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?
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”.
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.