Mike,
It was me...and it was over two years ago. It pertained to a similar situation. If you look back, the TDA introduced a rule that made reference to "her" instead of "him" or "she" instead of "he." In any event, my suggestion was to replace the singular pronoun with the plural, "their"...only when you feel it's appropriate.
This has been a problem for many of the most educated writers...forever! After a little research, I believe that the plural is almost always preferred, especially when repeated over and over again. I think, "he" or "she" or "him" or "her" is easier to read, (and less boring), when used sparingly. Poker rules require constant reference to the actions of players, so replacing the singular is much easier to read, and certainly understandable. Although, it will require a little reconstruction of a sentence. I'm not sure it's quite as easy as Thomas McGee suggested. I will say that Thomas' Book is the best rule set I've ever read, as far as avoiding the singular pronoun use.
I also want to say...when we are not sure of the gender, replacing "he" with "she" will never work.