Grandeur of the perfect sphere
Thanks the atoms that cohere.
—Ralph "Branca" Waldo Emerson

Perfect behavior is born of complete indifference.
—Cesare "Carl" Pavese

So who’s perfect? … Washington had false teeth. Franklin was nearsighted. Mussolini had syphilis. Unpleasant things have been said about Walt Whitman and Oscar Wilde. Tchaikovsky had his problems, too. And Lincoln was constipated.
—John "Kid" O'Hara

In the wake of Randy Johnson's perfect game a number of analysts have remarked on the fact that there have been eleven perfect games since 1961, the year of baseball's first foray into expansion. In the previous ninety years of major-league baseball, there were just five regular-season perfectos (plus Don Larsen's 1956 World Series gem).

Of course, they argue, expansion diluted the player pool making it easier for a dominant pitcher to shut down an inferior team 27 times in a row. Keith Emmer wrote a very good article on the topic that cuts through the bluster and uses common sense to look at the issue. He uses on-base percentage to evaluate this claim and finds that most of the post-expansion perfect games have been thrown against teams with better OBPs than those in previous perfect games.

Here is a table of the overall ratios (batting average, on-base percentage, slugging, and OPS) in the majors per decade. Also listed is the number of perfect Games (PG) per decade. (Note: Don Larsen's World Series game is included with the regular-season stats for the 1950s.) There is a grand total and a breakdown by pre-expansion and post-expansion era:

Yeah, there's a big difference in OBP since expansion. Well, two percentage points at least, is that it? It's also worth mentioning that decades with low OBPs tend to have more perfect games (1880s, 1900s, and 1960s) than those with high OBPs (1890s, 1930s and 1940s). However, the decade with the highest overall OBP, the 1920s, did have a perfect game and the high-OBP Nineties had four. So there is something more than just how often players get on base that affects the likelihood of a prefect game.

The next big factor and one that is so obvious that it's remarkable that it hardly gets mentioned. It's the number of games. As Emmer points out, the number of games per year has nearly doubled in the expansion era. There are 30 teams now as opposed to 16, and the schdule is 162 games as opposed to 154. If one views the likelihood of a perfect game being throw as a probability problem, each game pitched (except those pitched by Brian Anderson) has a shot at being a perfect game. The more games there are the better than chance that a perfect game is thrown.

I though that I would use the number of games and on-base percentage yearly to calculate the probability of a perfecto being thrown in that year. However, I wasn't happy using the standard OBP. For those of you unfamiliar with the stat, it is the sum of a player's hits, walks, and hit-by-a-pitch divided by his total plate appearances, i.e., the sum of his at-bats, walks, hit-by-a-pitch, and sacrifice flies. It was a decent predictor but had some obvious issues.

First, it counts sacrifice flies as a plate appearance (actually, initially the devisors of OBP went back and forth on this issue for a few years before settling on including sac flies). Now, obviously in a perfect game, there is no possibility of sac flies since whenever a given batter is at the plate there could not possibly be someone already on base, let alone at third, for him to drive in with a sac fly.

I thought about taking sac flies out of the plate appearance total, but decided against it for two reasons: 1) Sac flies were not officially recorded prior to the 1954 season. Prior to that they were just fly outs. If I removed them from the post-1954 plate appearances, it wouldn't be consistent with the pre-sac fly era statistical record.

2) I don't have a lot of faith in sac flies. Sure, if it's the bottom of the ninth of a tie ballgame and there's a runner at third with fewer than two outs, the batter is trying to drive the ball deep to the outfield to score the runner and win the game. That clearly is a sac fly (though often the outfield is drawn in and the ball drops for a hit).

However, let's say there are runners at first and third with one out in the ninth and the team at bat down by three. The batter drives a ball deep to left to score the runner, is credited with a sac fly, and receives high fives all around in the dugout. But that batter wasn't trying to sacrifice himself to score the run: he was trying to hit the ball hard possibly for a home run or a gap double to score the runners.

I just don't buy that the sac flies accurately capture a batter sacrificing himself for a run. The framers of on-base percentage didn't but it either. That's why they eventually included sac flies but not bunts in the total plate appearances.

