### 1st Down Game Theory: Equilibrium and Exploitable Strategies

by Mike Sommers

You are faced with a first and 10 on your own 20 yardline, what percentage of the time should you run vs pass? You are on the 40 yardline, what percentage do you run vs pass?

**Equilibrium Strategy**

This is a problem that is solvable, but it depends upon your objectives. I used average NFL data. The average pass attempt yields about 11.6 yards per catch with a 60.5% completion rate. I rounded this down to 11 yards and 60% completion rate. The average rush attempt yields 4.36 yards (2011 data) I rounded this down to 4 yards. Now if you approach the problem as an “equilibrium” strategy, your goal is actually to reduce the expected effect on the play to no change in score either way. For those familiar with “EP” (expected points), this doesn’t mean that the EP result would be zero, but that over the long run, the EP would not increase or decrease from it’s current levels. That is, no change in expectation of how many points you score, so if your expected points on first and 10 is .34, using an equilibrium strategy, it will remain .34. This is based upon the assumption that in the very long run, the opponents will either adapt or correctly anticipate how to adapt to exploit whatever strategy you come forward with, so the best plan is to just produce “average” results”. I don’t believe this to be true, but the equilibrium strategy is still valid as a “base strategy”.

By taking this approach, your opponent cannot easily exploit you, and if you effectively neutralize the offense and defense on both sides, your chances of winning are based entirely on execution except for the moments of the game that you deviate from the strategy. If you have an average team vs another average team, with this gameplan, and equal “variance” your win percentage will approach 50% over the long run. This means that if a run play puts you in a situation where you expect to lose about 0.05 expected points, and a pass gains .35, you need to run 5 times for every 1 pass or pass 1/6 or 16.667 percent and rush 5/6 or 83.33% of the time. This will “balance everything out” in the sense that the net gain or loss from any play is zero therefore you will not lose or gain, nor will your opponent if he is average. The equilibrium is a base strategy, from which you can occasionally deviate from in order to exploit an edge, but the strategy on its own if run on both offense and defense, would neutralize to give you a 50% chance of winning vs. an average opponent.

Why would you do this? The GMs and owners would like such a strategy so they can evaluate talent based on a neutral or “average” offense. Your QB should get the league average if the strategy is designed on offense to replicate the strategy. Additionally, if you also neutralize the opponent to “average” on defense and have an offensive strategy that does the same, the game will be close and competitive. A competitive, closely thought game is something the owner wants because it’s good for ticket sales, and allows him to remain competitive for a cheap price. From a coaches perspective, an “equilibrium” strategy is neither good nor bad, but it may serve you well to conceal a more effective higher yielding strategy which you may revert to once you pick up on tendencies or “tells” by your specific opponent that you can exploit.

In theory, if you pass too often or run too often, your opponent should be able to recognize it and change his strategy on a given down and distance in a given situation. If you try to deviate from “equilibrium” in theory you become exploitable. It’s possible that NFL defenses are far away from equilibrium and unable to adjust effectively, if you believe this to be the case, you should go with a more aggressive “exploitative strategy”.

**Exploitative Strategy**

The alternative is an “exploitative strategy” where you force the opponent to try to balance out the advantages. Well actually, a pure exploitative strategy would ONLY do either a run play or a pass play in a situation, never a balance. However, this method does use a balance and manages risk by distributing the frequency of the play proportionally by the degree of success. This is more easily approached when BOTH strategy yields positive results. If for example, a pass attempt yielded .30 expected net points and a rush attempt yielded .15, you would pass 2/3rds of the time or .30/.45 or 66.67% and rush .15/.45= 33.33%. If one play was negative that same formula would suggest over 100% of the time and negative a percentage of the time. It would suggest borrowing from your run plays to pass. This is impossible without borrowing from another situation in which the situation is flipped, and impractical.

Neither of these strategies can be formed accurately 100% of the time based on “average” expectations because sometimes you have positive expectations from either a run or a pass, other times you have negative expectations for either, and sometimes you have positive expectations for both. Equilibrium strategy only works when one is negative, and exploitative strategies only work when both are positive. Otherwise, you will get a number over 100% when running calculation. However, if we take the situation where a run strategy is negative, we can simply only choose to pass if we are using an exploitative strategy. Or we can choose an aggressive balance in situations that suggest more than 100% of either one. I went with a 90% of any one when the data suggests over 100% and 10% to compose the other side.

I constructed a table. From the tables I constructed graph charts.

Finally I used a weighted average of 50% for each strategy and produced a chart combining the two. Here are the results of the strategy charted by yardline on first down and 10.

The number on the X axis represent from a player’s own given yardline (with 90 yardline representing opponents 10 yardline)

Near the goal line I took 50% of the value from having the ball on the 99 yardline and 50% of 7 points the value of a touchdown plus extra point.

The real distribution of a play is less predictable than for a pass play, 60% of the time the play results in exactly 11 yards and 40% it results in zero yards… and for a run play 100% of the time it results in a 4 yard gain. It would be a much more complex problem if I wanted to really accurately model it and consider the percentage breakdown and include the probability that the given play lands on each individual yardline and weight the expected points by the probability of the event and add them all up. But theoretically if the 11 yard averages is typically only 5 yards but skewed by a bunch of really long plays for 7 points it could drastically change the expected points per pass attempt and resulting strategy.

