A Prisoners’ Dilemma in the NFL?

Last night, the Las Vegas Raiders faced off against the LA Chargers in a decisive and unforgettable end-of-season matchup. The game itself was exciting and suspenseful, but the reason this game will long be remembered is because of the unusual context in which it occurred. Both teams came into the game with 9-6 records and could clinch a spot in the playoffs by winning the game. But there was also another path to the playoffs for BOTH teams ... to tie!

The Wall Street Journal wrote an interesting article on this (in which I am quoted), claiming that the “game” created by this unusual situation is a Prisoners’ Dilemma, and there was plenty of fun chatter on social media. But then the fourth quarter ended in a tie, and the teams remained tied going into the final drive of overtime. Suddenly, those theoretical discussions were beginning to feel a bit more real ... or did they? Rather than taking a knee and ensuring that his team would tie the game and advance to the playoffs, Raiders coach Rich Bisaccia sent out his kicker Daniel Carlson to win the game---even though doing so risked the possibility that the Chargers might block the kick and run it back for a touchdown.

In this post, I offer some reflections on what happened (and didn’t happen) in this epic football game, from a game-theory perspective, focusing especially on three questions: (i) was this is Prisoners’ Dilemma; (ii) why couldn’t the coaches escape this PD; and (iii) was it “irrational” for Coach Bisaccia to kick the final field goal??

1.??????WAS THIS A PRISONERS’ DILEMMA (PD)?

First, let’s start with the basics. Was this actually a Prisoners’ Dilemma (PD), or were people misusing that term here? This is an easy one. YES, this was a PD.

Why? Recall that a PD is any situation with two features: (i) each decision-maker (“player”) has a dominant strategy, an option that is always best for them no matter what others do, holding others’ strategies fixed, and (ii) all players would be better off if they all could somehow NOT play their dominant strategies.

Here, the PLAYERS are Coach Bisaccia of the Raiders and Coach Brandon Staley of the Chargers. What about their OPTIONS? The actual game is very complex, as there are countless strategies that each coach could adopt, but for simplicity let’s reduce those options down to the most basic decision---are they going to “play to win” (trying to get first downs to maintain possession and score) or not (kneeling each play)? In this simplified game, there are four basic outcomes:

Both play to win: The game is very unlikely to end in a tie. Since the teams are evenly matched, each has approximately 50% chance of winning the game and hence advancing to the playoffs

Both DO NOT play to win: The game will end tied 0-0. The coaches will likely endure some jokes at their expense, and perhaps some grumbles from players who want the chance to showcase their abilities on the field, but they each advance to the playoffs.

Only one plays to win: The coach who plays to win will win the game and advance to the playoffs, while the other loses and fails to secure a playoff spot.

The PAYOFFS in this game depend on how each coach ranks these four outcomes. Assuming that each coach cares most of all about advancing to the playoffs but also enjoys winning:

·????????the best outcome is when only they play to win [win + playoffs];

·????????the second-best outcome is when no one plays to win [tie + playoffs]

·????????the third-best outcome is when both play to win [50% win + 50% playoffs]

·????????the worst outcome is when only the other coach plays to win [lose + no playoffs]

We can represent these preferences over outcomes using an “ordinal payoff matrix,” with outcomes ranked from best to worst. (The Raiders’ coach is the “Row player” with strategies, payoffs, and incentive arrows shown black, while the Chargers’ coach is the “Column player” in blue.)

Note that, no matter what the Chargers coach does---holding what Chargers do fixed---the Raiders’ coach prefers Play to Win. Thus, Play to Win is a “dominant strategy.” Moreover, if both coaches could somehow both commit not to play their dominant strategy, both would be better off. Thus, indeed, this is a Prisoners’ Dilemma ... or “PD” as I affectionately call it. ?

2.??????WHY COULDN’T THE COACHES ESCAPE THIS PD?

As students learn in my class, there are FIVE conceptually distinct ways to escape the PD. (Those who haven’t learned about these “escape routes” are encouraged to read my book “GAME-CHANGER: Game Theory and the Art of Transforming Strategic Situations”. This non-technical introduction to game theory is perhaps the most comprehensive book ever written on the Prisoners’ Dilemma---the whole book is about this one type of game and all the ways NOT to end up playing the lose-lose Nash equilibrium.) Unfortunately for Coaches Bisaccia and Carlson, none of these approaches were practical options in their context.

?

FIRST, “REGULATION.” Is there some third-party (“the regulator”) with the power to change the rules of the game or create new incentives, who could be persuaded to intervene and help the coaches coordinate on their mutually preferred outcome? NO! If anything, the powers-that-be in this context (NFL, TV stations, etc) have a strong incentive to stop the coaches from making a farce of such an important contest. As WSJ reporter Andrew Beaton noted: “The NFL might decide that such a sham violates the league’s competitive principles and force both teams a forfeit. The teams could face the social pressure of becoming laughingstocks.”

