Dave Cameron at FanGraphs writes about the Blue Jays adding Tulo to a lineup that already scores the most runs: "There are no diminishing returns to scoring more runs; there is no point on offense to where the marginal value of a run scored is worth less than preventing a run from being allowed on defense."
This is an excellent article with lots of research, so I hate to nitpick. But the above statement isn't completely true. For any team that scores more runs that it allows (which includes pretty much every team that makes the playoffs), preventing a run is more valuable than scoring an additional run.
It's because of the Pythagorean Theorem of Baseball, which states that the ratio of a team's wins to losses corresponds to the ratio of their runs scored to runs allowed (actually, to the SQUARE of these numbers, but that's not essential to this analysis).
Take a team that's on pace to score 600 runs and allow 550 runs. Their ratio of wins to losses should be (600 x 600) : (550 x 550) or 3600:3025. Or, to put it another way, their winning percentage should be:
(600 x 600) / (600 x 600) + (550 x 550) = .543
(that's a record of 88-74 over a 162-game season)
Let's imagine they have a choice to add a hitter that will give them 50 extra runs, or add a pitcher than will prevent 50 runs.
After adding the hitter, they score 650 runs but still allow 550. Their new projected winning percentage is:
(650 x 650) / ((650 x 650) + (550 x 550)) = .583 (94 wins)
After adding the pitcher, they still score 600 runs but now they allow only 500. Their new projected winning percentage is:
(600 x 600) / ((600 x 600) + (500 x 500)) = .590 (96 wins).
That's a difference of two wins. This may not sound like a lot, but the difference between making the playoffs and going home has averaged just 1.5 games in the American League over the last 4 seasons.
The interesting thing about this fact is that it doesn't matter if you are a great offensive team or a mediocre one. As long as you are a good team, one that scores more runs than it allows, it's always more valuable to prevent a run than to score a run.
And ... for fans of the Red Sox or Phillies ... the reverse is also true. If you are allowing more runs than you are scoring, you will improve your team more by adding offense than by adding the equivalent amount of defense.
Interesting, I hadn't heard about Pythagoras and his theories around baseball before. I quickly checked it against my local teams and it was almost exact in predicting the Giants wins, and unsurprisingly it was way off on the A's, who have won a number of their games by blowouts while typically losing close games. His article argues more about the value of adding another bat to a good hitting team, saying the gain is larger than one would predict, but I don't think it actually address the issue of the incremental value of additional runs. One of the articles I remember best from the Bill James book I read was along the lines that the first 5 runs that a team scored in a game added the same amount of additional win %, but after a point, there must be a decrease in the additional value of a run, because you can't win more than 100% of the time for any amount of runs.
Why is this true? I understand the basic assertion (this math predicts wins better than a straight runs differential analysis, and therefore based on past patterns a better than 50% team is better off with an improvement in defense than an equivalent improvement in offense), but I'm not clear on why that pattern would hold. Is it because increasing (already good) offense increases the number of blowout games, whereas increasing defense wins more of the marginal close games? E.g. you almost never score 10+ runs and lose, so increasing the number of games in which you score 10+ runs isn't very valuable, whereas decreasing the opposing score from 3 to 2 (or 5 to 4) is very valuable in terms of winning more games? That makes a certain amount of intuitive sense, but I still find myself wondering about it. Imagine that your team would, without changing its lineup, win 5 games by 3+ runs, 10 games by 2 runs, 15 games by 1 run (5 of which were in extra innings), lose 13 games by 1 run (4 of which were in extra innings), lose 8 games by 2 runs, and lose 3 games by 3+ runs, for a total record to the end of the season of 30-24. It can change its line-up by either increasing its average run production by 1 per game, or by decreasing its average runs allowed by one. It seems like those would have the same effect: the tied games that would have been lost in extra innings turn into wins, as do some fraction of the games that were lost by one run without extra innings (because they would be moved to being ties at the end of nine innings, and then have a slightly greater than 50% chance of being won in extra innings).
Hmm. Maybe it's because extra runs won't be evenly distributed? If you think of each extra run produced as a fraction of an extra run produced per at-bat, then because you get more at-bats in games when your team is already doing well, you get more of the extra runs in the games where you're already scoring a lot of runs. Conversely, the runs prevented will be concentrated towards when the opposing team is getting lots of at-bats. Because you're a better than average team, the games where your batting is above average are likely to be wins already, but the games where their batting is above average are likely to be losses or close--so increasing your defense is more likely to save runs in the games where it matters, whereas increasing your offense is more likely to add runs in the games which you win either way. That seems like a plausible explanation.
It seems like those would have the same effect: the tied games that would have been lost in extra innings turn into wins, as do some fraction of the games that were lost by one run without extra innings (because they would be moved to being ties at the end of nine innings, and then have a slightly greater than 50% chance of being won in extra innings).
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