Nash Equilibrium and 3rd Down Strategy

Today I’m following up on yesterday’s post.  I encourage anyone who hasn’t done so to read that before moving to this post.

Now, onto the important stuff.

I discussed how we can apply Game Theory to NFL 3rd Down Strategy, highlighting what appeared to be a large inefficiency in 3rd down play-calling.  Basically, we should expect the Run/Pass play call breakdown to reach a point at which both options are equally likely to succeed.

This is the point of equilibrium.

I did some more digging into the numbers, the result of which is better resolution and more actionable intelligence.  Yesterday we looked at all plays run on 3rd down with between 1 and 5 yards to go.

Given what we know about the NFL, this is a very wide range for yardage; teams should run more often on 3rd and 1 than on 3rd and 5.  Below, I’ve broken the data down into smaller parts.  The results are encouraging.  Here is the complete table.  Note that I’ve put this together under the assumption that the original ranges are inclusive of each smaller range (3rd and 1-5 INCLUDES the 3rd and 1 stats), if that’s not the case, then this data is useless.

Screen Shot 2013-07-18 at 10.14.35 AM

See why I find it encouraging?

Starting with the bottom row (3-5 yards), we see that the NFL has, in fact, reached the expected equilibrium (close enough).  In a 3rd down situation with between 3 and 5 yards to gain, run and pass plays are both equally likely to succeed.

Remember the Tecmo Bowl Model, though.  As we can see, the equilibrium point occurs at a Run/Pass split that his HEAVILY tilted towards the Pass (18%/82%).  This is expected.  The important part is that play-callers appear to be operating efficiently, that is, calling runs and passes with the theoretically correct frequency.

This could be a coincidence, but given that we made the prediction beforehand and its the logical extension of yesterday’s theory, that seems unlikely.

Now the important part:

Screen Shot 2013-07-18 at 10.21.30 AM

While the league looks to have hit its Nash Equilibrium in 3rd and 3-5 yard situations, it DOES NOT look to be operating efficiently in 3rd and 1 and 3rd and 2 situations.

Again, our theory is that at the league level (a very important point), the call rate on 3rd downs should naturally evolve to a point at which the average success of both run and pass are close to equal.

Given the information above, teams should be running a lot more often on 3rd and 1.  Even though they are already calling a run roughly 75% of the time, the success rate of such plays is still significantly higher than for pass plays.

So….teams should run more, which will cause defenses to adjust by “expecting” the run more (Tecmo Bowl), which will presumably lead to Pass plays being “less expected” and therefore succeeding a higher percentage of the time.  Over time, the success rates for both run and pass should converge until they reach equilibrium.

Looking at the 3rd and 2 situations, we can still see some inefficiency, though it’s not as severe as in the 3rd and 1 situations.  Again, the process should be the same.  Offenses should run more, defenses should then commit to the run more often, and pass success rates should increase (while rush success decreases) until both Run and Pass have an equal chance of success.

It’s very important to note that this is League-level data.  Therefore, applying it to any individual team is tricky.  We can NOT say, for example, that the Eagles should definitely run more than 75% of the time in 3rd and 1 situations.  All we can say is that the LEAGUE should run more than 75% of the time in these situations.  It’s certainly a logical step to then take it to the team level; but know that it becomes more difficult at that point, since you have to weigh individual strengths and weaknesses as well as the relative strength of the opponent.


– NFL offenses are not running enough in 3rd and 1 and 3rd and 2 situations.

– This is a short-term opportunity to exploit an inefficiency in the game.  Once offenses adjust, defenses will too, with the end result of equal success rates and no advantage.

– While I didn’t discuss it here, teams should likely ALSO be running more often in 3rd and 3-5 yards situations, though for a different reason.  I haven’t run the numbers yet, but my guess is that once we incorporate 4th down opportunities into the equation (especially 4th and 1), running will still carry the higher overall success rate (and therefore need to be used more until the defense adjusts).

– This, along with 4th down strategy that we previously discussed (see tab on menu bar), is an area ripe for a “forward-thinking” coach to take advantage of.


13 thoughts on “Nash Equilibrium and 3rd Down Strategy

  1. I can’t help but wonder if your definition of success is skewing your findings. I can’t argue with the notion that teams should run more with 2 or less yards to go, so long as success is defined purely as getting a first down. However, as you mentioned, teams like to “take a shot” when they can expect single coverage on a speedy receiver, etc. Such plays have an inherently lower success rate but the reward, in terms of yardage, is significantly higher, as the charts you posted yesterday show.

  2. Me again. Sorry. I saw something from Al Borges (OC at Michigan) recently that bears on this, I think.

    “People sometimes don’t understand the value of a failed play. Sometimes the defense overdefends a play and gives you another play by doing so. So you may run a run in there and it doesn’t gain anything, and obviously people say, ‘Quit running the ball up the middle!’ How many times do you hear that? ‘Don’t run the ball up the middle!’ Well sometimes running up the ball up the middle will afford you the opportunity to pull the ball out and throw the ball down the field, because people are so aggressive with playing that play up the middle. I call it the residual effect of football plays. What’s the leftover effect of what we just did?

