The Linebackers' Dilemma

Say there are two inside linebackers on the same team, John Andrew and Dan Farley. They’re considered equally good players, and both are in a contract year. Their team is good, and one of the early favorites to go deep in the playoffs this season. Each one has some choices to make over the course of the upcoming season.

Both Andrew and Farley want a big payday. In order to get as big a contract as possible, they need a lot of gaudy numbers to point to—lots of sacks, QB hits, forced fumbles. Millions of dollars are on the table, and the temptation is strong. However, leading their defense to a successful year and going deep in the playoffs will increase their values too. But that can only happen if both Andrew and Farley play selflessly and put their team first.

On any given play, a LB can either guess at the play call, trying to get a sack or stuff a run. Or, he can play his responsibility within his scheme by reading and reacting, doing exactly what the coordinator expects him to do. In other words, he can do what his team is counting on him to do, or he can gamble trying to make a play himself, exposing his team to giving up big gains.

Sometimes when defenders gamble, things turn out well. Sacks and stuffs in the backfield may certainly help a team win. But over the long run, too much gambling doesn’t pay off. Further, to even have the chance that a LB’s gamble will be successful, it relies on the other LB doing his assigned job. If both Andrew and Farley gamble, there’s no one left at home to cover the middle routes, and their team is going to get burned badly most of the time.

Additionally, neither one can afford an injury this season. If they practice hard, they will help their team be better, but it will also expose them to more opportunities for injury. Each player has to make a choice between what is best for himself and what is best for his team.

Let’s call the two choices team-first and me-first.

The "Game"

In this situation, the interests of the two players simultaneously conflict and overlap. If we could put a number on how good each potential outcome is for each player, we could suppose the following payoffs:

  • Playing me-first while the other LB plays team-first is worth a 5. Going for the big contract while the other guy plays his expected assignment isn’t very nice, but it’s the big payoff every player waits for his whole career. (After all, who are the guys that say money doesn't matter and all they want is to win? The guys who have already been paid, that's who.)
  • Playing team-first by playing assignment defense, while the other guy goes for all the sacks would be a 1. He’ll be re-signed by a grateful coordinator, but he won’t get a big contract without a lot of highlight film, plus his team won’t do well.
  • If both players go for the big contract and neither one plays his assignments, they both would get just a 2. They’ll have some nice footage for their personal highlight reels, but neither one will particularly shine because no one is home to guard against big gains, plus the team won’t have any success.
  • If both players play team-first, the team will do very well. After each game of mutual cooperation, they’ll be a step closer to the championship, which ultimately would be the biggest payoff if it really happens. They’ll get lots of attention but tend to get more modest contracts in free-agency. Let’s say this is worth a 4.
Don't get too wrapped up in the exact payoff values for each outcome. They're just abstractions. Just accept that the payoff for selfish play can be higher than for cooperative play. The worst thing is to try to cooperate but be taken advantage of, and the next worst outcome is for both players to try to take advantage of each other.

What we have then is a classic non-zero-sum game. There are two players each with two strategy choices: For every week of the season, each LB must make a choice. They can either play team-first by practicing hard and playing their assignments, or play me-first by skipping practice and risking long gains in exchange for the personal glory of making big plays. The chart below represents the choices and payoffs for the linebackers' dilemma. The payoff to Andrew is the first number in each cell, and the payoff to Farley is the second number in each cell.

AndrewTeam-First4, 41, 5
Me-First5, 12, 2

Those who are familiar with game theory will quickly recognize this as the classic Prisoners’ Dilemma, named after the situation in which two robbery suspects are separately offered a deal by the police to turn on his partner.

A tragic outcome

The tragedy of this type of game is that it is always better to play selfishly, no matter what the other player does. If Farley plays team-first, Andrew should play me-first because 5 > 4. And if Farley plays me-first, Andrew should also play me-first, because 2 > 1. And the same goes for the other player. Team-first/me-first is not a stable outcome. What ends up happening is that me-first/me-first is the only “rational” outcome. Technically, it’s the Nash equilibrium, a fancy way of saying it's the natural and stable state when interests simultaneously coincide and conflict. It appears that, mathematically, it’s always better to screw the other guy over.

If it were a one-game season, that’s probably how things would turn out. But it’s a 16-game season, and Andrew and Farley have to play their dilemma over and over. This is known as an iterated Prisoners’ Dilemma, and it changes all the rules.

For instance, if during the first game Farley plays me-first and Andrew plays team-first, it’s doubtful Andrew would continue to play the sucker. He would be likely to defect himself in week 2. After a 15 more games of mutual me-first play, no one comes out ahead. Totaling up the payoffs would give Farley a 34 and Andrew a 29.

But if they both have faith in each other, and they can cooperate for the whole season, each would receive a payoff of 64 (16*4). They might even be Super Bowl bound. It would be the best outcome, but at every week of the season, the temptation still exists for both players to what’s best for themselves instead of what’s best for each other the team.