Anyway, I kept sac flies in the total plate appearance totals.

That then leads us to sacrifice bunts. They are not in the plate appearance totals and I decided to keep it that way. Again, in a perfect game, there is no possibility of a bunt except for a base hit and then the at-bat would result in a hit, an error, or an out, but not a sacrifice bunt. Bunts have been recorded since 1895, and more importantly unlike sac flies they do capture an intentional sacrifice on the batter's part.

So bunts are not included in the total plate appearances calculations.

Next there is the issue of intentional walks. No one is going to give someone a free first-base pass when he is pitching a perfect game. Intentional walks have only been recorded since 1955, but I decided to subtract them from the walk totals, which affects both parts of the equation. Intentional walks will not be included in total plate appearances but will also be subtracted from the calculations for times on base.

Lastly and possibly the biggest issue with using on-base percentage, OBP ignores errors. Well, more precisely, if a player reaches on an error, it counts as an at-bat and therefore, a plate appearance but not as a time on base. However, an error would break up a perfect game. Therefore, I am including errors in the equation, which I will no longer call on-base percentage, since it is far afield of the official definition. I'll just call the new equation reached percentage—why not?

By the way, I will not be including passed balls, wild pitches, balks, or any other stat since they, for the most part, occur when there are already runners on base. (I know that a balk could be called with no runner, thereby charging the pitcher with a ball. Also, a dropped third strike could be called a passed ball if the batter reaches. The same goes for a wild pitch. But I think those instances are so rare that they can be ignored.)

The final "reached" equation combines batting and fielding stats: It's the sum of hits plus hit-by-a-pitch plus walks minus intentional walks, all divided by "modified" plate appearances (the sum of at-bats, hit-by-a-pitch, sac flies, and walks minus intentional walks). Given that it combines the offensive stats for the team at batter and the defensive stats for the team pitching, it can only be done at the league level. (Note, that interleague play complicates all of this, but instead of going to the major-league level as a result, I decided to keep it at the league.)

I then took the reached percentage and subtracted it from one to get the percentage of times that the player did not reach. I then multiplied raise it to the 27th power to represent the full complement of plate appearances in a nine-inning perfect game. That gave me the probability that a given game in a given year could result in a perfect game. I then multiplied that by the total number of games (one game per team) to arrive at the expected number of perfect games for a given league in a given year.

Now here are the results:

I even ran the numbers for the postseason, but won't try your patience by posting it all here. I'll just list 1956 and the overall numbers:

Anyway, if you look at the regular-season table totals, you'll notice that even though OBP has remained about the same in the expansion era as it was in the pre-expansion era, due to improved defensive play, fewer men, about 3.5%, reach base after expansion. The result is that a perfect game is 4-5 times more likely to be thrown. Add in the extra games, and the expected number of perfect games in the expansion era trebles the pre-expansion expectation.

Actual perfect games do seem to occur when the expectation numbers have a sudden increase (e.g., 1880, 1904, 1908, 1968, and 1988), but also seem to happen when the expectation level is relatively low (1922 and 1994). I take that to mean that probability plays a guiding role but that there's still a great deal of randomness to the whole thing.

You may also notice that the expected number of perfect games is about a quarter of the actual (about 3.25 to 15). Obviously, basing the calculations on the league average for defense and offense was not an entirely accurate model for reality. It appears that the advantage of a very good defensive team facing a very poor offensive team outweighs the disadvantage of a very poor defensive team facing a very good offense.

A study for another day might be to based the calculations not at the league level but at the team level and they formulate the expected number of perfect games based on the matchups in that year. The 1950 Phils offense against the 1950 Braves defense if they played 33 times as they did in the days of the 154-game schedule. Maybe that would give us a more accurate picture.

Then again there is the possibility that pitchers exceed expectations why there is a possibility of a perfect game. Look at that Kevin Costner movie after all (although how difficult could it be to pitch a perfect game if Kelly Preston is the prize that you'd win).

Whatever the cause, given that actual results generally follow expectations, I have to believe that there's a bit more than randomness in the mix.