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NFL Decisions: 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 cooperates 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.

.nobrtable br {display: none} table {border-collapse: collapse; border-width: 1px 1px 1px 1px; border-style: solid; } th {padding: 3px} td {text-align: center; padding: 3px;} #logocell {padding: 0px 3px 0px 3px; } #colorcol {background-color:#ffffe0}



4, 4
1, 5

5, 1
2, 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 defect-defect 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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