Austin Hedges had a disastrous season at the plate. He batted .176/.252/.311, good for a 47 wRC+ that placed him last among batters with at least 300 plate appearances. But Hedges had a remarkable season behind the plate. He was worth 27.3 defensive runs above average, second only to J.T. Realmuto and first on a rate basis among all players in baseball.

WAR gives us a handy way to convert this, and WAR was pretty happy with Hedges in 2019. He was worth 1.4 WAR in 347 plate appearances, which works out to 2.4 WAR per 600 plate appearances. An average player would be worth just under 2 WAR per 600, which means Hedges was above-average despite his sub-Mendoza batting stats.

But when someone’s batting line is as poor as Hedges’, there’s an impulse to say that WAR doesn’t capture everything. Hedges isn’t a bad hitter; he’s an exceptionally bad hitter. He had a 47 wRC+! Six pitchers had better batting lines last year. My former colleague Sheryl Ring pointed this out recently, and opined that NL teams simply couldn’t afford to carry a bat that poor in addition to the pitcher’s spot. This made me think several things at once. My first thought was “Of course not! WAR works.” My second thought was “Actually yeah, two pitcher-level batters in a row would be awful. I’m not sure that a linear stat could handle that.” My third thought was: time to do some analysis.

At a high level, it’s clear that wRC+ isn’t simply additive. Consider a team with four batters who always walk or hit a single, and five batters who always strike out. This is an extreme example, of course, but that team would have a roughly average wRC+. It would also be an incredibly potent offense if you stacked your four productive hitters together; it would average something like 7.5 runs a game depending on the timing of singles and walks.

Of course, if you sequenced your hitters poorly, it could also average zero runs per game. When we’re looking at extreme examples (and Hedges is extreme, though not nearly this extreme), the order really does matter. How should one deal with this? It’s a bit of a cliche, but: very carefully.

To work out what Hedges might do to your lineup, I simulated a variety of potential teams. First, I looked at the generic NL lineup. To do this, I took the aggregate batting line of every spot, 1-7 and 9. I don’t just mean the wRC+; I looked at the actual rates of walks, strikeouts, singles, and so on from each spot. From there, I ran two separate simulations; one of the lineup using a normal No. 8 hitter, and one with 2019 Austin Hedges.

In these simulations, I simply played out a nine-inning game, simulating each batter’s turn based on their rate of each outcome. I stripped out intentional walks in each case, but otherwise left everything as is. A million simulations later, we had some results. But before we get to those results, let’s take a quick look at the aforementioned generic batting lines, using wRC+:

Generic NL Lineup Batting Spot wRC+ 1 99 2 114 3 114 4 114 5 100 6 93 7 89 8 78 9 38

No surprise: lineups generally have their best hitters at the top and proceed downward from there. Per my simulation (which includes double plays and somewhat includes sacrifice bunts but excludes steals, pinch hitting, and the effects of pitching and defense), the generic NL offense scores 4.73 runs per game over a million trials. Happily, this matches the actual 4.78 number somewhat closely — closely enough that I’m happy with my simulation’s parameters. Onward!

Take a look at the relevant rate stats for Hedges and the generic No. 8 hitter, because we’re about to swap them:

Batting Lines, Hedges vs. No. 8 Batter BB% 1B% 2B% 3B% HR% K% In Play, Out % No. 8 9.0% 13.8% 4.1% 0.3% 2.8% 23.5% 46.5% Hedges 8.4% 10.0% 2.6% 0.0% 3.2% 31.7% 43.9%

Hedges makes a lot more outs, which is a good explanation of why he has a bad wRC+. In fact, plugging him into that generic NL lineup lowers their scoring output to 4.56 runs a game. He costs his team roughly 0.17 runs per game. Over 162 games (not that any catcher would ever play all of those), he would cost his team 27 runs relative to the average No. 8 hitter!

Next, we’ll repeat the process for an AL lineup. This time, Hedges will be batting ninth. The lineup looks like this:

Generic AL Lineup Batting Order wRC+ 1 109 2 114 3 112 4 109 5 103 6 97 7 86 8 86 9 72

That lineup scores 4.88 runs per game, almost dead on the actual AL average this year. There’s a problem, though: we want to swap Hedges into this lineup, but we need to compare apples to apples. In the NL lineup, he was replacing a 78 wRC+ hitter. In this theoretical AL lineup, he would only be replacing a 72 wRC+ hitter. As ridiculous as this sounds, we need to plug the NL eight hitter into the AL nine spot to make sure we aren’t giving Hedges too much credit.

