Last week, an interesting question got me wondering about Billy Hamilton and the new extra-innings rules. As it turns out, he’s a valuable runner to have on second base! So valuable in fact, that he projects to gain his team roughly 0.3 wins in a 60-game season just by being fast.
For the Giants, that’s great. For the other 29 teams in baseball (or 28 if the Dodgers end up rostering Terrance Gore), that’s no help. What should their strategy be in extra innings? I had all these run expectancy tables, so I decided to dive in.
First things first: let’s set the parameters of this discussion. I’m going to be considering two decisions. First, does bunting to lead off the inning make sense, and does that decision change based on whether you’re the home or visiting squad? Second, assuming bunting doesn’t make sense, what about stealing third? Presumably you’d steal with one out, what with not making the first out at third base and all, so we’ll focus on those two decision points: bunting to lead off, and stealing if the first at-bat doesn’t produce any advancement.
The value of being a home team is immediately evident when looked at through this lens. Consider a situation where the visiting team scores two runs in the top of the inning. Right away, a bunt goes out the window. That’s a big edge; in 2019, and excluding extra innings so that walk-offs don’t interrupt a team’s run scoring, teams that reached the position of a runner on second base with no outs scored two or more runs 29.1% of the time.
In other words, nearly a third of the time, bunting the runner over serves no purpose at all; your team will need two or more runs just to tie, so the position of that runner is nearly immaterial. Getting to act after knowing how many runs your opponent scored is huge.
Let’s talk tactics for the home team first, then. They’re more straightforward. When the visiting team scores two or more runs, the strategy is obvious: just play baseball. There’s a small subset of these where bunting might later make sense — consider starting the inning down two, then having your leadoff hitter smack a double — but those happen so rarely that we can round them down to zero. Instead, let’s focus on the times where the visiting team scores zero or one runs.
To get an idea of how often a bunt moves the runner over from second to third base, I looked at every time a batter bunted with a runner on second base, and only second base, since 2008. I excluded bunts with two outs, since those are aiming for a different outcome, but there were luckily few of those. 70.8% of the time, things went as planned — a successful sacrifice bunt. Another 14.3% of the time, something good happens; a bunt for a single, an error, or an attempt on the lead runner that isn’t in time. The remaining 14.8% of the time, it’s a disaster; an out without advancement. More specifically, the lead runner is out 4.1% of the time and the bunt results in an out without advancement another 10.7%.
Quickly, you might consider changing this data to account for the fact that pitchers won’t usually be doing the bunting. That’s fair, but the bunting data for non-pitchers in that spot is thin, and the batter data might be non-representative; a decent percentage of the bunts in that sample were surprise bunt attempts rather than telegraphed sacrifices. If you’d like, you can mentally make bunts more likely to succeed for the rest of this exercise. Onward!
When the game is tied to start the bottom of the 10th, playing the game “straight up” results in a win 60.8% of the time — the percentage of innings where a team scores at least once. If the bunt succeeds, that number ticks up to 64.8%, a meaningful improvement. By counting the odds of scoring at least one run after each of the possible results of the bunt, we can work out the overall odds of scoring at least one run, given a bunt attempt:
|After||Odds of Scenario||Odds of Scoring|
|Lead Runner Out||4.1%||27.7%|
|Bunter Out, No Advance||10.7%||40.8%|
After netting everything out, the result is clear. Given the first plate appearance being a bunt, and using the overall success rate for bunts with a runner on second, the home team will win 63.5% of the time it enters the bottom of the inning tied. That beats the 60.8% naive probability of scoring a run. A bunt is the way to go!
We can do the same thing for a one-run deficit, the only other margin that makes a bunt reasonable. This time we’ll simply use the same run probabilities, but assign the team a 50% chance of a win if they end the inning tied (it should be a little bit more than 50%, but we’re painting with a broad brush here). That looks like this:
|After||Odds of Scenario||Odds of Tie||Odds of Win|
|Lead Runner Out||4.1%||11.2%||16.5%|
|Bunter Out, No Advance||10.7%||22.4%||18.4%|
That works out to a 43.2% chance of a win overall. Playing it straight results in one run 31.7% of the time, and two or more runs a further 29.1% of the time, for a net 45% win probability. When down one run as the home team, bunting to lead off the inning is a marginally bad idea.
Whether you play it straight up or start with a bunt, one scenario is bound to eventually come up: when the first batter of an inning makes an unproductive out, the runner on second has an opportunity to attempt a steal of third base. With one out, this can be a big deal; complete the steal successfully, and your team’s odds of scoring at least one run in the inning climb from 40.8% to 64.8%. There’s a reason that the saying is “never make the first or third out at third base” rather than “don’t be out at third” — getting to third with one out is enormously valuable.