For those wondering, the reason the exploitative and equilibrium strategy are sometimes the same is because for example the expected point gain from a passing attempt might be .02 to a rush attempt of -.2 so you would need 10 passing attempt to equal 1 rushing attempt for equilibrium, but an exploitative strategy would also be a very heavy percentage of pass, if not 100%; so both might be around 90% in this case.

## 3 comments:

All the stat-oriented discussion of play calling I've seen pretends that plays are called as either pass, or rush in the huddle when, in fact, the quarterback frequently has the opportunity to make reads before the snap. Similarly, they assume that all rush and pass plays are equal, independent events, when, in fact, there's a huge difference in terms of expected outcome between say, a hail Mary and a bubble screen, and that play-action passing is silly unless you do sometimes run the ball.

The article mentions a Nash Equilibrium, but there's no mention about what the defense's choices, or the relative payoffs are.

As mentioned in the post, an exploitative strategy should never be mixed. (At best, it's ambivalent.) Yet the charts that are posted suggest that yours are.

Yes this is a pseudo equilibrium strategy assuming that you can have such a solution on other downs and that the defense can neutralize opponents edge as well.. It's based on several assumptions. It is certainly far from perfect. And yes, a pure exploitative strategy would be 100% pass or run. What I did was a model that uses one function of the simplified "Kelly Criterion" risk management strategy to exploit based upon long term growth. Pure exploitative strategies assume opponents won't make counter adjustments. So this one hedges against it but is maximally aggressive based on some risk management theory that probably doesn't entirely apply. I used it because I think always doing one thing is so entirely exploitable and obvious,particulately in football, that one would have to be brain dead not to adjust. However with some mixture weighted towards the strongest strategy to the exact degree that it has value over the other is the maximally aggressively "Kelly criterion" strategy. The idea is that you can exploit to the strongest degree in which you can still handle the volatility of opponents counter adjustments. You can also better disguise the fact you are exploiting due to variance. I feel anything beyond this would not be exploitable over the long run because the strategy would become entirely obvious, and vulnerable. Rather than label it "Kelly criterion based semi exploitative strategy" without taking a lot of time to get into Kelly criterion and it's applications, I just labeled it "exploitative strategy". I personally feel realistically it is the maximum exploitative strategy that you can put into practice in real life or very close to it. The balanced strategy is a less aggressive exploitative strategy that reduces variance. In areas the Kelly criterion is commonly used 1 half the Kelly criterion bet will produce 3/4ths the result as the full Kelly bet with 1/2 the variance. But this is a really I'm depth subject and there are entire books on it. It usually applies to money management, not expected points management and is not always used in game theory, except perhaps with regards to a poker tournament or blackjack card counting to a limited extent.

You are entirely correct about presnap reads, this is a subject I plan to get very in depth into. 8 men in the box or more makes passing easier and rushing more difficult in theory. I haven't seen an actual statistical analysis but it's pretty intuitive that there are solutions, yes passing plays has entirely different functions but if you want to get into specifics upon which one you have to read the coverage and know what percentage of the time the opponents defense actually sticks with the defensive coverage they show. Cover 2 would be vulnerable down the seems and the exploitative strategy would attempt the pass down the outside fade routes and corner routes underneath the safeties and 4 verticals against traditional cover 2, with a window underneath the MLB in a standard Tampa 2. Man cover two would require exploiting individual personel matchups, and depending on if the team uses inside or outside leverage might be rub routes (crossing drag routes) or whip routes (zig/zag routes). But there is also press man coverage and fade routes and slants or slants and gos work. Any natural pick routes works very well (Peyton Manning is the king of these combination routes that he will call at the line taking a very aggressive exploitative strategy that the

Peyton Manning got countered very effectively in the 2010 playoffs by the jets. What Peyton did to try to avoid the counter exultation was to call a bluff play, then call a play at the line after giving defense chance to stem coverage. the jets noticed in film that on average Peyton took around 7 seconds from when the OL was set until he hiked the ball, and they would stem coverage much later than normal, and as a result many of his intended exploits exploiting soft coverages suddenly would face press coverage, and press would suddenly face soft coverage look. His presnap reads could not be relied upon. the Falcons effectively mixed up the presnap look in 2012 season as well picking him off multiple times in the first quarter. Even so he is probably the best presnap exploitative QBs in the game. My "expected points" is based entirely on Brian's calculator which is entirely based upon past results. If NFL teams are making several mistakes this really is only a strategy that is I exploitable against the average opponent. In reality I think the expected points model will need to evolve to apply towards specifics. That is another thing I have worked on. A spreadsheet that allows you to customize individual probabilities of ending up in a given situation from another. That will determines probability of converting and then with some help of some other information determine your expected points on both offense and defense. You could then modify th spreadsheet based upon certain presnap reads and how your expectations change given that particular read. It would then produce "expected points" based upon more specific data. You can then develop strategies based upon some of these principals.... But there's a lot more explaining that would have to be done for people to really understand the functionality and how to estimate more specific conversion data. My hope is to publish a kindle book at some point, but I have a ton of editing and structuring to do.

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