Imagine a world without weekends. No days off at the end of a long string of days doing whatever it is you do to gather your thoughts and rest. (Or at least do a different kind of work. The lawn doesn't cut itself, after all.) The weekend is nice, because even if you aren't "doing anything," you still get a reprieve from your job or classes or whatever you do the rest of the week. But imagine that the weekend was taken away. You just became a professional baseball player.

Baseball is unique—and beautiful, at least from the perspective of the fan—for the fact that teams play almost every single day. I can imagine that players might have different adjectives for it. Players do get the occasional Monday or Thursday off, but it's often a travel day, and travel is hardly a relaxing activity. It's always game day. The term "everyday player" does not exist in basketball or football, because, well, football players have a game once a week. On top of that, the game of baseball itself is one that requires immense amounts of concentration. In the field, a player has to be ready on each pitch to spring into action, despite the fact that on most pitches, he won't be needed.

Baseball is also largely unique in being the one sport where individual players are given scheduled days off on days when the team is playing and they are healthy and would otherwise start. Bench players in baseball should more properly be called "occasional starters," because they get to fill in on the days when the regulars need a breather. Recently, I've been looking into the idea of roster construction and how teams might allocate their roster spaces (and payroll) in a more efficient manner, and this is an interesting area to look into. How important is it that a team have a good bench, not only for the inevitable injuries that will happen during the season, but for the purposes of giving its starters a rest every now and then?

First let's talk about why baseball players need days off. Baseball requires sustained attention in the absence of a stimulus. In other words, baseball players wait around a lot, but it's not like they can mentally check out of the game. (Oh, hi Manny.) Sustaining attention like that is the job of my favorite part of the brain, the pre-frontal cortex (PFC). Like anything else in the body, when the PFC has been overused (in this case, those neurons have been firing a few million too many times), it gets tired and needs some time for repair. We usually call that sleep, but sleep doesn't fix everything, especially when the PFC is going to get another workout in the next night's game. As I've mentioned in other articles that I've written, the PFC also controls things like pattern recognition, higher-order planning, and impulse control, things that after a long day at work, you lose the ability to do as well.

When the PFC is operating at a reduced capacity, the body does not shut down. In fact, the parts of the brain that are responsible for things like breathing, visual perception, and movement are located elsewhere. The PFC is the part of the brain that coordinates them in a logical way. A hitter will still see the ball and can still swing, but he may be working more from muscle memory than an ability to tend to what's going on and then execute a plan. That means that reaction times will be a little slower and decisions won't be as good. In a game that depends on quick reactions and good decisions, that can be a problem. Ever wonder why you're hearing more about the Adderall epidemic in baseball?

Let's look at what happens when players don't get the occasional day off.

Warning! Gory Mathematical Details Ahead!
I took a couple of different measures of how fatigued a player might be. I counted a player as having played in a game if he came to bat, which does exclude guys who played an inning or two as a defensive replacement. I counted the number of games that he played in the seven games prior to a given day’s game (and in the previous 14 and 21 days, although those both behaved in much the same way), the number of days since his last day off (even if a naturally occurring off-day), but capped it at 10 to make sure that it wasn't a DL stint (or a demotion to the minors). I also created a code for whether he had the day before off, whether through an act of the manager, an act of the schedule maker, or a snowstorm that happened to pop up. I also started all analyses in May to allow players to actually build up to being able to play three weeks in a row with little rest.

Using data from 2003-2012, I used the log odds method to control for the likelihood that a plate appearance would end in a specific event (strikeout, walk, HBP, single, extra base hit, HR, or out in play) given the pitcher-batter matchup, and whether the outcome was a general on-base event. I also controlled for the pitcher having a handedness advantage or not. I ran a bunch of binary logistic regressions and slipped the various "tiredness" indicators in to see what happened.

When running this many logit regressions, it's better to talk about variables that were consistently predictive, rather than individual results. For example, the number of days off before today was generally not a significant predictor, nor was having yesterday off. However, the number of games that a hitter had played in the previous seven days (or 14 or 21, it all came out significant) had an across-the-board negative effect on all sorts of hits (singles, doubles/triples, and home runs) and increased the number of outs in play that a hitter made. These variables actually had little effect on walks (there was some evidence that hitters walked a little more when tired) and surprisingly, strikeouts.

