There's been some discussion of late about the often repeated wisdom that "a run saved is equal to a run earned." For just about any reasonable decision-making process, I think that's a pretty safe rule to follow. Pedantic jerk that I am, however, I did a small amount of work to show how a simple and commonly used model (pythagorean win expectation) doesn't actually subscribe to the "run saved = run scored" statement. Before going any further, I want to point out that the results I show here rely on a trust in pythagorean expectation that is entirely unreasonable given the likely (in)accuracy of pythagorean expectation. But why would I let that stop me?

Pythagorean expectation is a simple way to predict a team's winning percentage as a function of runs scored and allowed. There are plenty of places to read about it, but wikipedia is probably good enough if you've never heard about it before. The resultant equation is very simple:

winning percentage = 1/(1+(RA/RS)^2)

where RA is the number of runs allowed and RS is the number of runs scored. The top left heatmap in the image shows the expected number of wins as a function of any reasonably achievable pair of runs scored and runs allowed. Hopefully the results of that plot aren't surprising: if you score the same number of runs as you allow, you have an expectation of winning 81 games (white); as you score more runs than you allow, you win more than 81 games (red) and vice versa if you are outscored (blue).

The rest of the plots show the results of a very simple experiment. Lets say that we are comparing the impact of upgrading the team by 10 runs. For simplicity's sake, let's pretend it's the AL and I'm deciding between upgrading a pitcher (and reducing my runs allowed by 10) or upgrading at DH (and increasing my runs scored by 10). The top right and lower left plots show the expected change in number of wins for these two options.

As you probably know, 10 runs generally means about 1 more win over the course of the season (white in both plots). For middle of the road teams that score and allow around 800 runs, the 10 runs = 1 win rule holds up pretty well. For teams that score a lot of runs already, adding an extra 10 runs on offense doesn't help as much (blue region of top right plot). Conversely, for teams that already allow fewer runs than is typical, saving 10 more runs is actually worth more than a win (red region of the bottom left plot).

The main point of this exercise is summarized in the bottom right plot. Here I've plotted the difference in wins added if you score 10 more runs rather than allowing 10 fewer. As you can see, for teams that score about the same as they allow, it doesn't really matter where you add the extra runs (white region on the diagonal). For good teams, however (ones that score more than they allow), a run saved is actually more valuable than a run scored (blue region). In contrast, for bad teams, a run saved is less valuable. To reiterate, pythagorean expectation probably isn't accurate enough for these results to be meaningful when making real roster decisions, but if you trust it completely, good teams should get more bang for their buck by adding pitchers and defensive wizards whereas bad teams should be targeting hitters (assuming all else is equal).