Save Your Closer! (Terms and Conditions May Apply) by Ben Clemens August 25, 2020 If you’re looking for a sure sign that a manager is thinking in old-timey baseball cliches instead of playing to win, listen for the words “save situation.” There are tons of reasons not to use your best reliever in a big spot — load management, handedness matchups, heck, maybe he ate some bad sushi last night. “We wanted to hold him for a save situation”? Nope! Bad management alert… or at least, it was until the rules of baseball changed. Love or hate the automatic runner in extra innings, it’s changed the tactical calculus of baseball significantly. Teams haven’t bunted as much as I predicted, which is fascinating in itself, but today, I’m more interested in which pitchers are doing the extra innings pitching. Before 2020, “saving” a pitcher for a lead was self-defeating, but that isn’t an automatic truth, simply a contextual one. Let’s delve into why that was the case, and why it might not be this year. To explain how this scenario worked in the past, I’m going to use a hypothetical situation. It’s the bottom of the ninth inning in a tie game, and the visiting team has two options for pitching: Nick Anderson or Aaron Loup. They’ll bring one of the two in for the ninth, the other for the 10th if necessary, and then completely average pitchers for every inning after that. More specifically, they’ll be bringing in pitchers with the career rate statistics that exactly match Anderson’s and Loup’s. This is an abstraction, so we’re ignoring opposing batters and handedness matchups, which a real-life manager would care about: for this article, we’re only worrying about whether bringing in your closer makes sense with everything else held equal. Here are those result rates: Outcome Rates, Career Outcome Anderson Loup BB% 5.8% 8.5% K% 42.7% 21.7% Single% 11.6% 15.4% Double% 3.8% 4.6% Triple% 1.0% 0.6% HR% 2.7% 1.9% Other Out% 32.4% 47.3% Anderson is clearly better. In fact, over a million simulated innings (every batter receives a random result from each pitcher’s result grid until there are three outs), he allowed 2.80 runs per nine innings, while Loup allowed 3.74 runs. Anderson was better in terms of the percentage of innings holding opponents scoreless, too: 80.3% of his innings were scoreless, as compared to 76% for Loup. That makes the downside of saving Anderson clear: an extra 4.3% of the time, your team loses immediately. The payoff comes in extra innings; if you escape the ninth without using Anderson, you have him for the 10th inning, when there’s a decent chance he’ll be protecting a lead. Not only that, but he’ll sometimes protect a lead of more than one run, and he’s naturally better than Loup at protecting large leads in addition to being better at protecting one-run leads. More specifically, here are Anderson’s and Loup’s runs scored distributions in a regular inning, as well as the overall league average using the same method we used for Anderson and Loup above: Runs Scored Distribution (per inning) Runs Anderson Loup League Average 0 80.4% 75.8% 73.0% 1 12.2% 13.7% 14.7% 2 4.8% 6.1% 7.0% 3 1.7% 2.6% 3.1% 4 0.6% 1.1% 1.3% 5 0.2% 0.4% 0.6% 6+ 0.1% 0.3% 0.4% With those in hand, we can see how likely the team is to win the game depending on whether it uses Anderson or Loup first in normal baseball, when extra innings don’t start with a runner on base. With Anderson on for the ninth inning, the game goes to extra innings 80.4% of the time. There, the visiting team scores according to the league average run distribution and allows runs according to Loup’s distribution. If the game is still tied after 10 innings, I gave each team half a win (look, it’s a simulation here). In that scenario, the visiting team wins 41.4% of the time. It’s bad to enter the bottom of the ninth tied if you don’t get to bat again, hardly a shock. When Loup throws first, the visiting team wins only 40.9% of the time. It’s a small edge, but using your better reliever first matters. Do something even sillier, like only using Anderson when you get a lead, and the visiting team wins only 38.8% of the time. Unsurprisingly, using your best pitcher is good — the more the visiting team uses their best pitcher, the more they win, all else being equal. In 2020, however, all else isn’t equal. Let’s look at those run-scoring distributions again, with one key variable changed: now a runner will start on second base in the 10th inning and beyond. The home team won’t bunt, in keeping with the way teams have behaved this year; instead, they’ll swing away using each pitcher’s regular distribution of outcomes, giving us a new runs scored distribution: Runs Scored Distribution (extras) Runs Anderson Loup League Average 0 46.8% 35.0% 36.7% 1 33.6% 41.1% 36.6% 2 12.1% 13.4% 14.5% 3 4.8% 6.1% 6.9% 4 1.8% 2.6% 3.1% 5 0.6% 1.1% 1.3% 6+ 0.3% 0.7% 0.9% It doesn’t make much sense to report runs allowed per nine inning totals in this circumstance, because all we really care about is the distribution, but if you must know, the league average RA/9 would come out to 10.1, Anderson would allow 7.6 runs per nine, and Loup would check in at 9.6. The run scoring edge from a bonus runner is no joke. Also no joke: Anderson’s advantage over Loup when strikeouts are imperative if you want to keep runs off the board. In a regular inning, Anderson is only the difference between no runs and runs 4% of the time. When a runner starts on second, that amount balloons to 