Is one of these preferable grammatically, or are both acceptable? (Or neither?)
Joe and Tom each won his final match to assure themselves of playoff spots.
Joe and Tom each won his final match to assure himself of a playoff spot.
I’m guessing they’re both OK, and it’s just a matter of whether “each” applies to “won his final match” or “won his final match to assure himself of a playoff spot.” Is there a technical name for an “each clause” like this – I’m not even sure “clause” is the right word, but I mean the stuff that “each” applies to. Also, how would you diagram these sentences (assuming I’m right and they’re both legitimate sentences)?
No – the second is correct — because of the “each,” you’re setting up a singular construction (or rather a matched pair of singular constructions) afterward. You’d use “themselves” after a “both.”
The only problem comes if Joe or Tom is a woman, or in a situation where you are not sure of the gender of those involved. If they are both women, you can write:
“Jo and Tammy each won her final match to assure herself of a playoff spot.”
But for mixed or uncertain gender, the easiest solution (in my opinion) is:
“The two players each won their final match to assure themself of a playoff spot.”
However, some authorities don’t like you using “themself” in this situation.