by Chris Bruce

*Editor’s Note: This piece was originally written in a distant time (about 72 hours ago) when the possibility of a Red Sox collapse still appeared unlikely. Whether or not the Red Sox ever get a chance to think about a game one, this post examines more generally the strategy of deferring an ace’s start until the second game of a series. *

If you take the current projections and ignore the scenario that everyone in Boston is dreading, the Red Sox will be facing the Justin Verlander-led Tigers in the first round of the playoffs. Verlander has gotten a lot of press this year for having a great season (even though he was statistically better in 2009 and has benefited from an exceptionally low opposing BABIP this season, but that’s a story for another time…) and carrying the Tigers to the top of the AL Central. So that begs the question, how should the Red Sox go about trying to face him?

Traditional baseball strategy has the best starter pitching the first game of the series, followed by the second best starter in the second game, etc., but given that Detroit will definitely pitch Verlander first to get him two starts in the 5-game series, could the Red Sox change their rotation to have a better chance of winning the series? What if they decided to just accept the loss in game 1 by pitching John Lackey, giving their top 3 starters a better chance of winning against Detroit’s bottom 3?

To get a better answer to this question we can look at the win probabilities for the potential pitching matchups. Using the FIP for each starting pitcher as a projection of runs allowed in each start, win probabilities can be found using the Pythagorean expectation for each matchup. I’ve also assumed that each starting pitcher will pitch 6 innings and relievers (using their overall FIP) will pitch the final 3 innings (with the exception of Verlander who averages 7.4 innings per start) resulting in a “Full game” FIP. You could also adjust for the difference in offensive efficiency between the teams, but for simplicity’s sake I have not done so – while some of the magnitudes below would change, the relative win probabilities and conclusions from the analysis stay the same.

So what does the analysis tell us? Let’s assume that the Red Sox put Lackey in for game 1, then start their normal rotation of Beckett, Lester and Bedard after that.

At first glance the strategy seems to be working – the Red Sox still have their lowest probability of winning against Verlander, similar chances of winning games 2 and 3, and a much higher chance of winning game 4 – but in reality the Red Sox’s chance of winning through the first 4 games hasn’t increased at all. Given that the pitchers’ performance (ie, FIP) doesn’t change depending on what game they are pitching, there is essentially a conservation of win probability. The increased probability of winning games 2-4 is compensated by the decreased probability of winning game 1.

Whether Lackey pitches game 1 or game 4, there is still a ~36% probability that the Sox will have won the series through 3 or 4 games and a ~26% chance that the Tigers will have. Actually the probability of the Red Sox winning the series decreased by a nominal amount of 0.3% – this relative equity is due to the relative parity of Lackey’s ERA, hold that thought. While an observer might intuitively write off the game against Verlander completely and assume you should maximize your chances in the other 3 games, the fact is that there is still a non-zero probability of beating Verlander in game 1 and pitching your worst pitcher in that spot will decrease that chance by about the same amount that you increase your chances in the other games. Interestingly, in the extreme case that you really should write off that game (ie, if your 4th starter has a FIP of, say, 100), then the probability of winning through 4 games actually does increase by 6.5% when you bury that pitcher in the first start. This is because there is such a small probability of this pitcher winning any game, whether it is game 1 or game 4, that it does make sense to match the top 3 pitchers against their worst possible opponents.

Why is there this conservation of win probability? To understand it better we can look at it graphically. Below is a graph of the win probability for Boston against each of the Detroit starters, with each line representing one starter, and on the x-axis is the number of runs that Boston may allow. The area highlighted in blue is the range of Full Game FIP for Boston’s starters, and the win probability of any matchup would be found by finding the point on the Detroit starter’s line that corresponds to the Boston starter’s FIP on the x-axis.

Within the range of the Boston pitchers (in the blue box), the win percentage functions are essentially linear and parallel. This is what leads to the effective conservation of win probability – no matter how you arrange the Boston pitchers to intersect with these lines, what is gained in one place is given up equally in another place. But, if the range of that box were wide enough (or the Detroit pitchers FIPs were disparate enough) such that the lines were not linear and parallel, then the opportunity for optimization would exist.

