Complete Games

So tfloyd asked on Monday about the probability that Bryce Elder was going to throw a complete game against the Padres. With the benefit of hindsight, I can tell you that the probability was 0. Exactly. Many of you, nay I suspect most of you, would have come up with that estimate. But that’s because you’re a normal human being. For those of us doomed to a combination of abnormality and lack of crystal ball access, I posited about 200:1.

But I want to take a moment to talk about the estimation of binary probabilities from sparse data. (Oh, goody, I hear from most of you. Note that there’s a section below this one in which I discuss the game. It will hurt my feelings if you go straight there, but since when have you cared about my feelings?)

Let’s start with the facts, updated to include Monday. Bryce Elder has started 98 games and thrown one complete game: it was his last start in his rookie season: an 8-0 shutout of the gNats which gave them their 100th loss of 2022. In the other 97, he has another two in which he pitched 8 innings, though, including once this year.

So, the simple estimate is about once every 100 games, or 1%. (I’m not going to quibble over two starts, and I include his playoff start, by the way. If your estimates are very sensitive to what starts count, you’re doing it wrong.)

So why did I say 200:1? First, because 0 is an embarrassing estimate. But that’s not all, and for those of you preparing to wager your life savings on Kalshi (pro tip: don’t do that) I can show you some fancy handwaving that justifies it.

First, let’s talk about how little information is actually present when all you know is that you’ve gotten 1 complete game in 98 opportunities. Lots of little things contribute to a complete game, and if Bryce had had just one more, it would have doubled his rate to close to 2%, or 50:1. It coulsn’t have been any worse than zero, but there’s a substantial chance he might have even had 3 complete games in his career to this point, which would triple the probability to about 33:1.

Even more to the point, Elder had one complete game in 9 attempts in his first year. Taking that seriously would mean you expected him to have an 11 percent chance of a complete game in 2023. You shouldn’t. Eleven tries is far too few to estimate a tiny probability, since, if the true probability were 1 in 200, either 0 or 1 complete games would be the only things you’d have any real chance of seeing.

We can make this concrete with something called the Wilson score interval. Under some assumptions (that I’m not going to make the least effort at justifying) we can create a formula for the underlying true probability of a Bryce Elder complete games (based on 1 success in 98 attempts) that will generate an interval that brackets the true rate 95% of the time. It turns out that the Wilson score interval for this probability is embarrassingly wide in odds terms. At the high end, it turns out to be 5.6% and at the low end 0.18%, for a range of 17:1 to 553:1. Because statisticians have to earn enough money not to starve, and because I know that someone offering to pay me for the odds of a Bryce Elder shutout is going to be unsatisfied with the answer “Oh somewhere between 17 and 555 to 1,” we need to do more math.

So we need to do something to make the comparison set larger than 98 attempts. There are a lot of ways we can do that, but one is to take all starting pitchers over some historic period (say 2022-2026) and try to place Elder’s performance in context. We have lots of starters over that period: some have started one game, some have started 150. The match combines all these experiences to get a mean rate of 0.55%, about 182:1. Because Elder’s experience is actually a little better than the average starter, we average his 1 complete game in 98 tries with an implied data over all pitchers which turns out to be a little under 1 in 160 tries (the method allows fractional tries: for those who actually know beta-binomial models, a=.88, neff=a+b=160). Adding in Elder’s experience, we now get an estimate of 0.729% (131:1) but still embarrassingly wide if we want to be reasonably confident: the 95% confidence interval now runs from 47:1-1245:1.

So we gained a little bit of information, but the comparison set is a little too broad: Bryce Elder is not Framber Valdez (8 complete games in 148 starts) but he’s not some bullpen game one inning starter either. So in our last attempt, we’ll restrict the sample to pitchers who have roughly the staying power of Elder. Our sample now consists of all pitchers in the 2022-2026 sample who have innings per start within 0.4 innings of Elder’s 5.51, which is in the 89th percentile of all starting pitchers over this period. That gives us almost 200 pitchers to compare him to. (Actually, 198). We repeat the same exercise as the previous paragraph. These are somewhat better pitchers: the estimated probability of a complete game is 0.65%, but greatly increases the effective sample size, since these guys are much more consistent with each other than the full sample is. We now add Elder’s 1 CG in 98 starts to an effective 3.6 CG in 550 starts, giving a new estimate for Elder: 141:1. The range is now narrowed, but still pretty large: between 67:1 and 468:1.

