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June 8, 2010 Expanded HorizonsPerfection
What moment defines a perfect game? Is it the clinching pitch, the fistpump, and the dogpile? Is it the camera shot of the players in the dugout avoiding the starter like he’s got bad news? What about the sensational defensive play that typifies the way perfect games are not simply individual performances? No. The defining moment of any wouldbe perfect game is the moment it dawns on you. Say, you think to yourself as you glance down at your scorecard, how many hits has he given up? No walks, either, huh? From that moment on, the narrative of the game changes. It’s a thing to be protected, a delicate embryo with a nineinning gestation period. By the time it reaches its third trimester, only the most hardened hometown supporters are rooting against it. Most of them, though, don’t last. They become stillborn ideas of a nearly impossible feat—a feat so rare you couldn’t even field an entire roster out of players who had accomplished it. At least, that’s how it used to be. In this yetyoung season, there have for all intents and purposes been three perfect games. This is a mindboggling unlikely occurrence that demands explanation. It’s like buying a Honus Wagner T206 card for over a million dollars and then finding one, nearmint, at the rummage sale down the block. You’d feel excited, sure, but more than a little cheapened and confused. When Dallas Braden threw a perfect game, a whole lot of fans thought, “him?” When Roy Halladay did it just a few weeks later, it made more sense, but it was also suspicious; two perfect games had only occurred in the same year once before, and that was when Rutherford B. Hayes was president. When Armando Galarraga retired every batter he faced a matter of days later, people were as dumbfounded as they were enraged at the call that kept the game out of the record books. Modeling the Mystical For some reason, I always thought it was better when perfect games happened during the day. It’s an oldtimey kind of thing to have happened during a baseball game, its recent resurgence notwithstanding. The image of a perfect game commands images of handoperated scoreboards populated by zeroes. It is necessary, of course, that any pitcher who throws a perfect game be dominant, but that dominance is often casual and understated. The calmness and composure of the pitcher belie the incredible feat he performs. So perfect games are as close to mystical events as baseball gets during the regular season. Mystical events are difficult to fathom, and attempting to understand them can often lead to simple reductionism. But without a model, it may never be possible to wrap our minds around them. So here’s a simple model of a perfect game. A perfect game is a series of nine consecutive perfect innings thrown by the same starter in the same game. This is not a completely accurate model of a perfect game. We have ignored the possibility of a tie or extra innings, both of which could confound this model. Nevertheless, the nice thing about a simple model is that we can learn things from it. Let’s start simple. What percentage of starters’ innings are perfect? When we ask our resident Jane Austen scholar, he gives us an impressive data dump, which we can then render graphically.
The chart above shows the percentage of perfect starter innings divided by total starter innings, going back to 1974. That is a period over which runs per game have increased at a relatively steady pace. It’s also a period over which the percentage of innings thrown by starters has steadily decreased. And yet, the ratio of perfect starter innings to total starter innings has remained mostly flat. If anything, it has declined very slowly, from about 40 percent in 1974 to about 38 percent last season. Let’s use the higher of those numbers (40 percent, because I like easy math) and see if we can’t figure out the likelihood of a perfect game in a given year. Let’s assume that this 40 percent probability is constant across all pitchers and all innings, and that the probabilities are independent from one inning to another. Now, I know these are patently false assumptions, but the distortive effects shouldn’t be too bad, at least not orders of magnitude so. After all, the good pitchers ought to mostly cancel out the bad, and few managers have the guts to take the ball from a guy with a perfect game going. Again, we’re going for simple math here. As we noted above, we need nine consecutive perfect innings to get a perfect game. So we take (.40)^9, multiplying the probability by itself nine times, and we get a very small number: 0.00026. That would be the probability of any of our identical starters throwing a perfect game on an individual night. But each team plays 162 games, and there are 30 teams, and if we multiply that out we get 1.3, suggesting we should expect more than one perfect game thrown per year! Of course, that’s not at all what we have seen:
