Last night, the Red Sox won 3-1, and are headed back to Boston with two shots to win one game. They are now the heavy favorites to end the season as the World Series champs, thanks in large part to Jon Lester outdueling Adam Wainwright for the second time in this series. The Wainwright/Lester match-ups, on paper, favored St. Louis, but Boston was able to beat the Cardinals best pitcher because Jon Lester threw two brilliant outings in this series. But, for some people, the thing that they’ll remember most about last night’s game isn’t Jon Lester’s pitching, but instead, Jon Lester’s hitting.

Or, at least, Jon Lester being sent to the batter’s box with a bat in his hand; I don’t know that you can call what he does up there “hitting”. In his career, including the postseason, Jon Lester — career AL pitcher — has walked up to the plate 43 times, and in those 43 opportunities, he has made 43 outs. Back in 2009, he drew a walk, the only time he’s ever reached base successfully, but he made up for it in 2012 by hitting into a double play, bringing his totals of PAs and outs back into equalization. 21 of his 43 plate appearances have ended with a strikeout. He is, maybe, the closest thing baseball has to an automatic out.

And yet, with runners at second and third, in the 7th inning of a one run game, Jon Lester was allowed to hit. With Mike Napoli sitting on the bench. With Adam Wainwright tiring on the mound. The Red Sox had a chance to turn a close game into a pretty sure victory, but passed on the opportunity for a big inning in favor of keeping Lester in the game for a couple more innings. And that decision is essentially a microcosm of how baseball is managed.

I don’t want this post to be too mathy, because this decision wasn’t really based on running the probabilities. But, briefly, we can lay out what the math says about these trade-offs, so we can understand what the Red Sox were giving up by not letting Napoli pinch-hit.

From last night’s play log, we can see that the run expectancy when Lester came to the plate was 1.34 runs. That would be what you’d expect with a Major League hitter at the plate, anyway. With Lester hitting, that number is much lower. It’s not zero, because if he strikes out or otherwise avoids a double play, Jacoby Ellsbury would hit with runners still in scoring position, and even if Lester can’t hit, there’s always some chance of an error or something wacky happening that allows him to reach base through an unexpected turn of events.

We know that the RE of Ellsbury hitting after Lester made the out was 0.57 runs, and since the out wasn’t a guaranteed outcome, the actual RE of having Lester hit in that situation would be higher than that. Maybe it’s even as high as 1.00, once you account for the fact that maybe he could have gotten down a squeeze bunt or something; he does have 5 sac bunts in his career. But, again, we’re talking about a guy who has never gotten a hit in the big leagues, and most of his at-bats end with a strikeout or a groundout. The odds of a positive event occurring with Lester at the plate were very, very low.

Alternatively, Mike Napoli is a pretty terrific hitter. Even with the pinch-hitting penalty, you’d expect him to perform above the level of an average hitter in that situation, so the run expectancy with Napoli hitting is probably in the range of 1.40 to 1.50 runs, depending on how much of a penalty you want to add for Napoli having sat on the bench up to that point. The actual numbers aren’t that important, as any reasonable calculation is going to result in a massive difference in expected offense with Napoli hitting instead of Lester.

Just for sake of argument, let’s assume that there’s a 0.4 run difference between having Napoli hit and having Ellsbury hit. How much worse would you think the relievers being asked to get the next six outs would have to be from Lester just to even that decision out?

Let’s use some names to make this more clear. Over his career, Clayton Kershaw has allowed 2.82 runs per nine innings pitched. He’s the best starting pitcher in baseball, and on average, he’s allowed 0.63 runs for every six outs he’s gotten. Let’s say the Red Sox would have given the ball to Brandon Workman and asked him to get two innings, because they don’t trust Craig Breslow right now and because Junichi Tazawa died in between innings or something. And we’re going to ignore the fact that Workman’s peripherals were actually decent, and that he worked as a starter during the regular season which would underrate his performance as a reliever, so we’ll use his 4.97 RA per 9 as the basis for his expected performance.

With those poor assumptions about Brandon Workman’s abilities skewing his value downwards, he would be expected to allow 1.10 runs per six outs. The difference between Workman’s regular season RA9 and Kershaw’s career RA9 is 0.47 runs per six outs, or almost exactly the same difference as we would estimate the gap was in letting Lester hit versus having Napoli hit.

