Shortly after Super Bowl XLIX,  there was an interesting opinion piece in the New York Times by Justin Wolfers. The subject was Game Theory application in the waning seconds of the recent Super Bowl. You can access the article here:

http://www.nytimes.com/2015/02/03/upshot/game-theory-says-pete-carrolls-call-at-goal-line-is-defensible.html?_r=0&abt=0002&abg=1

I agree with the main premise of Mr. Wolfer’s position – namely that from a game theory strategy perspective a “mixed strategy” (i.e., one without a lean in either direction – run or pass – and indeed one which is applied randomly when outcomes are likely to be similar) is the best approach. There are however additional game theory points to be made and a more advanced application of Game Theory methodology sheds additional insights.

First, a few indisputable points …

• Time was of the essence. This was a function of a conscious decision by Belichick to not call a time out and let time run down. The result is that there was time for three plays of which two could be run plays and possibly… but not with certainty… three run plays.
• The ball was at the one yard line and it was 2nd down with the clock running.
• Marshawn Lynch was roughly 1 for 5 at “punching the ball in” from the one yard out during the preceding games for Seattle in this season. However, New England was not necessarily known as a stalwart defensive line team … the lone exception being nose guard, Vince Wilfork.
• An analysis of all NFL games for the season indicated a very similar rate of success (defined as a touchdown) from the one yard line, whether running or throwing. That is to say the “expected payoff” of either a run or a loss from the one yard line was virtually the same.
• Seattle had one timeout remaining and trailed 24-28 with 26 seconds remaining.
• A touchdown (and extra point) wins the game and would require a Patriots Field Goal for overtime. Merely kicking a field goal does not earn enough points to go into overtime and as such the option was fully off the table. Basically, the game was on the line (figuratively) and on the one-yard-line (literally).
• New England’s defensive posture was a heavy package with run-stopping lineman and linebackers. Essentially New England was posturing to defend heavily against a QB sneak or a straight power run (presumably via Lynch). New England wanted to hold or push back the line of scrimmage at the expense of more space and opportunity on the edges of the playing field and at the expense of a more structurally effective pass defense.
• In game theory terms, this is a simultaneous move scenario with a maximum of three sequential moves given the time constraint. This is a zero sum game – one team wins and one team loses.
• In game theory terms, there is neither Nash Equilibrium nor strictly/strongly dominant strategy, nor is there a weakly dominant strategy. Basically there are pros and cons of running or passing on offense and stacking the line or nickel/dime packaging in defense. No clear play or strategy on either side exists.

two more salient points …

• He believes that a big part of Lynch’s success in the regular season & playoffs – and indeed that of the Seattle offense – stemmed from Seattle having a strong “mixed” (i.e., random) balanced approach to offense. Basically, Seattle was likely to run between the tackles (again, Lynch) or use a short-pass or edge run plays (QB Russell Wilson scrambling, read-option run, screens, etc.) or use a medium- or long-passing strategy. This mixed and balanced strategy was largely evident when down and distance did not necessitate a type of play call.
• He is also skeptical – as am I – that Lynch gets one yard on a single given play. The fact that he had running success earlier in the game is not conclusive; those yards may have been gained against a lighter line/linebacker package. The one-for-five for the season seems suspect as a data set, and I believe that New England’s run-stop/heavy package was not their strength and may well have been inferior to the average NFL team. I believe that Lynch – In full beast mode – gains one yard 50% of the time, if both teams are postured for a power run.

Now for the fun part … 

 1. The role of Credibility and Marshawn Lynch. One common game theory application is the classic “prisoner’s dilemma” whereby two partners in crime are separated. If they both keep quiet, they both walk free. If they both incriminate themselves and the other charged prisoner they both face moderate punishment. If one talks and other does not then one gets a light punishment and one gets a heavy punishment. What makes this work for police forces is the separation of the prisoners so they cannot collude and the credible threat that each will self-optimize.

We now return to the Super Bowl …

It was absolutely credible that Lynch might be chosen to run the ball one yard, and he broke the huddle as part of the offense, so at least initially it was credible. New England’s decision to go “heavy package” endorses the assertion that running was initially a credible threat.

2. The role of making a commitment. When New England committed to a heavy package defense, they sought to seize a first mover advantage. This was ultimately significant as it likely factored in the decision by Seattle to pass the ball.

