When I was a kid, I played a lot of sports board games, from Strat-O-Matic (MLB baseball, NFL football, NBA basketball, NHL hockey and even a card and dice NCAA football game), to APBA (NBA basketball and Saddle Racing), to Statis-Pro (MLB baseball, NBA basketball). I played Speed Circuit (an under-rated game in my opinion), Avalon Hill’s USAC Auto Racing, Win, Place & Show, Paydirt… it’s a long list. In addition to all of that, I once played a World War II war game by Metagaming Concepts called “Hitler’s War.” I bought it because it was a “pocket game” and therefore not very expensive.

I am a huge fan of replays (even though I never made it past a few games when I played regularly as a kid) so usually when I played any of the sports games I owned I played solitaire and didn’t really care who won. I was much more focused on the stats, which I assiduously scribbled down on a scoresheet or, more often, a sheet of notebook paper.

##### Metagaming

Hitler’s War was different. I played against my uncle and brother and I really wanted to win. Indeed, I wanted to win so much I spent the night before preparing. And I don’t mean merely thinking through scenarios, I mean calculating the dice probabilities associated with various strategies. I was a “metagamer” before I even understood what that meant.

These days, I view metagamers and metagaming in a pretty negative light, even if I understand the desire to do it. (For the record, the game dragged on through the night and into the early morning, my early success rolling through Poland was halted and I grew irritable as my losses increased. So much for metagaming!)

Interestingly enough, a number of games seem to encourage it, including the Strat-O-Matic game company which publishes a baseball ratings guide

This article is all about metagaming. In it, I explore Strat-O-Matic’s pitcher fatigue system and compare the performance of a pitcher who has reached his point of weakness to his nominal performance when not fatigued. While I will utilize an equation or two here this is not intended to be anything more than a crude estimate.

This discussion will make more sense with an example so I will be referring to Randy Johnson’s 1995 Strat card throughout. Since I don’t have permission to include his card here, I’ll be sure to discuss only the relevant details.

*Against right-handed batters*, Johnson’s card includes two strikeouts that are changed to singles once he reaches his point of weakness. These occur in columns 4 and 6 and correspond to dice rolls of 7 and 9, respectively. The dice rolls are the same against lefties but occur in columns 5 and 6 respectively.

Now, I *could* spend a fair amount of time “reverse-engineering” Johnson’s card to show the probabilities associated with each result on his card and from these determine the probability he allows a walk, single, double, etc. I did a lot of that in preparation for this article to make sure the numbers I was calculating seemed reasonable. They did.

##### The mathematics of fatigue

Fortunately, trying to assess the effect fatigue has on his performance isn’t so complicated; we need only focus on the outs that get transformed to singles. To do so, we’ll need to consider how the probability of a right-handed batter getting a hit off Johnson increases when he is fatigued. This isn’t hard to do. The number of singles Johnson allows when fatigued that he would not have allowed otherwise is 6 + 4 = 10. This is due to the fact there are 6 chances of rolling a 7 and 4 chances of rolling a 9. This is from a total of 36 + 36 + 36 – 7 = 101 chances. Note that I am subtracting 7 in this case because Johnson allows a walk when a 7 is rolled in column 6 and walks don’t count as at bats and are thus ignored when calculating batting averages. At first glance, this would seem to indicate that Johnson allows opposing RH batters to bat 10/101 = .099 ≈ 100 points higher when fatigued. But this isn’t quite right.

Strat-O-Matic results are obtained from the pitcher’s card only half the time. In general, the formula looks like this:

*BA*_{nominal} = [(*BA*_{b} + *BA*_{p}) / 2]

*BA*_{b} is the batter’s batting average and *BA*_{p} is batting average allowed by the pitcher. (Note: These ** are not** raw averages. They have been adjusted to ensure players will duplicate their real-life statistics when facing the same level of competition).

Here is how things look when Johnson has reached his point of weakness and is said to be fatigued.

*BA*_{fatigued} = [(*BA*_{b} + *BA*_{p} + 0.099) / 2] = (*BA*_{b} + *BA*_{p}) / 2 + .099 / 2 = *BA*_{nominal} + 0.099 / 2

The last bit is important. It shows that Johnson will allow opposing batters to hits approximately 50 points higher when fatigued. I might have just said that a few paragraphs ago— it seems intuitive— but I feel it’s important to be deliberate in these cases since our intuition is sometimes wrong. (Note: The same holds true for left-handed batters, whom Johnson faced far fewer times).