In other words, to justify the decision from an empirical standpoint, you’d have to believe that Lester was capable of throwing the next two innings like the best starter alive — and ignore the fact that he was going through the order a third time — and that Farrell would turn to a roughly replacement level reliever to get six outs without going to Tazawa or Uehara if Workman started struggling.

These are not reasonable assumptions. Lester isn’t Kershaw, and Workman isn’t a true talent 5.00 RA9 guy, especially coming out of the bullpen. And Farrell wouldn’t turn to Workman for six outs without going to Tazawa or Ueahara, both of whom are pretty excellent relievers and would immediately skew the results back towards pinch hitting. But these are the assumptions we’d have to make in order to make this a math-neutral decision.

So, yeah, this wasn’t about playing the probabilities. This was part of the manifestation of baseball’s emphasis — really, this happens in every sport — on lead preservation. This decision is not that much different than the way that the evolution of the closer has taken place over the last 30 years.

There’s little question that a relief ace can make a larger impact by coming in to a tie game, where the game could be lost on one swing, than he could in getting three outs to protect a three run lead. Nearly any reliever in baseball can usually protect a three run lead with one inning to go. Yet, because that setting is defined as a “save situation”, teams employ their best relievers to protect leads that would almost always stand up anyways. It’s not a decision that works from a probabilistic model, but instead, one that is built around avoiding as many memorable collapses as possible.

It’s risk aversion, essentially. There was absolutely a risk involved in lifting Lester after having just thrown 69 pitches and replacing him with a series of less trusted right-handed relievers. If Napoli had failed to produce, the Sox would have likely have had to throw a series of right-handers at a line-up that is much better against right-handed pitching — since Breslow has pitched himself out of the Circle of Trust — and would have set up Matt Adams to be able to get the platoon advantage when he pinch hit. Betting on Napoli comes with downside, especially because his most likely outcome at the plate is making an out and the move not producing any positive results.

But it shouldn’t be enough to just say that a move is “too risky” without weighing the offsetting benefit of taking that risk. Even if you think the cost of removing Lester and replacing him with relievers (seen to be inferior) is extremely high, the benefit of having Napoli hit with two men on base is also extremely high. Mediocre relievers protect three run leads at a higher rate than good starters protect one run leads. The value of adding on, in that situation, was remarkably high.

That isn’t how sports teams are managed when they have a late lead, however. In football, teams with the lead often still punt on 4th-and-short, when going for the first down would allow them to retain possession and run valuable time off the clock. They trust their defense to hold, even when the alternative would be to not have to trust their defense to begin with. In basketball, players protecting a late lead often will pass up a chance at a guaranteed two points for the right to dribble around for an extra second or two and then have to earn those free throws at the line, even though raising the scoring margin would make it much less likely that the opposing team could close the gap in the remaining time.

This isn’t a John Farrell is stupid thing. This is how sports teams operate when protecting a lead. It’s why we have the modern day closer, even though it not an optimal usage pattern for a team’s best relief pitcher.

From a probabilistic mentality, these decisions are kind of nuts. Increasing your lead is just as valuable, if not more valuable, than maximizing the skill of your defenders in protecting the lead, but that isn’t how teams operate. When they have a lead to protect, they get defensive, and cost/benefit analysis becomes simply a cost analysis, with the benefit of being aggressive in expanding a lead taking a back seat to maximizing run prevention.

I don’t know that we’re going to see changes to these mentalities any time soon, because in reality, these strategies work more often than they don’t. Jon Lester held the lead, and the Red Sox won, so besides the segment of seamheads who really enjoy discussing the theory of baseball, this isn’t even an issue to be discussed. Had Lester blown the lead, well, Farrell trusted his ace, and you can’t blame him for that, right?

I think this is one of those inefficiencies that we’re probably just going to have to accept as part of baseball, at least for this generation. Pinch hitting Napoli for Lester almost certainly would have increased the Red Sox chances of winning the game, but it’s the kind of move that simply isn’t part of the sport’s culture, or any sport’s culture, at this point in time.