had three options on first and 10: balanced (base 4-3), run defense (8-9 man front), and pass defense (7 man front). The team that averages 6 yards a rush and 7 yards a pass may face the balanced defense. Then, when the rush improves to 7 yards a carry, and the offense begins to run more and more, the defense then stacks the front and the O now must face the run defense.

The expected values against the run D, even with the improved run game, could end up being something like 5 yards per rush and 9 yards per pass, in which case they would pass more and run less than they did when their running game was not as effective. I suppose one could kind of label this a theoretical approach to "take what the defense gives you."

Anyway, your point is well made though and I do agree, and is still fairly counterintuitive: if your run game improved by 10%, you'd run more, but by some percentage less than 10%.

Anon: I briefly looked at some NFL teams and their passing premiums range quite a bit higher, but I think there are reasons for this. On a cursory review of the playoff teams, the premium appears to hover around 3 yards, with 4 being the high (Colts) and 2 being the low (Seattle). This does imply that NFL teams undervalue the passing game, which is consistent with what some of the other forward looking stat people have been saying, like Football Outsiders, etc.

That said though, I cross checked a couple stats. First, it appears passing is riskier in the NFL from an interception standpoint: NFL teams had interceptions per pass ratios of around .04, give or take, while even teams like Texas Tech in college had ratios around .02, or half that. Also, injury problems at the QB position plague NFL teams. More passes exposes QBs to more hits and more injuries, and this alone might be enough to tilt the playcalling to running.

Nevertheless, the most balanced teams had the best offenses, like Seattle. The outlier was the Colts, but their passing game has been so good they score plenty of points, they won so many games and put teams away early (which is rare in the NFL), and seemed to be preparing all year for a possible cold/bad weather game in NE.

Anonymous said...

The other missing variable is the role you expect your offense to take. Some offenses serve as almost placeholders aiming to hold the ball for as long as possible while grinding clock and hoping the other team makes a mistake.

Those teams should have lower per-play yields--like Urban's offense, for example. There's no way they should approach Tech's YPP on passing ever, since they're designed to pound on the ground and nibble away yards with short passing.

Anonymous said...

Chris, I can see how the defenses could be a little lumpy and how that maybe could lead to a reduction in running, but I think that assumes that you can only play one defense every down. I think if instead the defenses response is just to increase the number of plays that it calls the run defense relative to the pass defense I think the response gets nearly continuous and a lot less lumpy.

I think you actually wrote an article about this so called "mixed strategy"

Theoretically in this situation the number of runs should increase, but in practice Defensive coordinators may take more of an all or nothing approach. If they took that strategy or something that approached it you are correct that the number of runs could also go down. A defensive coordinator employing this strategy likley would not be acting optimaly but there is nothing in practice to say that your opponent has to be fully rational.

Anyhow, not meaning to pick nits, but just enjoying the discussion.

Anonymous said...

I've got to be honest, I think this data is useless. The theory itself may have some merit, but to come to a conclusion that Minnesota should have passed more due to their yards per pass play being greater than yards per running play - well that is ridiculous and does not take into account the poor abilities of the quarterback, and the fact that Minnesota would often run for very short yardage late in games to run the clock out.

Minnesota isn't the only team like this, the old Nebraska teams would be extremely successful at running the football and just good enough at passing to be dangerous. Similarly, I would expect NU's yds/pass play to be greater than yds/running play.

Another thing is that I don't think you can manipulate these numbers in order to score yourself more points per game (3, as you noted). That seems like a huge reach. We all know that defenses toughen up in the redzone and taking advantage of this information isn't going to somehow help you convert a 3rd and goal at the 1, turning a FG into a TD, for example.

Chris said...

One note I'd like to make is I'm not entirely sure what the correct "passing premium" should be. I just know there should be one, and that several major college teams had rather small ones. The NFL guys had premiums around 3, though the better offenses were closer to 2 yards.

At some point I'd like to hang a number on a passing play's "turnover risk" per play, where turnover risk is the risk of an interception, the higher than normal risk of a fumble, and an incompletion's contribution to a higher chance of a long-yardage 3rd down that won't get converted. Once at least a rough value for "turnover risk" can be established then I think we can get closer to figuring out exactly what the passing premium would be (another way is to just run regressions of the passing premium to total points and wins).

Sto: Unsurprisingly, I disagree with you. First, I just pulled very rough numbers and I admitted that the numbers may be off since this analysis really matters most for 1st and 2nd down. That said, I don't think the 20 3rd and shorts per season will materially affect the numbers for a team that ran as much as Minnesota, and you failed to mention all the 7-8 yard draw plays they ran on 3rd and 10. Last, the QB's lack of ability did not get translated into the stats, so maybe, just maybe, they underused him by a couple passes a game.

Second, my theory applies quite fine to the great Nebraska teams. According to cnnsi, in 1997 Nebraska had a 6.2 run average on a whopping 755 rushing attempts, and a 7.9 pass per play average on 182 attempts. A passing premium of 1.7 makes perfect sense to me for a team that ran for 6 yards a carry. I'd bet that with Frazier in '95 NU's rushing per play was even higher. NU hardly disproves this approach, in fact it reinforces it. NU was a lot better at running and offense than Minnesota was last year, and--surprise!--even more balanced.

Anonymous said...