After accomplishing that bit of legerdemain (it’s easy when your players are just lines in a spreadsheet), our new AL lineup projects for 4.92 runs per game. From there, we simply replace that hitter with Hedges and re-run the simulation. This time, the team’s runs scored projection drops to 4.75 runs per game. That’s also a 0.17 run drop, though it works out to 28.1 runs over 162 games (due to rounding).

That’s a very unsatisfying answer. His impact on both lineups is roughly equivalent, and the actual math of it is hidden in a black box. Taking an average team and plugging in Hedges costs you nearly 30 runs if he replaces a No. 8-caliber hitter, regardless of the people around him. It feels like there should be some rally-killing stacking effect of Hedges and a pitcher in sequence.

But digging into the numbers slightly, a countervailing effect emerges. In the “normal” NL lineup, the pitcher makes the third out while leaving runners on base roughly once every five games; he strands 0.2 runners per game. That number falls to 0.18 with Hedges hitting eighth, because he’s simply on base less often to get stranded. Thus, the pitcher’s poor batting line gets de-emphasized by having a worse hitter in front of him. Over a 162-game season, that’s around three fewer runners stranded on average. That’s right — Hedges actually makes the pitcher’s spot less deleterious by being awful!

We can take an extreme approach to this. What if the No. 8 hitter never got on base? How many fewer runners would the pitcher strand? Taking a normal NL lineup except with a literal black hole in the eight spot, I project merely 0.12 runners stranded per game, a 13-run difference over a full season as compared to a generic lineup.

On the other hand, we could put a great hitter in the eight spot to see how much more damage a pitcher could do with more men on base. Rather than a complete zero, let’s replace the eighth spot with the second spot’s line; a 114 wRC+ with a good OBP. That barely increases the number of runners the pitcher strands; just over 0.2 per game.

The true extreme would be if the eighth hitter always reached base via a walk; that would result in nearly half a runner per game left on base with two outs by the pitcher. Essentially, the eighth spot is counterweighted by the pitcher; the better your eighth hitter is, the higher the pitcher’s leverage is in their average at-bat, and therefore the more the pitcher’s spot hurts your offense.

Despite that slight detour, we have an answer; carrying a tremendously low batting line on your team costs AL teams slightly more than NL teams, though the effect is vanishingly small. But that’s boring. There’s something else we could answer here, while we’re doing the lineup math; we can attempt a sort-of proof of WAR.

First, let’s imagine an exactly league average player. His batting line works out to a 100 wRC+, and he has the offensive rate stats of the average non-pitcher. His defensive and baserunning values are both exactly average; that means that he gets a 0 BsR and 0 Def by FanGraphs’ definitions.

Next, we’ll plug him into the NL lineup. With a 100 wRC+, he should bat sixth. This bumps the six and seven hitters down a slot. That team projects to score 4.84 runs per game. If we assume it allow an NL-average amount of runs per game and plug the results into a Pythagorean predictor, this team is an 82.8-win team. That’s neat — a perfectly average player is worth 1.8 WAR more than the generic eighth hitter we replaced. That roughly tracks with the intuition that an average player is worth 2 WAR per 600 PA and that a No. 8 hitter is slightly above replacement level.

Next, we’ll plug in Hedges. We know from before that his team is only going to score 4.56 runs per game. He was worth essentially 0 BsR, so we can ignore that. This isn’t quite right, because it over counts double plays, but the difference is around a run a year, and there are limits to how useful it is to parse his stats. All we have to do is add in his defense — and what sterling defensive value he had in 2019! He was 27.3 runs above average on defense in only 813 innings, which works out to about 0.30 runs saved per game.

With all of that accounted for, the team projects to score more runs than they allow. In fact, they work out to be an 83.3-win team by the projections — better than the team with the average hitter added. I also put the two hitters into the AL, for completeness’ sake. There, replacing your worst regular with Hedges yields an 83.1-win projection, while plugging in the average player instead gets to 82.8. Even with the disastrous offense, Hedges is again more valuable than a league average player.

This answer, like this whole article, probably isn’t very satisfying. The answer to “can WAR capture these extreme cases” seems to be yes. But I reached that answer via a simulation, which is more telling than showing, and I know that doesn’t make for good reading. For the most part, I can only say: tough. When you’re trying to get an idea of the value of non-linear effects, you need to simulate them, which makes for an opaque model.

But if you feel like playing around with the parameters, feel free. I uploaded the code to GitHub here, and you’re welcome to dabble as desired. It’s certainly not elegant, but it got the job done.

And overall, I’m happy that the result of playing out the games, with sequencing and runners on base and all, mirrors the results of WAR, which is context-neutral. Hedges is a truly extreme player, the worst batter in baseball and the best defender, and even then, WAR does a pretty good job of estimating his value relative to an average player. That’s a satisfactory result for me, even if it doesn’t answer the central question of whether he would hurt an NL or AL lineup more.