me: the individual fumble is not break even, but there's a very non-zero chance it doesn't affect the outcome of the game
Gary: That seems counter-intuiti
me: sure, but look at facts
Gary: Winning Turnovers + Big plays has a high correlation with winning the game. That's my mental model
me: if your points for = points against, you expect to be 8-8. For every 2 points per game you gain or lose, your expected record changes by 1 game. This is actually a very strongly correlated line and that's the exact value. So the question is, how many points per game does a 1per50 fumbler cost you?
assuming he gets about 15-20 carries a game, that's 1 fumble per 3 games ...
Gary: But not all carries are equal. If you believe that fumbles in the 4th quarter hurt more than in the 1st, and that "the winning team" rushes more in the 4th ...
me: a fumble in the 4th could also be during a blowout when it definitely doesn't matter. In the aggregate, 2 points per game = 1 win change over a season. The intuition gets messed up cuz it's really easy to pin a loss on a fumble, but a lot harder to pin a win on the 4.5 yards per carry on the other 49 carries. However, they both matter. Think of it as an egg race: one guy is faster, the other guy never drops the egg. If the faster guy drops the egg, you think "my god, if he could just hold on to the egg". Assuming he only drops the egg once per 3 races, he still beats the slow guy. That 1 fumble may lead to a loss, sometimes a heart breaking one, but the better rushing average, in the aggregate, will lead to wins [unless the fumble rate is abysmal].
This was an ad-hoc conversation stemmed from the Arizona Cardinals' running back situation. New coach Bruce Arians has called out that the Cardinals' existing running backs are poor in pass protection. I was suggesting they should bring back Tim Hightower (who could probably be had on a minimum contract, and let's ignore his health status for a second). Gary argued that Hightower has serious fumbling issues. I was pointing out that Hightower averaged something like 4.5 yards per carry while the Cards' top backs this year were under 3 per attempt (and let's even ignore that both Wells and Williams struggle, current tense, with fumbles).
The true cost, compared to no fumbles, of Hightower's approximately 1 fumble per 50 attempts is not that much. Given his expected fulltime work load of 15-20 rushes per game, this breaks down to 5 or 6 fumbles over a full season. Each fumble might prevent his offense from scoring a touchdown or field goal, or give up the ball deep in his own territory, leading to an easy field goal or touchdown. Assuming the absolute worst case of every fumble being a touchdown swing, his fumbling would account for a net loss of about 40 points per season. A more realistic scenario is that he costs his offense a touchdown and a field goal, gives up short field position a couple more times (say, another touchdown and field goal) and the remaining 1 or 2 times the turnover leads to no points, nor was the offense near scoring position anyways. This would be a net swing of about 20 points per season, or about 1.2 points per game. Using the regression line, an entire season's worth of fumbles might cost the team a win. The only points you can absolutely pin on fumbles are impending points for the offense. There's no guarantee the other team wouldn't have scored points driving 80 yards instead of 30, after all.
This does not account for the 'momentum' factor, which is basically impossible to quantify. So let's stick to the numbers and conclude that a back with apparent fumblitis might not actually be that big a liability**
I didn't break down the expected outcome of running for 4.5 yards per carry instead of 3, but the former theoretically allows a team to advance the ball and convert on 3rd downs, while the latter leads to a lot of 4th and 1 situations. Judging by the performance of the Cards' offense in 2012, it doesn't work.
** - except in the post-season when "performance in the aggregate" stops being relevant in favor of "nothing matters but this game". Just ask Ray Rice.