3. The role of a conditional response. This was a result of the aforementioned commitment by New England. Seattle set up in a bunch-set wide receiver formation and moved Lynch a bit wide in the backfield. The lack of heavy package substituting by Seattle – which would have taken valuable seconds off the clock or forced Seattle to spend their last time out – suggested a pass on the play (second and goal from the one-yard line, 26 seconds remaining).

4. Reputation. (A) Coaches: In game theory terms, both coaches lean toward a mixed strategy, which contributes greatly to their reputations as innovators or even geniuses.
(B) Teams: Seattle’s trading away key receivers this year established a bit more of a “run-first” and “be physical” team ethos. Conversely, Tom Brady and the short-passing game to the little sprites running across the middle had more of a scheme and “finesse ethos” for New England.

5. Real risk versus reputation risk. For both teams, a very real risk of losing the game existed. Reputation risk – akin to ensuring that Michael Jordan takes the last shot when you are down by one point in basketball – can be strongly controlled by play selection. Even if Jordan misses the shot a coach could hardly be faulted for putting the game in the hands of the best clutch player in basketball history.

And then, for Seattle, it all went to hell …

• The decision to pass was fundamentally sound, but the play selected was simply awful. By passing first, an incomplete pass would stop the clock and would still enable two more plays with the benefit of a time-out still in pocket. Seattle coach, Pete Carroll, could go incomplete pass, stuffed run (time out), stuffed run. Game Lost. Sadness but not criticism of the coach. Reputation risk would be almost fully mitigated. Three plays would be guaranteed. But the pass play itself was bad on at least two levels.

1. The bunch set of wide-receivers and the moving of Lynch a bit to the outside removed the credibility of the power run and tipped off the New England Defensive Backs. A play-action pass likely would have caused the linebackers to not commit as quickly to pass rush/coverage responsibilities. This would have kept the linebackers from gathering in

the middle as they did. More critically, it would have helped establish credibility of run for the next two plays (in which Seattle may choose to pass given the New England alignment, package, etc.).

New England Defensive Back, Malcolm Butler, made a tremendous play but he also knew from the set-up of the WRs and Marshawn Lynch where he needed to focus. He jumped the route due to no disguise … and then he also made a magnificent play on the ball.

2. If not a play-action pass, then the decision to not use a very capable, mobile quarterback was poor. I can think of no quarterback better at the run/pass option than Seattle Quarterback, Russell Wilson. If this is done to the side with a Tight-End (the strong side) then one could make the case for an equally likely power run (with Wilson behind the offensive tackle and tight end) or a sprint edge-run or short-pass via the roll out. This decision would also enable a sequential mini-game-theory game between the QB and a subset of the players (on both teams) where a first-mover advantage could likely be gained by Wilson via a pump fake or hip fake, reading the other team’s defensive end (and that is one of the advantages of running an option).I do concede that a Lynch play-action followed by a multi-option roll out would likely fail … a play like this has many moving parts and takes time to develop, and on the one-yard line, the field is compressed. It is probable New England would have effectively defended, resulting in a sack, loss of yards, loss of precious time, Seattle to using their final timeout, or any combination thereof.

As such and from a game theory perspective, a choice of either a play-action pass (run fake to Lynch, lined up in a traditional running formation) or the QB option were better plays.

• From a reputation risk, a safer throw (not over the middle in a packed area) would have been much better. When the ball is thrown, barring penalty, one of three things can happen: A. The ball is caught by a receiver, B. The ball is not caught by the receiver, and C. The ball is intercepted.More so than the ill-fated slant pattern Seattle chose, a fade route to the corner of the end zone may have resulted in an incomplete pass, but it also would have mitigated the risk of interception. Furthermore, and incompletion would enable two more (likely running) plays.On the assumption that a single, one-yard, run play was 50% likely to succeed with both teams featuring a heavy package of linemen and Lynch running for Seattle, the expected value of getting the one yard over three plays would have been over 80% for the three-play series. Had Carroll and his staff chosen a three-play series as outlined above the result, at its worst, would have been losing the Super Bowl with no reputation risk. I’m convinced Seattle’s coaching staff squandered an approximately 80% chance of success.

The Final Gun …

While taking nothing away from an outstanding interception, anyone wanting an example of poor game theory put into practice need look no further than the final 30 seconds of the Seattle Seahawks 2014-15 season.