Greater than the sum of its parts

You've heard people say that a team can be "greater than the sum of its parts." The linebackers' dilemma shows why this can be literally true and mathematically valid. The "parts" are the self-interested individual players represented in the me-first/me-first cell, represented by 2, 2--which sums to 4. But playing selflessly is represented by 4, 4, which is plainly greater. The team-first/team-first cell totals 8, which exceeds the total of any other cell, and is the best outcome for the group as a whole. 

This is why trust and faith are so important on a team. And this is also where leadership can make a crucial difference. Part of a coach’s job is to convince his players to put the team first, essentially lifting the team from the me-first/me-first equilibrium of self-interest to the team-first mindset of champions.

A profound metaphor

Prisoners' Dilemmas surround us in everyday life--merging into traffic, conducting business deals, upholding contracts, criminal acts, keeping confidences, staying faithful in a relationship, or anytime you hear someone refer to a "win-win" situation. Prisoners' Dilemmas also a large part of international relations and global issues, involving topics such as free trade, over-harvesting of fisheries, pollution, and military standoffs.

The Prisoners' Dilemma is in many ways a metaphor for civilized life. It's difficult to overstate how profound it can be, and entire books have been written on the subject. It's often posed as the fundamental tension between self-interest and doing what is best for the group as a whole, but I think that misses a subtle but important point. It can also be thought of as the conflict between short-term self-interest and long-term self-interest. The Prisoners' Dilemma shows why long-term cooperation and foregoing the short-term gains of selfishness is the path to make yourself better off, which, as a happy byproduct, makes others better off too.

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16 Responses to “The Linebackers' Dilemma”

  1. Aaron Gordon says:

    Perhaps, but smart teams would notice a player engaging in a me-first strategy, which obviously decrease their value on the open market; it is bad business to invest in a me-first player. Obviously, it only takes one oblivious owner, which more often than not is either Dan Snyder or Al Davis. However, I would argue this is not a good example of a prisoner's dilemma, since every year we see players get big contracts who don't have gaudy numbers because they play team-first: Scott Fujita, Bart Scott, Jonathan Vilma to name a few.

  2. Jeff Clarke says:

    Quite a lot has been written on this subject in basketball blogs. The me-first players take a ridiculous number of stupid shots. They make a relatively small number of them but thats enough to get them high point totals. The team-first players make smart shots and pass when they don't have a reasonable percentage of getting it in.

    In theory, teams should recognize and reward team guys. In reality, there is a much greater correlation between points and salary than there is between efficiency and salary.

  3. Anonymous says:

    My only complaint with this article is that I don't think the pay off rates or correct for many schemes, to the point where for some teams, having a me-first + team-first gives a higher pay off than team+team.

    For example, I think some players like Asante Samuel, Darren Sharper (especially last year), Albert Hainsworth, and most 3/4 OLBs would have inflated me-first pay offs in the 7 or 8 range and/or reduced team-first pay offs.

    Haynesworth might be the most extreme case, with at least an 8 for me-first, and literally a 0 for team-first (IE won't be playing at all).

    It might be interesting to identify team-first players paired with me-first players, and examine their salaries. It would be very, very subjective, but potentially interesting.

  4. Alex says:

    The Prisoners' Dilemma is obviously a pretty powerful economic game, but I have to say that I have never once thought that 'greater than the sum of its parts' ever referred to playing within a team framework. I always thought it meant that a team did better that expected given who was on the team. Usually this is attributed to everyone playing a team game, as opposed to going for their own stats, but I've never seen strong evidence for it when players are evaluated in any kind of numerical sense, as opposed to commentators trying to explain why they didn't know before the season that team X would do really well.

  5. Anonymous says:

    Hi Brian,

    Will you be posting your weekly probabilities for the 2010 season on the New York Times' Fifth Down blog, or anywhere else, this year?

    Office pool launches soon. You're a great resources.


  6. Jeff Clarke says:

    Haynesworth is a really interesting case with this sort of thing. Basically, the only threat a coach has against a me-first player is to bench him. But when your own job is in jeopardy (and nearly all coaches lack job security), benching one of the fans' favorite players is an extremely risky move. As we all know, coaches are typically very risk averse.

    You know these guys are actually hurting your team but reporters are likely to seem them as the "best" player since they have the best stats. In most circumstances, coaches will ultimately have no choice but to stand down. This erodes their authority with the rest of the team.

    I think the Shanahan situation (and last year's McDaniels/Cutler feud) is all about establishing immediate authority. A new coach can do something like this, since his job is likely not in immediate jeopardy. It allows him to more effectively bluff when he wouldn't actually follow through in later situations.

  7. JMM says:

    A very interesting study on the iterated prisoners dilemma:

  8. Brian Burke says:

    Forgiving tit-for-tat takes the cake. I read that in a really old book somewhere.