The difference between a hitter who had played five games in the last seven days (a day off, plus a Monday travel day?) and a hitter who had played in seven games in the last seven days was about three points in OBP. That's not a lot, but of course, over 600 plate appearances, it's about an extra two on-base events, and that's worth a run or so. Over multiple lineup spots, it can build into a lost win really quickly. A manager who does not rotate his players a bit may be bleeding away value and not even realizing it.

I also took a look at whether a player's age moderated any of these relationships. I ran a series of regressions in which I added in an interaction term between age (as of April 1st) and the tiredness factor under question. Maybe younger players are able to recover more quickly? The answer was no. While, for the most part, the "past seven days" factor continued to be predictive, the interaction term did not hold sway. Older players and younger players seem to experience this brain drain at equal levels.

What it Means
There are a couple of interesting takeaways from these analyses. One is the obvious implications for roster construction. A strong bench not only means that you have guys who can slot in and pick up most of the production that the regular would give you, but it also makes the regular better in the games that he does play.

But let's extend this a little more. Last week, I talked about players who are capable of playing more than one position and who hit well enough to start, and tried to measure the value their versatility can bring to a team. One way that a multi-positionalist can affect a team is to allow the better bench bats to get more playing time—the manager doesn't have to play the guy who hits barely over the Mendoza Line just because he's the only one who can handle shortstop. A multi-positionalist allows the manager to rotate guys in and out of the lineup with the knowledge that he doesn't have to give up as much today just to get his starter a rest. And the more he's able to give his starters a rest, the less value he bleeds away in tiredness. I also argued last week that multi-positional players make platoons easier to construct. While platoons have their own advantages, they also naturally rest the members of the platoon. Maybe we need to revise our valuation of multi-positional players upward.

Another interesting wrinkle (maybe for another episode) is the idea of AL teams forgoing a traditional designated hitter and using the position to give their regulars a rotating "day off," at least from field duties. It's not clear whether DHing actually has the same effect as a day off, but if it does, it's an interesting question as to whether a team would be better off paying for a regular DH (now the position with the highest salary in MLB) or for a really good 10th and 11th man to see semi-regular duty, whilst the team saves on the fatigue penalty paid by its regulars.

There are also the obvious lessons around managing players' workloads. Players need a day off every now and then. If not, they bleed away real value. But there were a few interesting properties of what happens when players don't get rest. One is that while playing more games in the space of seven or 14 or 21 days does sap a player of some of his mojo, there was little evidence that simply giving him the day off allowed him to press the reset button. You can't expect that the next day, he'll improve by leaps and bounds. Fatigue is a cumulative effect. A day off is more akin to recharging a battery than a reset button. The good news is that the extra battery life appears to be available for several days down the road.

Another interesting thing that came out of these analyses was the fact that the effect did not vary by age. Both older and younger players felt the effects of playing in a lot of games. While physical recovery times are generally shorter for younger men, the issue of pre-frontal cortex over-stimulation is one that would likely be felt equally across ages (at least within the 20-39 bracket). I can't state this definitively with the data set that I am working with now, but it leads me to wonder if this is a problem of the brain, rather than the brawn.

Finally, when the results were broken down by event type, strikeouts and walks, often the first outcomes that analysts look at, were unchanged from expectations by a heavy workload. In fact, players who have been playing a lot lately, if anything, tend to walk more. (The effect was not significant.) The effect came from hits that were turning into outs in play. Fly balls that used to go over the wall ending up in the right fielder's glove. Ground balls that used to be smacked hard enough to get past the shortstop now end up in a simple 6-3 putout. Baseball is a game where a small fraction of a second can make a huge difference on a ball. Miss on timing that swing by a little bit, and you'll roll the ball over to the second baseman.

Fatigue, whether physical or mental, is a real issue in the game of baseball. It's hard to do something every day that requires such intense concentration and not get a break from it, and it affects a player's performance. It's always been something that's been known in the game, and now we have some idea of how big the effect is.