In a 1989 study, college students were offered three chances to acquire two goods: coffee mugs and chocolate bars. In the first stage, they were given mugs, then offered the chance to exchange them for chocolate. In the second, they were given chocolate, then allowed to exchange for mugs. In the third, they were simply given the choice to take either one. In all three stages, students were ultimately making the same choice: mug or chocolate. Yet, 89 percent of students concluded the first stage with a mug, whereas only 10 percent concluded the second stage with a mug. (Fifty-six percent of the students chose the mug in the third stage.)

These choices can be partly explained through the economic principle of loss aversion, which says that people prefer keeping what they have rather than gaining something new, and they're pretty pervasive in our everyday decision-making. (Remember this next time you're offered a free trial.) Football, in particular, is rife with these kinds of tradeoffs, and coaches often don't realize the fallacy of their decision-making.

On Monday, NFL commissioner Roger Goodell brought this concept to the forefront when he floated the idea of eliminating the formality of extra points, which would make a touchdown worth seven points and a two-point conversion an attempt to gain another point while risking a one-point reduction.

Of course, this doesn't change the basic math at all. Coaches would still be making a choice between risking a point for the chance to gain two, the same calculation since the two-point conversion was instituted in 1994. The only thing that may change is the way we think about the situation.

Currently, six points are added when a touchdown is scored. Technically speaking, at least, coaches aren't "sacrificing" a point for the chance to go for two, since they haven't acquired that point yet. The proposed rule would change that, so coaches would risk literally taking points off the board. You may think it's ridiculous that this simple reframing of the problem often changes people's choices. It is ridiculous. But it happens all the time.

Closely related to loss aversion is framing, or the psychological concept of how choices are presented to us. More people will elect to have an operation with a 90 percent survival rate than a 10 percent death rate. More students will register early for a class if it carries a $20 penalty for late registration than if it has a $20 discount for early registration. The way problems are framed carry great importance in human decision-making, and coaches and fans are people, too.

An offense generally perceives a five-yard run on first-and-10 as a success, while a defense often considers a five-yard pass on first-and-10 a success. This makes no mathematical sense; how can one play be a success for the offense and the other the defense if they have the exact same result?

What affects our analysis is the way we think about those plays. Running the ball is often considered to be a low-risk, low-reward call. Once the ball is handed off, we intuitively know that an average running play yields between three and four yards, so anything more than that is icing. Plus, the visual of a man running into a giant pile of other men and emerging from the other side gives the visceral impression of success.

Passes are riskier, but have a higher per-play yield (NFL quarterbacks this season had a yards-per-attempt range from 6.3 to 9.1). When the quarterback drops back, our expectations suddenly increase, and a five-yard gain is actually below average, so the defense is pleased. Likewise, a five-yard completion falls in front of the defense, so visually we have the impression that the defense "stopped" them, since the ball did not advance beyond the defense's structure. Nevertheless, both plays result in a second-and-five.

Also consider the fourth-down attempt, a favorite of economists for decades that is just recently beginning to gain traction among NFL insiders. In 2012, Brian Burke of Advanced NFL Stats put this in the context of prospect theory, which is a related concept to loss aversion:

*"Prospect theory simply observes that we are about twice as upset to lose something as we would be happy to gain the same thing. If you misplace a $20 bill, you'll be twice as pissed at yourself as you would be glad to find a twenty on the sidewalk. Experiments show that this is a universal human tendency.*

*…*

*Coaches see failing on fourth down as losing the $20 bill and succeeding on fourth down as finding one. They look at that single decision in isolation from the larger game, magnifying the relative consequences of failure before they make their decision."*

This framing has a lot to do with the game's tradition. We view punting as the conventional play, so any deviation from that must be to benefit. As Burke has often said, it's easy to imagine a hyper-intelligent alien creature learning football's conventions for the first time and remarking, "Wait, you mean, you want me to *give up* one of my four chances to gain 10 yards and just give the ball back to the other team in order to gain a measly 40 yards? Are you crazy?"

The field goal is another prime example of this kind of odd thinking. Back in the days where extra points weren't completely automatic (the most extra points missed by one kicker in a season is eight), a field goal was essentially worth half a touchdown (and still is, if you think about it). Yet, most football fans -- and every commentator -- will tell you that, given a chance to attempt a field goal, you "take the points." Of course, this ignores the basic logic that if your chances of converting the fourth down and going on to score a touchdown are greater than 50 percent, the field goal is almost always a bad choice. But coaches still take the points because field goals are the mug you already have.

This mindset plays into the larger metaphor of football field as battleground: never yield territory, never give an inch, etc. For coaches, the game isn't so much about amassing points as it is hoarding them. (If the goal were to gain as many points as possible, the term "protect the lead" or "run out the clock" would be ridiculed.) Rather, coaches often think of football as getting a lead, then protecting it. This means they avoid risks as often as possible when sure points present themselves. This framework is very much supported by the fact that you cannot have points subtracted. Whatever you earn, you keep.

That's what makes Goodell's proposal so intellectually challenging. With a failed two-point conversion, points are literally taken away, a concept remarkably absent from major professional sports. Perhaps this will lead coaches to consider points in a different way, something advanced stats have been doing for some time with the concept of Expected Points, which considers the down, distance and field position and compares that situation to the historical result of those drives. For example, first-and-goal on the one will have very close to six Expected Points since teams often score a touchdown, whereas first-and-10 on your own 20 will be about a half a point.

This is a whole different framework for considering scoring in football, because it turns this mug/chocolate mindset against itself. Rather than considering drives worthless until a field goal attempt is possible -- then suddenly assuming you have three points until/unless a touchdown is scored -- the Expected Points model implicitly assumes points every possession. It's easy, then, to see why stats people love going for it on fourth down: Possession implies points, so why would you willingly give up possession?

With the tide already turning on fourth-down attempts, some advanced stats proponents have turned their attention to advocating for more two-point conversions. Their argument is that, quite simply, two-point conversions succeed more than half the time. It was always going to be an uphill battle to convince coaches of this fairly basic calculation, but it will be even tougher if it means they have to take a point off the board. After all, they do love their coffee mugs.