This chart shows the number of perfect games (blue) and nohitters (red) each year over the same period as above. Recently, perfect games have tended to happen once every few years. So what gives? For one thing, our model is simple, but it is also very sensitive to inputs. That nine exponent—you know, the little hitch about a baseball game being a rather drawnout affair—means that if we change the probability, we can drastically change the result. What if, instead of 40 percent, we use the more recent 38 percent figure (simplicity be damned and dust off that calculator)? Well then the probability is cut nearly in half, and the expected number of perfect games per year falls to 0.8. Oh, Great, It’s Bad Benny Tonight But it gets more complicated still if we let the probability of throwing a perfect inning vary over the course of a start. Let’s take two pitchers, Chaz Consistency and Benny Badinning
Notice that both Chaz and Benny have the same average probability of having a perfect inning. If the entire league were made up of Chazclones, we would expect just over one perfect game per year. If, on the other hand, it was nothing but Bennies, we’d expect just under one perfect game per year (it’s approximately 1.0 versus 0.9). And it’s much more likely that the probabilities vary wildly from inning to inning. Why? Because managers lump their best hitters together in the lineup, meaning that one third of the lineup is the thumping heart of perfectgame murders, just waiting to destroy today’s bid for unlikelihood. And the more the probability varies from one inning to the next, the less likely perfect games get, even if we keep the average probabilities the same. What that means is, without beating you over the head with more numbers, is that it’s perfectly reasonable to expect a perfect game to happen once every few years. And if that is the expectation, it’s possible that you’ll get three in the same season, because three in the same season really isn’t all that different from one every three seasons. However simple this model may have been, it won’t overwhelm the feeling of recognition when I look at the scorecard and see nothing but zeroes. As someone who sat in the stands and watched a complete oddity of nature throw one, no amount of number crunching could take that away from me. Question of the Day If you want to know my favorite culprit for the slight uptick in perfect games over the last century, it’s expansion and the longer schedule. There are both more games and more pitchers, meaning there are concomitantly more opportunities for perfect games. The fact that the last two guys to accomplish the feat were Dallas Braden and Roy Halladay demonstrates just how much it is about randomness, and more games means more opportunities for that randomness to pay off. What is your guess? 18 comments have been left for this article.

Tommy, I think you might have a problem with your 40% or 38% figure. You are probably counting three onethird of innings as one inning, rather than three innings, to begin with.
As for the chance of a perfect inning, why not do it the even easier way, and take OBP (the real OBP, one where reaching on error is a good thing, etc). Chance of a perfect time at the plate for a pitcher is around 66%, so a perfect 123 is .66^3 = .287. A perfect 27 is .66^27 is around 1 in 75,000.
Presuming that perfect games occur in slightly more conducive settings, say a true OBP of .300, that makes the above calculation as 1 in 15,000.
With about 350,000 starts in MLB since 1900, that works out to 22 expected perfect starts.
Tom, can you explain what you mean by:
"You are probably counting three onethird of innings as one inning, rather than three innings, to begin with."
I excluded all partial innings from the data I ran for Tommy  a pitcher had to retire all three batters he faced that inning to receive credit for a "perfect" inning pitched.
Colin, I meant for the denominator (opportunity). The way I read Tommy, if someone pitched three 6.1 innings, that would count as 19 innings, when it should count as 21.
I see what you mean, Tom.
It's an interesting question, really, and one that I think comes down to your choice of models. As I hoped to make clear in this piece, it's hard modeling something complex like a perfect game, and a lot of your answer is going to depend on your model. I offered a rather simple model, with plenty of assumptions that aren't particularly rigorous.
So while I agree the denominator I used (the sum of all starters innings) is probably artificially low, I'm not sure round up every partial inning is the best way to do it either. The reason is because many pitchers are taken out earlier if they have previously given up a hit, walk or run in the game than they would be if they had a perfect game intact. That means that it's not always the case that fractional innings aren't completed due to failing to be perfect. Something in between would probably best capture the truth.
Right, the correct answer would have to be between the two. But, I would think it's far closer to me, because I would be shocked if you can find 10% of starting pitchers who were removed midinning who did not allow a runner to reach base.
So, the reality is that virtually all midinning removals occur when the inning is no longer perfect, and therefore, you have to count that as "1" opportunity, and not "1/3" or "2/3" as your model would specify.