When Should Teams Press the Advantage?

When the Nationals took an early lead in the World Series, there was a popular cry for the team to knock the Astros out while they could. Expend resources you were planning on saving for later in the series, turn Patrick Corbin into a reliever, maybe bring back some starters on short rest: what does it matter if you hurt your chances of winning Game 7, the thinking goes, if Game 7 never happens?

A softer version of this came up as the Cardinals walloped the Braves in Game 5 of the NLDS. The game was already decided. Why not pull Jack Flaherty so that he could pitch Games 1 and 5 of the NLCS rather than Games 3 and 7? It’s not an identical situation, but it relies on the same logic: earlier games happen more often, so get your pitchers into those.

Tomorrow night, there will be yet another version of this discussion. The Astros are a win away from ending the series. If the game goes into extra innings, say, or Justin Verlander gets knocked around but the offense keeps the team in it, would Houston use Zack Greinke in an attempt to end things right then and there? And should they?

While these questions are similar, they’re not identical. Does this reallocation of win probability matter? The answer, as it often is, is “it depends.” I believe the answers to these three questions are “not much,” “not at all,” and “more than you’d think,” respectively, and I’ll attempt to lay out why I think that is the case here.

First, a note of caution: I ignored a few things in this article. For the most part, I’ll cover them in each example, but let’s get one out of the way up front. I ignored momentum, which won’t surprise you, this being FanGraphs and all. Maybe there’s a psychological advantage to going ahead 2-0 above and beyond the fact that you’re up 2-0, but I’m not searching for evidence of that in this piece. I would caution against putting too much weight on that argument (the Astros literally just came back from down 2-0 to be up 3-2), but hey, you can do whatever you want in your own time. I won’t be covering it here, is all.

Let’s think about the Flaherty situation first, because it’s the cleanest example. Imagine a team with a great starter, a good starter, and two okay starters, like so:

This is an abstracted version of the rotation the Cardinals had to deal with. Play a seven-game series without home field advantage (which I applied as a fixed constant in each game) and this rotation stacked top-to-bottom in order, and the Cardinals win the series 49.5% of the time (I solved this via Monte Carlo, but you could mathematically solve this equation as well).

That’s the “ideal” distribution of starters. When you plug in what the Cardinals did instead (Good, Okay, Great, Okay) in order and re-run the simulation, their odds of winning fall imperceptibly. To be precise, they were 49.528% in the first case and 49.496% after re-ordering over 10 million trials each. This difference is too small to be noticed, though it of course ignores one benefit to using your best starter in Game 1 — the Game 7 relief appearance.

That’s worth something, and makes it preferable, but calculating its value is difficult to say the least, and it’s also a small effect. If you think that an on-demand inning of Flaherty increases the team’s odds of winning a random game by 1%, that bumps the team’s odds of winning the series up to 49.8% — a measurable advantage, but a tiny one.

I’ll admit that 1% is merely an educated guess, but it’s not really the point of this article. Consider instead this: the order your pitchers go in basically doesn’t matter. Because this is just a computer simulation, I could line up Flaherty to pitch Games 1 and 2. Rest is just a construct in this digital world. In fact, let’s make two more strategies: Shock and Awe, where the team’s two best pitchers pitch the first four games, and Best for Last, where the best two pitchers start the last four games:

It just doesn’t matter. Why is that? Why doesn’t stacking a bunch of good pitchers in the games that are guaranteed to occur boost your odds? It’s maybe easiest to think about it through abstraction. Imagine a series where you are guaranteed three wins and three losses, and then play a 50-50 game as the seventh game. How do you want to order these? Whichever way you do, the outcome of the 50-50 game determines the series.

Why doesn’t the average length change? Think of the same example. If you stack your wins at the front, you win the series in four half the time and lose in seven half the time. Stack your losses at the front, and you either get swept or win in seven. That’s the same average length. Math is cruel that way sometimes.