    “If both plays don’t work, then you probably have a problem. Either the plan wasn’t good or your execution’s off. There’s only two ways plays fail. The plan isn’t good or your execution is lousy. Overdefended, underexecuted. That’s why plays fail. But you have to understand that a play, just because it fails, doesn’t mean it’s a bad play. It may give you something down the line. For example, if you ran the ball into the line of scrimmage and gained a half a yard. But the play-action pass off that play gained 35 yards. What’s the average of the two plays?”

    … 17.75?

    “Would you take that?”

    I’d take that.

    “Not a man in the world wouldn’t. And that’s why you have to understand, that’s how it works sometimes. It costs something at times to get to that 35-yard gain.”

    • Good points. The success issue is real (i.e. teams not valuing 1st downs as highly as I expect).

      That, of course, brings up another point: How highly should teams value 1st downs (my answer would be VERY).

      I think a lot of coaches sometimes get seduced by the notion of “setting things up”. Reid, in particular, seemed to try very hard at this. It’s definitely an important strategy (goes into the whole “expectation” game), but I’d question the value of say, running a sub-optimal third down play in exchange for potentially setting up a bigger play later.

      In the example from the coach above, I think it’s a really foolish response. Not all yards are created equal. I’d rather have two 8 yard gains that both result in first down than a 17 yard gain couple with a failed 3rd down conversion.

      For example, look at the first down numbers I showed a couple weeks ago. New England and Indy consistently gain first downs with short plays, as opposed to the “big-play” offenses.

      This response itself is probably worth a whole post.

      Also, lots of other variables here, and im currently digging around more for other explanations.

      On Thu, Jul 18, 2013 at 1:36 PM, Eagles Rewind

  3. “That, of course, brings up another point: How highly should teams value 1st downs (my answer would be VERY). ”

    Very? We expect you to do better than that! A huge challenge, that: how to value a first down. (Hey, it was your idea, and I would love to see you lay out your thought process.)

    In fairness to Borges, he was talking about the trade-off between a no gainer vs.a 35-yarder, not two 8-yarders vs. a 17-yarder.and further, that when a defense over-defends in one place it must, of necessity, under-defend somewhere else.

    Getting back to the “very ” business, can it be correct to select a play that is less likely to get a first down than another that is more likely? I think the answer is “yes” but i don’t know how to measure it. I think your last two posts are appoaching the answer but you’re not there yet.

    BTW, I’m loving this your line of analysis.

    • Yeah, definitely didn’t intend to leave it at that!

      Borges’ example was different, but I thought mine was a clearer example of the “not all yards are equal” aspect. Needs a lot more work. I’m sure it’s true, but the trick, obviously is calibrating it.

      Glad you’re enjoying it though. Importance and value of first downs is in my head, ill post something when I get a good angle on it.

      On Thu, Jul 18, 2013 at 2:38 PM, Eagles Rewind

    • Nice, thanks for the link. Ill take a look. I alluded to it in the post, but I expect that incorporating 4th down options will skew the equation more towards the run (definitely an extension of the expected points angle, since that’s what my 4th Down Strategy chart is built from)

  4. I think Justrelax is missing a big point. This is an analysis of Third down. You don’t set up that potential 35 yard play action with a no gainer up the middle on 3rd and 1-5.

    Brent, can you apply this analysis to red zone/goal line? I’d love to see that.

    • Well you shouldn’t, thats for sure. Doesn’t mean coaches don’t do it. Im going to try to mine this particular line of thought for a lot, so im sure red zone/goal line will come up.

  5. Bill, I was quoting Borges, who did not make mention of 3rd and 1. Nor, in fact, had I.

    In terms of the Eagles, and this is so small a sample size as to be purely anecdotal, two years ago they had the league’s best 3rd-and-1 conversion rate. Last year it bottomed out, this with the same HC and OC.

    Did the coaches forget how to run on 3rd-and-1? No. They knew how. They also knew, however, that it was Demetress Bell or King Dunlap at LT instead of Jason Peters. That made a world of difference. With Peters, the run was optimal; without, it was, um, suboptimal.

    Brent’s analysis is, of necessity, an all things being equal approach (10 years 32 teams, aggregates) and it is a beautiful thing. He’s got me thinking in ways I had not done before. I am not quite yet convinced that his breakdown, even of 3rd-and-1 or 3rd-and-2, tells the whole story. He makes a compelling argument, but not quite yet (for me) a convincing argument. Expected points is a good concept, but I am suspicious of it, simply because it is all after-the-fact analysis. It smacks too much, to my limited brain, of coaches looking at a chart to tell them when to go for 2. I don’t know how to say this except to say that I feel there’s an invisible straw man in this whole argument…and I can’t see him. (Well, duh, that’s because he’s invisible.)

  6. Pingback: Run/Pass Game Theory; Optimal 3rd and 1 play selection | Eagles Rewind

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