A few weeks ago, the topic for the BP Lineup Card was "Unanswered questions for the second half." I noted that at the time, the Cardinals were several games behind both the Pirates and the Reds in the NL Central standings, despite the fact that they had a better Pythagorean record than either. In theory, the Cardinals should have been atop the NL Central.

As my father is fond of pointing out, everything works in theory.

The question that I posed was whether Pythagorean records were really a "true" gauge of a team's quality. It's a topic that has never been fully resolved. Are differences between actual record and Pythaogrean records a matter of luck, or do they reflect some underlying property of a team?

Both logic and previous research tell us that there's a really easy way to outperform your Pythagorean record: win a lot of one-run games. In one-run wins, the Pythagorean calculations see only that a team scored one more run than its opponent. The standings see that it won the game. So, are one-run wins a skill? The Orioles have a negative run-differential overall this season, but are an astounding 23-6 in one-run games so far this year, and as a result, are in the playoff hunt. Are they extremely lucky, or are they just gritty like that and "know how to win" the close ones?

There are a lot of theories on the issue: teams with good bullpens (or good closers) can protect close leads. Teams with good starters can overcome a bad offensive day. Teams with crafty managers can win because the manager can expertly push all the buttons. Then again, no one ever takes the contrary position. A team with a bunch of one-run wins might belie a bullpen or a manager who can bungle a five-run lead into a one-run lead, barely hanging on to fly the W flag over the stadium.

After studying the issue, I can boil down the recipe for winning one-run games to three words: wear white pants. Want to know how I got there?

Warning! Gory Mathematical Details Ahead!
First off, let's consider that there is a provision in the rules of baseball that encourages games to end with a difference of one run in the score. In the bottom of the ninth (or an extra inning), as soon as the home team takes the lead, the game ends. While the final margin of victory can be as many as four (game tied, home team hits a grand slam), the most common case is that the home team takes a one-run lead and then throws pies at each other.

In fact, while home teams win roughly 53-54 percent of all games, they won 61 percent of all one-run games from 1993-2011. Games won by the visiting team are decided by one run about 23 percent of the time, but home teams collect 31 percent of their wins by a single tally. Overall, 28.2 percent of all games are decided by one run.

The thing about a walk-off is that, by definition, the ninth inning (and game) ends before three outs are made. Suppose for a moment that we made the home team keep going until three were out. Indeed, a home team that protects a one-run lead in the top of the ninth leaves a whole bottom of the ninth on the table. If the home team took its turn as it was supposed to, it might score a few more runs… and suddenly, it's no longer a one-run game.

There are a few other interesting things about one-run games that might be instructive. First off, they are generally lower scoring than other types of games, with one-run games featuring an average of eight runs between the two teams vs. 10 for games decided by other margins. Then again, a good way to make sure a game is not a one-run game is for one team to score 12 all by itself. I restricted the non-one-run-game sample to games decided by either two or three runs. These games featured, on average, 8.6 runs, and the difference was still significant.

Not only that, but games decided by one run tended to be closer-played games, even after the first inning. In a game that is eventually decided by one run, the average deficit between the two teams after the first inning is .72 runs. In all other games, it's one run even. Restricting the non-one-run sample to games that end with a two- or three-run win (.82 runs, for the curious), the difference is still significant. One-run games tend to be low scoring and more closely played, even from the outset. A good starting pitcher might be helpful there.

But hold on: about 30 percent of one-run games are shootouts featuring at least 11 runs, and 10 percent feature 15 runs or more. An offense that can keep up with the Adam Jones-es would be helpful in these games, because the pitching staff isn't having such a good day.

What about the theory that a good closer is the key? Let's take a look at what games that are eventually decided by one run look like headed into the ninth.

  • Fifty percent enter the ninth inning with the team that eventually wins by one run up by one run. Most commonly, this represents a scoreless ninth. This is the work of a good closer. Occasionally, it's the mark of a closer who gives up some runs, but his offense bails him out (or if he is the visiting pitcher, his team scores a few runs in the top of the ninth, and he does his best to give them back.)
  • In roughly a quarter, the teams are tied after eight innings. Here, both the pitching and the offense need to come through.
  • In 14 percent, the team that wins by one was actually ahead by more than one, meaning that the closer gave up runs.
  • In 11 percent, the team that won went into the ninth trailing. Here, the heroics will have to come from the offense.

The thing about all these numbers is that you can make a case that strengths in certain areas of the roster would lead to success in one-run games. It's hard to argue that roster talent would be inconsequential to winning one-run games, but it's harder still to say that a general manager would get the most bang for his buck by focusing on X.

For a moment, I want to focus on the 25 percent of one-run games that go into the ninth tied. In some sense, from this point onward, the fundamental characteristic of the game has changed. From this point onward, the game is a series of one-inning sudden-death games. Prior to this, if the opponents scored in the fifth inning, you could make it up in the seventh. But no more. From a sampling perspective, we've gone from a nine-inning sample of team quality on that day to a one-inning sample. Smaller sample sizes mean more variance: anything can happen in extra innings. More than that, we're talking about a handful of plate appearances. Even a good reliever can have a bad day.

I once found that Pythagorean residuals—that is, how much a team over- or under-performs its Pythagorean record—are correlated with the team's overall winning percentage, although only moderately. Teams that win 100 games have somewhat of a tendency to out-perform their Pythagorean records, while 60-win teams tend to under-perform, but there are plenty of counter-examples. At the time, I wasn't sure what that was about. I'd hazard a guess now that what we're seeing is that teams that have good players are more likely to win games in general. These are players that you would want on the field in a close game where the game hangs in the balance. So, there probably is an advantage to having them that shows in the residuals. The problem is that as the game reaches its autumn, the element of randomness creeps in all the more. In extra innings, all you need is one good (lucky?) swing of the bat. Good players are more likely to take that swing, but even bad teams score runs now and again.

But let's do one more test. Let's look at how reliable performance in one-run games is. I'll use the KR-21 formula that I introduced a few weeks ago. For my sample, I'll look at all games between 1993-2011 that were decided by one run or more. For each franchise, I took a five-year block of games (1993-1997, 1998-2002, 2003-2007, 2008-2011… yes, I know that last one is only a four-year window) and treated them all as big long seasons. This was to increase the number of one-run games that I could look at: since real teams play only about 40 one-run games in a single season, I needed more data to work with. There are plenty of problems with this analysis strategy, but the idea is that even over the course of a few years, a team maintains a lot of the same players (usually). It's not perfect, but it's about the best we have available.

I looked at the reliability of a team's record in a single one-run game, then in two one-run games, then three, all the way up to 100. The KR-21 reliability settled in around .30 when I got all the way to 100 one-run games. At 40 games, which is roughly the number that most teams play over the course of a season, reliability was a paltry .17.

This suggests that the one-run records of most teams are not stable. If we gave the Orioles another 29 one-run games, they are not likely to go 23-6 again. Sorry, Orioles fans, those orange uniforms are probably going to turn back into pumpkins.

The Conclusion
To say that there is no skill in a team winning one-run games would be wrong. Teams that are good at scoring runs and preventing the other team from doing so will have a better chance at winning them. The problem is that one-run games actually happen in several different ways, and winning them would rely on the abilities of different parts of the roster. The way in which they unfold, often involving extra innings, adds an extra layer of variability over and above that of a normal game. Baseball is a game with a lot of randomness in it already, and that randomness overwhelms the effect of skill. Based on this, I wouldn't recommend reading much into a team's one-run record.

If your goal today is to win a game by one run, may I suggest the most predictable way of all to ensure that it happens: wear white pants.