Anyway, back to my point: we weren't arguing about analytics or intuition. You were saying that momentum exists, and my point was that, even if it did, why would it argue in favor of one decision or another, since any decision you make could cause some sort of momentum swing?
I think the reason it matters is because the outcome doesn't provide equal momentum swings. Just as
@Neil Beaufort Zod pointed out with his pizza example, the momentum gain by picking up the 4th down at that point in the game with that lead wasn't as big of a swing in overall momentum as the gain the 49ers would receive by stopping Detroit from picking up the first down. So part of the choice that should factor into the decision to go for it is what kind of swing in momentum could occur if we don't get the 1st down. Is that gain by your opponent worth the risk of going for it over attempting the FG?
Lets say momentum has a scale of 100% Detroit all the way to 100% SF and it's a sliding scale. So if at the point of the decision for the 4th down attempt I would estimate that the point of momentum on the scale was at 45% on the Detroit side of zero. SF had just kicked a FG after a decent drive to bring it to a two score game. Det still had control but SF was gaining ground on the MoScale. By my estimations going for the 4th down and getting it would slide the scale to 50% Det....a 5% gain. On the other hand, a 4th down stop would be a 40% gain by SF bringing the scale all the way down to 5% Det. They would still have the lead by two scores but a big ***** in the armor occurred as the SF defense showed they could hold the offense that gambles and wins at 75% on 4th and 3 on the year. Big Win for SF. As a coach the law of diminishing returns comes into play and while the analytics shows a slight advantage in EV to go for it, it gets outweighed by the downside of not making it as it gives SF a boost that I would want to stay away from.
Compare that the FG attempt. Det is at the same 45% on the scale and I would say making a FG increases the MoScale by 2% while missing it would be be bigger for SF but not as big as a 4th down spot. Because the defense did hold them....the didn't really stop them. So I would put that MoScale adjustment to a swing of 20% shifting the scale to Det at 25%.
Now in order to quantify these analytics we would need to go over play of every game and decide on an average percentage point gain/loss for every scenario there is. Bring in psychologists, ex players, coaches, fans (because fans do contribute to momentum) to figure out how this calculates out. You would need to know each players internal drive/emotion etc. It would be interesting if it could be done. I think this is what the analytics side is trying to get at. How do you quantify this swing for each outcome so it can go into a mathematical formula to come up with an ultimate "right" decision tree. But because there are 100's of humans involved in this it makes the variables incredibly hard to quantify. Each human's makeup is different so there isn't a way to know how much confidence (momentum) a stop gives to SF vs loss of confidence to Det. It's very complicated for sure. But it does exist.
NOTE: Percentages in the example are estimated based on nothing but my football viewing/playing experiences over the years. It did not include any actual historical data. No animals were harmed in these studies.