Okay, so that’s the Flaherty situation covered. That one is pretty straightforward — order doesn’t really matter much in a series, only who pitches. Start switching pitchers between games, however, and things get a little more complicated. Take the situation in Game 1 of the World Series, which I’ve already written about. Patrick Corbin as a reliever added to the Nationals’ chances of winning that game, and the consequences wouldn’t really be felt until Game 7.

In that article, I didn’t really discuss the issue of pushing win percentage across time. Let’s delve into it now. To take a really high-level view of the series, I assigned some win probabilities that worked out to the Astros winning 59.2% of the time, then adjusted Game 1 to the state it was in when Corbin entered the game.

For a no-Corbin state, I bumped the Astros’ odds of winning by 2% from neutral to account for Washington’s tire-fire bullpen. For a Corbin appearance, I dropped Houston’s odds by 1%, as well as by 0.5% in Games 2, 3, 5, and 6 to account for having Corbin available in relief. Finally, I docked Washington’s odds by 5% (the total I added to them in this exercise) in Game 7 when they wouldn’t have Corbin to pitch. Is this perfect math? Definitely not. But it’s pretty good for a first approximation, I think.

With no Corbin appearance, the Nationals win the series 55.6% of the time from there. Beating Cole in Game 1 is a big odds flipper. Turning Corbin into a relief pitcher, the Nationals would win the series… 55.6% of the time. Perfectly balanced!

Now, it’s fair to question my assumptions. Maybe having Patrick Corbin pitch fewer innings in the series overall somehow helps the Nationals’ odds of winning individual games, though I’m skeptical. But the argument that deferring the cost to Game 7 (a start that would be Corbin’s if he didn’t throw any games of relief) makes the tradeoff somehow better doesn’t hold water.

Finally, there’s the case of what the Astros can do tomorrow night, and this is actually a very different situation. In the other examples, the goal was symmetrical: play seven games, try to win more than you lose. For Houston, the goal isn’t to win more than they lose — it’s to not lose both of the next two games.

Take a very general example. Let’s say that the Astros are 50% to win each game. That would make them 75% to win the series. If they were 80% in one game and 20% in the other, they’d be 84% to win the series, despite winning half of their games on average in both examples. When you’re merely trying to avoid an extreme result (two straight losses), you benefit from unbalancing your odds.

That’s a neat little trick of geometric averaging, and it informs both teams’ optimal strategies. The Nationals, needing to win both games, can’t feasibly borrow from one to help the other. If they do something like use Sean Doolittle for two innings in Game 6, unless there’s a big leverage benefit, the Game 7 cost will be unattractive.

Assuming the matchups in the series are Verlander/Strasburg and Greinke/Scherzer, the above model gives the Astros a 78% chance of winning the series (close to ZiPS’ 78.8% odds, which is a nice sanity check). That leaves us with one interesting question: how much loss should the Astros accept in Game 7 to increase their odds in Game 6?

This varies a bit based on the exact odds you assign, but it looks roughly like this: if the Astros increase their Game 6 odds by 5%, they can afford to have their Game 7 win expectancy drop by 6.8% and break even. If you prefer Dan’s game odds rather than mine, 5% in Game 6 is worth -6.2% in Game 7. The Astros should treat Game 6, where they’ll be most favored, as the time to go all out, mathematically speaking.

Of course, this is just numbers on a spreadsheet. The game is played on the field, and the small edges I’m talking about, perhaps a 1% increase in series odds, are small compared to the change in odds that will arise if Justin Verlander is feeling a little off, or if Trea Turner has a particularly good breakfast and hits a home run to lead off the game.

But the math is interesting to me all the same. I like to think through the strategic implications of team’s machinations, and this one is particularly appealing to me. Should you punt Game 7 because the present demands more attention? No! But also, yes! And sometimes, it doesn’t matter! It’s entirely contextual, and that’s neat.