Of course, given the low impact of the rotation order in the first 4 games, where the decision ends up having the most effect is game 5, where if Lackey has to come back against Verlander. Here, Boston is at a much larger disadvantage than if Beckett were pitching. Overall, if Beckett starts games 1 and 5 Boston has a 53.0% chance of winning while if Lackey starts it is reduced to 49.0% (these are calculated using the above numbers – Red Sox fans, you can stop freaking out – making a simple adjustment for the Red Sox’s better offensive efficiency increases their probability of winning to ~69%). Whether the 4th starter is John Lackey or some guy who managed to make it to the majors with a FIP of 100, through the entirety of a 5 game series a team’s chances of winning are maximized by starting their best starter in game 1, no matter who the opponent. So, next time you see your team’s ace losing to the opponents’ best guy in the postseason don’t spend too much time wondering what could have been, it was the best shot you had.

Critical flaw in the analysis – Brad Penny will not make the Tigers playoff roster. Tigers will pitch Verlander in game 1, followed by Doug Fister in game 2, Scherzer in game 3, and Porcello in game 4. Doug Fister’s full game FIP is 3.10 – which is better than every Red Sox starter. If its Tigers/Sox – the Sox have their work cut out for them.

You’re assuming that Brad Penney will be the fourth started, when Jim Leyland has already gone on record as saying that Doug Fister will start alongside Verlander, Scherzer, and Porcello. Penney isn’t even a lock to make the postseason roster.

Sorry, meant to say that Penney will be the fourth starter.

It’s not really something that would change the broader conclusions here, but it’s worth noting that Doug Fister (and not Scherzer) will be starting Game 2 for the Tigers. He’s made fewer starts with the club, but his stats have been superb (sub-3 FIP). That clearly impacts your probabilities in Games 2 and 3, and in this case likely makes it an even worse decision to defer Beckett/Lester. Incidentally, the Sox probably won’t have a choice–Beckett is throwing Monday on 3 days’ rest, Lester is expected to throw Wednesday, and neither is likely to be available for Game 1.

Either way, it seems intuitively clear that the quality of the opponents’ Game 2 starter is a fairly important lever here. I’d be interested to see what the statistics say about that intuition.

Another Sox-related curiosity: per CoolStandings (http://www.coolstandings.com/collapses.asp?i=1&sn=2011), if the Sox miss the playoffs, it would be the 3rd-worst “collapse” in MLB history. But given the Sox’ slow start, if they were to hold on and qualify for the playoffs, they would also qualify as the 78th-best “comeback” (http://www.coolstandings.com/comebacks.asp?i=1&sn=2011) in league history. They reached a low “playoff probability” of 8.2% on April 15, and a high of 99.6% on September 3. Just a strange, strange season.

Good post, but one thing is missed….Doug Fister is the Tigers 2nd game pitcher. Since coming over from Seattle, he is 7-1 with a 2.02 ERA and .898 WHIP. I think it even makes it more difficult for Boston to win no matter what strategy they use.

http://www.baseball-reference.com/players/f/fistedo01.shtml

Why does this analysis leave out Doug Fister? Not sure that it would matter that much to the overall analysis, but he is expected to pitch in game 2 of the playoffs for the Tigers.

Not that it affects your point — great article by the way — but Detroit has a guy named Fister, too.

Guys, fantastic points by all of you, you’ll have to excuse my ignorance with the Tiger’s playoff rotation – I’ve clearly been spending too much time thinking about my Braves’ collapse. Also, as I was writing this a few days ago it was still unclear who from the Sox would need to pitch the late games in the regular season and potentially have to pitch later in the playoffs, or off 3 days rest, etc, so it had to be more of a theoretical discussion.

I do think the more interesting analysis comes from the “one ace” scenario and whether it’s actually better to put your top guys against the 3 non-aces in a rotation because this is the strategy that intuitively comes to mind in a situation where you have the opportunity to win the 3 games needed in a 5 game series – and less so if a team has two aces. That said, the principle stays the same. With Penny in for the Tigers, the “series win %” for the Sox through 4 games goes from 36.2% with Beckett leading off to 35.9% with Lackey (essentially the same) and goes from 53.0% to 49.0% through 5 games. When you put Fister (3.10 FIP on the season, so “Full Game” FIP of 4.14) in the 2nd starter spot and take out Penny those numbers change to 29.9% and 29.9% (so still essentially the same through 4 games) and 47.0% and 43.0% (so still a 4.0% drop in win % when starting Lackey first). This is due again to the linearity and similarity in slope of the win% lines among all the starters within the range of Boston’s expected runs allowed. All in all the conclusion is the same – not advantageous to start Lackey in game 1.

Again, thanks for the comments and keep ’em coming.