SECOND, “CARTELIZATION.” The essence of “cartelization” is that, afterwards, players seek to maximize their collective benefit, rather than their own individual benefit. In this context, that would mean somehow changing how the coaches feel about the game so that their best outcome is when both advance to the playoffs. That might happen naturally if the coaches were close friends (or brothers, as happened in 2013 when two Harbaughs squared off in the Super Bowl). But unfortunately, that doesn’t seem to be the case with Coaches Bisaccia and Carlson---based on how the actual game ended, with Bisaccia twisting the knife with that final field goal!!

THIRD, “RETALIATION.” Do players have the ability to detect and quickly respond to the other player’s move, like the US and Soviet Union could during the Cold War when it comes to launching nuclear weapons? In that context, each superpower chose not to launch nukes but could have, on a moment’s notice, if the other side launched first. ... This is more promising for us when it comes to the NFL, since the coaches could indeed switch their strategy during the game. For instance, suppose that Bisaccia takes a knee on the first drive and punt it away, but then Carlson were to drive down the field on the second drive? Well, now all bets are off. Bisaccia will naturally respond by playing as aggressively as possible, throwing them into an uncertain outcome in which either could wind up winning. Could fear of such “mutually-assured destruction” keep both coaches in line?

Unfortunately, this also isn’t going to work, since the potential to retaliate dwindles as the game nears its end. And anticipating that cooperation is going to falter near the end of the game, each coach will have an incentive to break the truce even earlier. As Yale professor (and game theorist extraordinaire) Ben Polak noted to the WSJ: ?“There’s an incentive to defect from that agreement and try and win at the last minute.”

FOURTH, “PROMISE.” In settings where the players make their moves in sequence (“sequential-move games”), the players can escape the PD if the last-mover has sufficient credibility to commit ahead of time to how they will play the game. (For details, see Game-Changer or this article that I wrote on the power of promises for Frontiers for Young Minds: “Game Theory and Cooperation: How Putting Others First Can Help Everyone”.) Unfortunately, that isn’t relevant here, since the coaches are making their choices at the same time.

FIFTH, “RELATIONSHIPS.” This particular game is just one of many that each of these coaches will play during the course of their careers. In situations like this where the future matters to players, their incentives in a given game can sometimes be changed if (i) their behavior in the current game can be observed by those who they will interact with in the future and (ii) they will be better off in those future games if they choose NOT to play their dominant strategy in the current game. Such “relational concerns” are crucial in many games and undoubtedly play a role in the NFL coaching world as well. But unfortunately, if anything, the relational concerns that matter here act AGAINST throwing the game for a tie. In particular, if Coaches Bisaccia and Carlson were somehow to succeed in making a farce of the game and never running a bona fide play, their players might lose respect for them and/or feel embarrassed at the outcome, hurting morale and perhaps ultimately undermining their ability to having a winning season next year.

All in all, then, it’s no surprise that the Raiders didn’t take a knee on the first play of the game, and that the Chargers responded with their own best efforts to win. But what about how the game ended? Here we saw yet another puzzle.

3.??????WAS IT “IRRATIONAL” FOR COACH BISACCIA TO KICK THE FINAL FIELD GOAL?

With just one second on the clock in overtime, Raiders’ Coach Bisaccia could have had his quarterback take a knee. This would have GUARANTEED that his team went to the playoffs, whereas kicking the field goal introduced the possibility of disaster---that somehow the kick would be blocked by the Chargers and run back for a touchdown. What explanation could there be? Why would a coach make a decision like this that, on the face of it, would seem to go against the best interests of his team? Was this “irrational”?

A decision-maker is “rational” if they make their decisions in pursuit of an objective, i.e., if they are maximizing some objective. When people claim that someone is “irrational,” what they often are really saying is that that person’s behavior doesn’t seem to be maximizing the objective that they THINK that person should be pursuing. But in fact, people often care about many things and weigh competing interests. Moreover, there are often complexities and nuances beneath the surface---including other interconnected games being played!!---that create new concerns and interests.

For instance, in this case, Coach Bisaccia’s decision to kick the field goal would be “irrational” if his objective was solely to maximize the probability that the Raiders go to the playoffs. But in reality, Bisaccia undoubtedly cared about several other factors, e.g.:

·????????team morale: Nailing the coffin on the Chargers with that final field goal undoubtedly left the Raiders’ players pumped up and excited for the playoffs. Playing for a tie might have maximized the Raiders’ likelihood of making the playoffs, but decreased their likelihood of making it to the Super Bowl.

·????????desire to avoid a rematch: If Bisaccia goes for the win, the New England Patriots will be going to the playoffs instead of the Chargers. If the Raiders are better matched against the Patriots, this would benefit his team.

·????????personal animus: Perhaps Bisaccia dislikes Staley, or was pissed off in the moment and wanted to beat him. (The on-air commentators discussed this possibility, suggesting that Staley’s decision to call a time-out with about 30 seconds left might have affected Bisaccia’s thinking. Watch the video starting around 52:00 mark)

·????????personal embarrassment: Perhaps Bisaccia doesn’t like the idea of having his name forever linked to a quirky one-of-a-kind game, or feels pride in always playing for the win.

If any/all of these factors were in play, then Coach Bisaccia’s decision to go for the field goal would have been a rational move.