Chris, this article got me thinking about the passing premium last night and the cost of a turn over. In your opinion what is the equivalent yardage that you should penalize a QB's passing yardage for an interception. I imagine using this number to subtract from total passing yardage to create a more apples to apples yardage comparision with running. My guess is that 50 yds is about the right deduction for every turnover. Using that number you could adjust the yardage so that the passing premium might disapear and also explain why a particular team might have a higher or lower premium given their propensity to throw interceptions.

Check out this site I found this morning. It seem to indicate that interceptions and fumbles are about equal.

It also supports my 50 yds assumption. They estimate the value of a turnover at ~4 points. Since previous work indicates that every 14 yds of field position increases your expected points on the drive by a point I get 4*14=56 as the appropriate yardage deduction for a turnover.

Anonymous said...

Chris, this was a very interesting article that I read while sipping coffee during my vacation; very thought provoking. While your stats may indicate that there should be a premium placed on offensive balance (on the whole, something that I would agree with), I think that your numbers do not account for D&D and field position. Football is an entirely situational game, unlike baseball or any other sport. As a coach, I can say with absolute certainty that the value of yards gained in certain situations may be higher than another. For example Most statisticians will view two yards gained as two yards gained; nothing more and nothing less. However, the situation in which the two yards gained place a different value on those yards. Two yards rushing is more valuable on 3rd&1 as opposed to 3rd&9. The other situational factor left unaccounted for is one's field position at the beginning of the posession. From a raw numbers standpoint (raw yardage), a team that consistenty gets the ball at mid field or in their opponent's territory is generally going to have a lower output of yardage. This is even more pronounced in the red-zone, where the field is constricted and the advantage tactically goes over to the defense. Again, a two yard run on the +2 yardline is of higher value than a two yard run on 3rd&9 from the -28yardline.

With those facts in mind, as a coach, I am more interested in seeing how efficient my team's offensive output is relative to what the D&D situation is(run/pass separated). On 1st and 10 I want our gain per call to be at least 4 yards. On all 2nd downs, I want our gain per call to close no less than half the distance needed to convert. On 3rd & 4th down, we want to gain the yards needed for a 1st down or a touchdown. Any turnover or penalty counts as a point against. Although this analysis fits my needs from a general standpoint, I will also break this analysis into field position categories: 0 to -10; -11 to +30; +29 to +15, +14 to +6; +5 in. On the whole, an offense that is efficient in all situations is going to be on the winning side more often than an offense with a high output of yardage.

In closing, while I'd like to state again that although I thought your analysis is very thought provoking, I think that your numbers do not factor in the value of efficiency in situational yardage; and those situations are precisely what detemines whether or not an offense is contributing to a winning performance of a team.

Zennie said...

Hi Chris,

This is really interesting, but as I do with any argument I look at the initial premise because all of the evidence presented takes off from that.

I then look at what the premise is "missing" in its logic versus expected logic from actual conditions.

To cut to the chase, you very neatly wrote this while forgetting about time. It's as if you were thinking in two and not three dimensions.

In other words, runs are also called with respect to clock management and score. Coach Walsh was known for calling a variety of passes, working to establish a big lead, then running to "kill the clock."

If I look at the Niners and Stanford teams from your "two dimensional" perspective, then they look like balanced teams. But the reality is that's not true.

I think an adjustment in your otherwise well considered take is in order. I would restict my analysis to the first half of each contest to gain a better perspective of play calling patterns when the games are still in doubt.

More later.


Anonymous said...

When's more of this coming? This is great stuff. Much better than "we're going to go out there and pound the footbaw...etc." Really seeing what's happening between the stats is great. I've just started reading so I don't have anything intelligent to discuss yet.

Ted Seay said...

Chris: Just going over this post yet again, and reflecting how your thesis not only stems from Nash, but also how the same phenomenon is reflected in the strategic world by what Luttwak calls "the paradoxical logic of conflict":

The best road between two enemy cities is often the worst road for an invading army to take, since it lies directly along the expected axis of approach that the enemy has prepared for...

Anonymous said...

Just wanted to say that reading your stuff is outstanding and I am a VERY regular reader of your "columns". I was wondering if there was any way that you could give some kind of update, (maybe once a week? every two weeks?) to let us know what you are working on or let us know when some new stuff will be posted. I check your blog at least once every few days and find myself saying- " Please let there be another article, please let there be another article....." Again, I love your stuff and heave learned a TON from you. Thanks.

Anonymous said...


I just picked up a copy of Football Outsiders 2006 NFL Prospectus. One sectin of the prospectus contains league wide statistics going back to the 1940's. One interesting thing that I noticed was how constant the "passing premium" was usually between 2.5-3.0 yds. This held even when many fewer passes were attempted and when the average yards per play were lower. Thought that you might be interested in this since your article mentioned the higher pro passing premium.

Anonymous said...

If I had seen this before I published this recent article, I would have cited you. Nice work

Anonymous said...

The numbers for a truly dominant running team could arguably be flipped.

Better running average, combined with more attempts, and just enough passing average to move chains more than back teams off the ball.

Depth, formations with multiple backs, this would fit into the plan without meeting the road blocks or mind sets we develop off the stats we see.

The conceptual egg is after the chicken, in either case.

"These two stats converge in the most important first and second down stats, which are average yards per play."

These or correlative indicators.

First down success means you can take more chances on second down and that the opponent has less likely odds of stopping the play.

Still, this should be an agreed upon barometer that helps measure overall success. It certainly helps draw attention to the things teams need to focus upon coming into games from both sides of the ball.

Efficiency of offense, and for the defense, a way to take the other team out of the comfort zone of play calls and situations that form the basis of their playbook.