That’s after a successful steal; after an unsuccessful steal, it’s bad times. Two outs and nobody on is, well, a tough way to score runs:
Armed with that knowledge, we can do the math to see how often a steal needs to be successful. In a tie game, it’s simple: successfully stealing raises your odds of scoring in the inning by 24 percentage points. Those wins are replacing ties, for a net win probability gain of 12 percentage points. In other words, a successful steal increases your odds of winning from 70.4% to 82.4%.
On the other hand, teams only score any runs at all 7.7% of the time when they face a two out, bases empty situation. That produces a total win probability of roughly 53.9%. Win 12 percentage points on a successful steal, lose 16.5 percentage points if caught; you need to be successful 58% of the time to make the math work. Given that runners were successful stealing third 74.4% of the time in 2019, going a little more often makes sense. How much more often? That’s up to teams to figure out, because there’s no formula for how much more often you’ll be thrown out when you run more often. Suffice it to say, though, that you should be far more willing to steal. In an early-inning situation, steals of third need to be successful 71% of the time to break even. Home teams should be running wild, in a relative sense, when the game is tied.
Switch the scenario from tied to down a run, and it stops making so much sense. Moving from second to third increases your odds of winning from 29.6% to 42.4%. Getting caught lowers your odds to 5%. That works out to a 66% breakeven rate; still lower than the leaguewide success rate, but not by enough that I’m confident saying a team should attempt meaningfully more steals. Every deficit beyond one makes a steal worse and worse.
Why did we spend so much time going through the home team’s decisions? Because they inform the visiting team’s decisions. Here’s a grid of the home team’s win and tie probabilities for a given deficit, assuming they bunt when tied and swing away when down, the optimal strategy:
|Deficit||Odds of Scenario||Tie %||Win %|
The “Odds of Scenario” column shows how often the visiting team will amass a lead of that many runs, assuming no bunting. To work out the totals, you can simply multiply the odds of each scenario occurring by the win and tie probabilities given that deficit and add them up. In aggregate, that works out to 36.9% wins, 27.2% ties, and 35.8% losses for the home team. That’s a 50.7% win rate for the home team — that 0.7% edge comes from the correct application of bunting in the home half of the inning. With those odds set up, we can work backwards to find the value of a bunt for the visiting team. For example, after a successful bunt, the visiting team’s run scoring distribution changes, which makes the home team’s win probability look like this:
|Deficit||Odds of Scenario||Tie %||Win %|
Sum those up, and that works out to a 52.9% win percentage for the home team after a successful bunt. Wait, uh, what? Getting the bunt down as planned doesn’t work? Bunting as the visiting team is bad! Why? Think of it this way: when the other team starts with a runner on second, you really need to score two runs to be comfortable with your lead. Bunting lowers your odds of scoring two or more runs from 29% to just under 20%. Don’t do that!
I’ll spare you the fancy tables for what happens when the bunt doesn’t work as planned. Instead, I’ll just present the unified table. After taking into account all possible outcomes of a bunt attempt and the likelihood of each score after that, here’s the final breakdown:
|Deficit||Odds of Scenario||Tie %||Win %|
That works out to a 52% win probability for the home team, against their naive 50.7% odds that come from the visiting team simply playing baseball. Again: don’t bunt as the visiting team. It simply doesn’t do what you want it to do.
There’s one further matter: should you steal third? As the home team, the breakeven got quite attractive in a tie game. As a visitor, your odds of winning the game are a mere 33.6% with a runner on second base and one out. At the risk of repeating myself, that’s because scoring twice is where you want to be. The odds of scoring two or more runs after an out are quite poor.
We know from before that the visiting team wins 47.1% of the time after a successful sacrifice bunt (the inverse of the home team’s 52.9% win probability). That means a successful steal increases the visiting team’s odds by 13.4 percentage points — stealing third with one out is equivalent to successfully sacrificing the runner over. Unfortunately, making an out at third is about as bad as you’d expect — it lowers the visitors’ odds to a mere 6.1%. That breakeven, 67.2%, means that steals of third in extra innings, for the visitor, are no different than steals of third in regular innings. Do so at your own peril, and only if you’re likely to be successful.
There was an absolute ton of math in this article. Some of it was messy, and I’ve no doubt explained it poorly in parts. Here, then, are some rules of thumb to follow:
- Visiting teams shouldn’t bunt to lead off extra innings
- Home teams should bunt to lead off extra innings only in a tie game
- Visiting teams should steal third at the same rate they would in regulation innings
- Home teams should aggressively steal if the game is tied with a runner on second and one out
It’s really as simple as that! If your team follows those four rules, they’re doing extra innings right. If they don’t, they have some explaining to do to the probability gods.
Ben is a contributor to FanGraphs. A lifelong Cardinals fan, he got his start writing for Viva El Birdos. He can be found on Twitter @_Ben_Clemens.