##### Did Strat-O-Matic get it right?

Does Randy Johnson allow opposing hitters to bat 50 points higher when he is tired? We don’t really know. According to the data (which includes *both* left- and right-handed batters), Johnson’s performance drop-off occurs during innings 4 thru 6, not innings 7 thru 9, but we don’t know if he reached his point of weakness in any of the 23 games he lasted into the seventh inning or further; we only know that his Strat endurance factor inning is the seventh inning. Of course, with the statistics available today, it would possible (though tedious) to calculate how many innings he pitched in real-life when he would have been considered fatigued by Strat-O-Matic rules but I won’t bother to do that here since the point isn’t to show whether or not the point of weakness rule is realistic or not. I happen to like it and suspect a big reason for it is to encourage realistic usage, which I support.

**Table 1: Randy Johnson’s 1995 Performance By Inning**

Split |
G |
IP |
ERA |
PA |
AB |
H |
BB |
SO |
SO/W |
BA |
OBP |
SLG |
OPS |

Innings 1-3 |
30 | 90 | 1.80 | 357 | 328 | 60 | 26 | 130 | 5.0 | .183 | .245 | .265 | .510 |

Innings 4-6 |
30 | 87.1 | 2.68 | 359 | 323 | 69 | 33 | 112 | 3.4 | .214 | .290 | .337 | .627 |

Innings 7-9 |
23 | 37 | 3.65 | 150 | 141 | 30 | 6 | 52 | 8.7 | .213 | .260 | .312 | .572 |

I did want to further explore how me might analyze the effects of fatigue on Johnson’s performance. We know that he will allow opposing batters to hit for a higher average but haven’t yet considered what that implies. I would like to know how many more runs he might be expected to allow under those circumstances. How much higher will his ERA be when fatigued?

##### The effect of fatigue on run-scoring

The table above provides an answer but it isn’t one I’d entirely trust. Most of Johnson’s stats are better in innings 7 thru 9 than they are in innings 4 thru 6 so it isn’t clear to me whether he was just extremely lucky to have allowed so few runs during those middle innings or simply unlucky in that regard in later innings. I’ll approach this question by tossing up a graph and a few equations that will make it seem like I’m doing real science when really all I’m doing is making a crude guess.

The graph below shows how a pitcher’s WHIP is related to runs scored during the 1995 American League season. Without delving into the details, I eliminated players with fewer than approximately 152 at bats and estimated runs scored using sabermetrics. Next, I plotted the relationship between WHIP and runs scored (see below). It’s far from perfect but it provides at least a crude estimate. The linear regression equation in the white box on the right shows how changes in WHIP lead to more or less runs. For example, a 1.0 change in WHIP leads to 5.33 more or fewer runs.

Let’s try applying this equation to Johnson. Here are his statistics against right-handed batters in 1995:

Split |
PA |
AB |
IP† |
R |
H |
BB |
BA |
WHIP |
R/G |
Estimated R/G |

vs RHB |
754 | 688 | 186.2 | 54* | 145 | 59 | .211 | 1.093 | 2.60 | 3.14 |

* The number of runs allowed vs RHB (54) and the number of runs allowed vs LHB (7) don’t add up to the 65 runs Johnson allowed. Thus, the R/G could be between 2.60 and 2.80.

† The number of innings pitched were estimated by multiplying the proportion of plate appearances versus RHB by the total number of innings pitched, 214.1.

They suggest that the linear regression equation, applied to Johnson, overestimates the number of runs he should allow by anywhere from about a third to a half of a run per game. While this isn’t the purpose of the equation, it’s important to keep in mind we’re pouring various quantities into beakers and test tubes, not performing real science.

So how do things change when Randy Johnson is fatigued? Since the only thing that changes is the number of hits (specifically singles) Johnson allows, that is all we’ll calculate: (.099 ÷ 2 × 688) ≈ 33.7 additional hits allowed. This produces a marginal WHIP rating of approximately 0.181. Since 0.181 × 5.3302 ≈ 0.96 runs per game, we conclude that Johnson’s ERA will be nearly a run higher when fatigued, an estimate that compares favorably with the actual data, which suggests a 0.97 rise in ERA from innings 4-6 to innings 7-9 (refer to Table 1 above).

##### Much Ado about nothing?

It obviously isn’t necessary to expend this sort of effort calculate a pitcher’s estimated performance when fatigued, but I wanted to provide a rough assessment of how fatigue effects even a pitcher of Johnson’s caliber and, as we observed, the effects are considerable. Beyond that, knowing how a pitcher performs when tired is really only useful when comparing him to relief pitchers who may replace him, and for this, it is probably enough to glace at their respective cards, noting that Johnson’s strikeouts will count as hits.