11.8%. Now using your best pitcher in the ninth inning isn’t quite so obvious. In fact, the opposite is true. Enter the 10th inning with your opponent using an average pitcher and you using Anderson, and you’ll win the game 56.3% of the time. In a world where 10th innings are far higher scoring than ninth innings, retaining a hammer is incredibly valuable. What about in the scenario before, when we have a choice of Loup or Anderson? Pick Anderson, and you’ll still reach the 10th inning 80.4% of the time. With Loup in, the team then wins 50.34% of those extra inning games, after accounting for both offenses. That works out to a 40.5% chance of winning the game. Switch the order, and things get better. With Loup in to secure the bottom of the ninth, the visiting team takes it to extras 74.8% of the time. We already know, from above, that they then win 56.3% of those games. That works out to a 42.1% chance of winning, far better than the reverse. We can rule out strategies where the team saves Anderson for a lead even in extra innings; his biggest edge is in stopping that runner on second from scoring, which matters as much in a tie game as it does with a one-run lead. What if the visiting team has a one run lead heading into the bottom of the ninth? We can simply use the tables we already have to calculate which pitcher gives the team a better chance to win. Pitching Anderson first wins the game then and there 80.4% of the time, loses it in the ninth 7.4% of the time, and sends it to extras 12.2% of the time, where we already know the win percentage with Loup: 50.34%. That gives the team a total win probability of 86.5%. Bringing in Loup first wins the game 83.5% of the time using the same style of calculation. With a lead, the visiting team should use its closer, hardly a shocking conclusion. Sadly, it’s not as simple as that. The other team has agency too; they can bring in their own Anderson or Loup equivalent in the top of the ninth or 10th. Things get complicated quickly. This might sound like a game theory problem: both teams making decisions that affect the other. It’s not, however, because the decisions aren’t made simultaneously, which makes our job far easier. We already know optimal behavior for the visitors; save your best pitcher if the game is tied, use them if you enter the bottom of the ninth with a lead. All that remains is to work out home team behavior from that. To make things simpler, I cloned Anderson and Loup and put them on the home team. They’d have their own closer and other reliever in real life, but we’re looking for symmetry here. This underscores something counterintuitive about the new extra inning rule: because which pitcher you use depends on your lead, entering the ninth inning tied actually gives the visiting team the advantage (ignoring bunting). The home team must choose first, and the visiting team can counter based on the game state after the top of the ninth. For example, if the home team uses its copy of Loup first, we already know the results: Loup goes a scoreless inning 75.8% of the time, allows one run 13.7% of the time, and so on from the table earlier in the article. The visiting team can then use its own Loup in a tie game (75.8% to escape the inning unscathed, and a 50/50 game in extra innings with both teams pitching a copy of Anderson). With a lead, they can bring in Anderson to protect it. That works out to a 50.5% chance of victory for the visitors after accounting for all of Loup’s runs allowed distribution. You might think that if Loup gives the home team a less than 50% chance at winning the game, they should bring in Anderson. You’d be wrong, though; that suffers from the same problem as the visiting team bringing in Anderson in a tie game. Using your own Anderson while allowing the visitors to go into extras (assuming they survive the ninth) with a decided pitching advantage is a disaster, resulting in the visitors winning 52.9% of the time. That’s right: ignoring other tactical decisions, the new extra inning rule gives the visiting team a mathematical advantage if the game enters the ninth inning tied. It’s tiny assuming both teams behave optimally, small enough that the home team’s ability to bunt or otherwise play to the score likely offsets it, but it’s a real effect. Saving your best pitcher for the higher run environment of extra innings is simply a no-brainer when the game is tied in the ninth. Looking for caveats to this rule? There are many. I ignored other tactical choices — bunting, playing the infield in, intentional walks, and so on. That was intentional, because I wanted to focus on pitching only, but it’s a countervailing effect that favors the home team. I also ignored future days; there’s a real edge to saving your closer for the next day in lopsided games, though those aren’t a huge percentage of the probability mass in games that enter the ninth inning tied. Facing a particularly left- or right-handed chunk of the opposition lineup matters too, should either of the pitching options have a platoon split. Pitcher rest matters — a gassed Anderson is probably best avoided even if he’s better overall. For the most part, though, the rules of engagement in the ninth inning this year are simple and controlled completely by the high-leverage environment of extra innings. Tied? Save your closer for when you most need them, with a runner 180 feet away from scoring. Winning? Put your foot on the gas and try to end it right there.