  9. Unknown says:

    I'll see your Haynesworth with a Shawne Merriman; Merriman and (Shaun) Phillips loved (Wade) Phillips' schemes because, as Shaun once said, on obvious passing downs they were both just told (or screamed at) "Go get the quarterback!" Maximizing their individual payout was built into the defensive scheme, there were sacks, everyone was happy.

  10. chem says:

    Perhaps this book:

    a good one. Not just game theory, but history.

  11. JMM says:

    Or perhaps: Metagical Themas: Questing for the essense of mind and pattern. Douglas Hofstadter

  12. James says:

    @ Jonathan, the statistics which Merriman and Phillips were allowed to accrue came at the expense of production from other players, such as their ILBs (all mediocre producers besides Donnie Edwards way back) and their DEs (Olshansky and Castillo were regularly asked to just hold gaps rather than make plays).

  13. Brewmaster EB says:

    Is there any merit to a front office trying to plan around their LBs (or any position) having contract years at the same time to avoid this situation? It may be too difficult because there are so many variables- cuts, injuries, extensions, etc. Or do some front offices already do this?

  14. Jason D says:

    We Seahawk fans have discussed this very situation with our linebackers. David Hawthorne made a lot of big plays (5 takeaways) last year when substituting for the injured Lofa Tatupu. But our keen analysts on Field Gulls (the blog/discussion site) observed that Hawthorne was much less assignment-correct.

    I wouldn't necessarily judge Hawthorne's play as selfish, because it could be a combination of poor awareness and bursts of enthusiasm. But the front office seems to realize the difference-- they offered Hawthorne a paltry 1-year contract in the offseason.

    And the defense looks better with Tatupu back, too.

  15. Anonymous says:

    Prisoner's dilemma is a flaw in thinking. It is an egotistical thought that is self servicing and one of a mindset that "people are out to get you" so it works well in the situation that its actually used. However, the greater result for humanity is always the right choice, except in "theory" using principals of game theory.

    An autoimmune disorder occurs when the bodies cell engages in "prisoner's dilemma" style 'consciousness'. Yes consciousness on a limited level. Even electrons are self-aware so they don't collide into each other and cells are made up of tissues which are made up of molecules which are made up of atoms... and electrons. The cell will attack itself believing it's own cells to be an "enemy" rather than working in coherent harmony with the other cells based upon trust...

    When someone else operates outside of that "trust" they are hurting everyone, and although that is their mistake, you get hurt because of it. But I maintain that operating as if the other participant would collude to produce the best joint outcome is the right one. It certainly is the "healthy" and "harmonious one". And there are checks and balances to see to it that any system works efficiently and in harmony. Prisoners certainly are "unhealthy" in their thinking and reasoning. But even they have "enforcers/mafia" on the street as a "counter" to the "prisoner's dilemma"...
    In the NFL, coaches and teams are smart enough to see what's going on. They will yell and scold the players, and they will often bench them or not sign a more talented selfish team player for this reason.

    The major flaw in this is actually that it only looks at the immediate result. The negative things that can happen are that the "system" will self correct" and you will be the odd man out for taking a selfish view. You will get labeled a "cancer" in the locker room, a "selfish player" and you will have a harder time having a long career. You will lament not being on a winning team, and there will be someone on a different team that not only displays sound stats but also technique and team work which shows up on the tapes by which scouts look at. Over time, the "cancer consciousness" players that partake and subscribe to the "prisoner's dilemma" thinking will have worse stats in terms of how many wins they have and how they look on game tape, and "although they may win the battle they will lose the war."

  16. Anonymous says:

    I got a little off topic with autoimmune disorder, but the principal is that the body will look at other cells as intruders and behave as if the other cells are a "threat" to the resources. IN harmony the body would stay alive but when cells start attacking each other, they create a self fulfilling prophecy and they eventually kill themselves off and body either dies or has "spontaneously remission" where it heals and stops engaging in that behavior. In game theory you might say it discovers "colluding". But the principal is that if one engages in that prisoner's dilemma, it usually will be to it's own downfall and everyone's around it. A cancer celll grows too fast to the point that it eventually consumes all cells until either chemotherapy kills the cancer mostly and also the healthy cells or until the cancer consumes the entire body and grows so fast it hinders it's own ability for growth and "goes down with the ship"... Not behaving as if you would if you were to collude is a mistake. Since there is more than one instance of this "game", behaving in one's own best interest will FORCE the other party into behaving in his best self interests, and you will both "go down with the ship." However, if you behave as if you would if you were to collude as long as your teammate does the same, it is a much healthier and superior option. Of course, towards the end of the season, it may be worth "guessing" if it either appears evident that your team definitely is, or definitely isn't going to make the playoffs. Additionally, it is worth doing if you think it can get enough of a boost to get more tackles, but not if it won't. In reality, a team has no place for players that act in their own best self interest to the degree that they put themselves over the team, and the players that last in this business are team players.

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