The Red Sox have a lot of right-handed hitters. Look at the expected starting lineup, and you’ve got Dustin Pedroia, Mookie Betts, Xander Bogaerts, Hanley Ramirez. Sandy Leon masquerades as a switch-hitter, but his splits both in 2016 and before would suggest that he, too, is a righty—just one who happens to take some swings from the left side on occasion. On the other side of things, we have only Andrew Benintendi, Jackie Bradley Jr., and the third baseman, whoever that happens to be (Shaw bats lefty, and Sandoval is the left-handed equivalent of Leon). Even Boston’s best bench bat is the lefty-crushing Chris Young.

It’s a situation which has led some to say the Sox should prioritize a left-handed bat in the offseason. Or, if not prioritize, at least weight slightly in favor of lefties in looking to fill the vacant DH spot.

Weirdly enough, though, when you actually try to build the lineups, it’s not the lineup against right-handed pitchers that’s hard to fill, but the one against lefties.

Here’s the thing: the Red Sox’ righties are actually pretty balanced hitters, or at least fairly capable when it comes to hitting same-handed pitching. Pedroia, for instance, has a .797 career OPS against right-handed pitchers, and hit them better than lefties in 2016. Hanley is a monster against southpaws, but hits righties at a similar pace to Pedroia. Bogaerts’ career split isn’t pretty, but has scaled up nicely with his overall improvement. And Mookie? Mookie is Mookie. Even Leon would be perfectly acceptable if he hit to last year’s levels, though, y’know, that’s hoping for an awful lot.

Against righties, then, something like this:

Dustin Pedroia Xander Bogaerts Mookie Betts Hanley Ramirez Jackie Bradley Jr. Andrew Benintendi Pablo Sandoval (DH) Travis Shaw Sandy Leon

(Order them however you like. Throw Brock Holt in there if you want to.)

That’s not really as bad as it seems. Yeah, Sandoval and Shaw are both in there and nobody’s going to be excited about that. Even in his horrible 2014 season, Sandoval hit to a .744 OPS against right-handed pitchers. Even as his numbers tumbled in 2016, Shaw’s line against righties stayed respectable, finishing at .257/.325/.437. This is a lineup that might lack peaks without David Ortiz, yes, but it’s also one that just doesn’t really end. It’s a monotonous drone of solid-or-better bats that never gives the opposing pitcher a break. And if Sandy Leon can’t hit like he did in 2016, maybe Blake Swihart can pick up the slack against RHP.

On the other hand, against lefties...

Dustin Pedroia Xander Bogaerts Mookie Betts Hanley Ramirez Chris Young Uh...Sandy Leon? Crap. Let’s go Benintendi. His splits are small sample size. Well, Jackie Bradley’s got his glove, at least. Wait, that’s four outfielders. Chris Young is the DH? Oh God this is a 3B.

It’s ridiculous to start. But it dies off hard at the end. Benintendi has a decent chance to prove that his very small sample size splits are completely misleading, but he is a left-handed hitter who might take some time to get up to speed against major league lefties. Jackie Bradley Jr., on the other hand, has shown pretty significant splits over a long enough period to start taking them seriously. And the third base situation is a total disaster when it comes to hitting lefties.

That’s not to say the Sox are going to be in trouble against lefties or righties. You can score with the more even-keel approach vs. right-handed pitchers or a top-heavy attack as we see against lefties above. But when it comes to choosing a bat to add at DH, you have to consider what your alternatives are, and how much improvement they’ll bring to each lineup.

Picking up a left-handed bat basically lets them pull out Sandoval or Shaw from that lineup. That’s the sort of move that offers a lot in terms of catharsis and less in terms of actual upside. Especially given the market the Sox seem to be targeting for DH, you’re just not going to find someone who’s heads-and-tails better than a .750ish OPS against right-handed pitchers. They can find an upgrade, sure, but it’s more likely to be a marginal one than anything else unless Sandoval or Shaw fall off even harder in 2017 (and Yoan Moncada can’t pick up the slack—he scouts better from the left side).

On the other hand, picking up a right-handed bat lets the Red Sox do some shifting in the outfield when necessary, with Chris Young moving out of that DH slot and into the outfield for either Bradley or, if he struggles against lefties, Benintendi. And where it’s hard to find bats who are particularly good against righties because, given the number of righties in the league, that generally makes them simply good hitters, the Sox wouldn’t be as hard-pressed to find a bat that’s a substantial upgrade over the sort of numbers they might expect from some of their weaker hitters against southpaws in 2017.

Unless they’re going to come out of left field with a move that makes huge, surprising changes to the roster, the Red Sox are going to have a top-heavy offense against lefties and a more balanced lineup against righties. They’re going to struggle mightily to find production at 3B against lefties, barring minor miracles. And when it comes right down to it, the bat they add in David Ortiz’ place won’t make a huge difference. It’s a question of a smaller boost in a larger portion of games, or a larger boost in a smaller portion, sacrificing some defense in the process.

The argument here is not, then, that the Sox need to add a bat that can hit left-handed pitchers. Nor is it that the Sox need to add a bat that can hit right-handed pitchers. There are fairly balanced pros and cons in both directions. And while that might not leave the Red Sox with an opportunity to find the perfect fit who they can make better use of than any other team (in the way Chris Young kind of was for them going into 2016), it does free them up to take the best player on offer, or at least the best player who fits into their financial plans now and in the years to come. Lefty, righty? It doesn’t much matter. Certainly not as much as all those right-handed bats on the roster might lead us to believe.