By the way, none of this accounts for the sort of direct stuff you’d want to adjust for, like “how good is the team he’s facing” or “how has he done in the last three starts” or platoon splits or anything else. These methods really assume that every start is an underlying constant probability of a complete game. That’s surely too simple; on the other hand, we don’t even have enough data to evaluate that probability, much less adjusted for opponent’s quality. (Note the way you’d do this is to have a model for innings pitched in a game.)

So why did I say 200:1 if the “true” number is 141:1? Well, I made another adjustment. The overall rate has been coming down. The following table gives the complete game rates across all pitchers:

20220.71%
20230.71%
20240.57%
20250.63%
20260.35% (so far)

2026 is only a partial year, but it’s well over a half year. The rate is tanking, and I bumped the odds accordingly. 200:1. Or zero, if you know what actually happened.

The Game

One of my college roommates, who hailed from Acton, MA, saw me after spring break one year and said: “I met a girl from the South over spring break.” I said: “Oh, yeah? Where from?” His answer: “Baltimore.” My jaw dropped well before he could embark on his Mason-Dixon line justification. Those of us who don’t credit the District of Columbia’s occasional identification with the South will certainly not extend that designation to DC’s northeasterly neighbor. Their most famous citizen, Edgar Allan Poe, settled in Baltimore after starting out in the unambiguous North (Boston), moving to the unambiguous South (Richmond and Charlottesville) and ending after much perambulation, as is appropriate for Poe, in the uncanny valley in between.

The Orioles have not won a World Series since 1983, and have made the playoffs seven times since then without advancing any farther than the ALCS. Camden Yards still looks cool, though. (I once attended a game in old Memorial Stadium. It was not cool.) The Orioles City Connect jerseys, featuring BMORE across the front, reminds me of the day when the Nationals misprinted their jerseys as Natinals. Diner is a great movie, and The Wire and Homicide are two of the greatest TV series ever.

Trevor Rogers against Sherlocks. Holmes gave up two runs in the second which show the influence of luck and the ABS system. A challenged strikeout was overturned on a <0.1 inch determination. Two more hits later it was 2-0. Last year that would have been a clean inning that wouldn’t have even been remarkably umpired.

A sac fly cut the deficit to 1 and a two-out single from Austin Riley evened it up in the top of the 4th. (Brandon and CJ are pulling so hard for Austin. I mean, we all are, of course, but I don’t think they do Austin a lot of favors regaling every single as if 2022 was only yesterday.)

The tie didn’t last long. A sac fly unknotted the tie in the bottom of the inning, but the net difference was reduced to 1. Holmes had to tapdance through the fifth, but a nice play by Jorge Mateo and the tortoiselike velocity of Chadwick Tromp kept it 3-2.

Danny Young replaced Holmes and got in more trouble, replaced by Tyler Kinley. Tyler has not performed well in the last month, and he induced Tromp to bloop what what would have been a hit but-for a fabulous catch by Mauricio Dubón, once again saving a run.

The bottom of the 7th saw the Braves debut of Elieser Hernández, the Braves designated Carlos Carrasco. In fact, he comes to the Braves from the same Korean team that just signed Cookie. I defy anyone to peruse the career of Elieser Hernández, including his 76 innings in Gwinnett this year and see anything that makes you say — yeah, this is what we need. So of course he pitched a perfect 7th inning. Honestly, I suspect the rehiring of Carrasco had become such an embarrassment that they simply decided to hire a Carrasco clone to shut people up. The big difference is that Carrasco was once really good. Hernández has never been any good. (It goes without saying that I’m just trying to make him prove me wrong.)

Drake Baldwin tied the game with a right-field homer in the 8th. In the bottom of the inning, they left Hernández in. He walked Colton Cowser who got to third on a stolen base and throwing error by Joey Bart (to be honest, Jarvis should have at least blocked it) but completed his second scoreless inning and we were tied going to the 9th.

Tyler Wells, a small Texas town of 13,582, was brought in to face the Braves in the top of the inning and yielded a solitary single. The bottom of the 9th saw Hernández try a third inning. A two out walk followed by a Polar Bear single and an intentional pass created a moment of truth for Hernández against Taylor Ward. After a 3-0 count, Elieser threw two strikes and then induced a grounder to Riley that he bobbled but still managed to get to first in time. Honestly, I think my typing my doubts created this result.

The tenth saw the Braves plate the Manfred Man on a terrible error by the second baseman leaving second and third with no outs. Strikeouts of Baldwin and Albies led to a two-strike three-out homer by Matt Olson.

Dylan Lee arrived to try and not give up 4 runs. He gave up 3, but retired Hall of Fame linebacker Sam Huff to end the game. Huff died in 2022 at the age of 87, so it’s amazing he can even swing a bat.

Bryce Elder with a chance at another complete game tomorrow.