Stand at the head game


From Fuller's "The holy state" (1642):

[Paula] built one mo∣nasterie for men, and three for women. It will be worth our pains to take notice of some principall of the orders she made in those feminine Academies; because Paula's practice herein was a leading case, though those that came after her went beyond her. For in the rules of monasticall life, Paula stood at the head game, and the Papists in after ages, desirous to better her hand, drew themselves quite out.

I have already received clarification that "drew themselves quite out" means later papists did all they can to improve what she did. But the general sense of the passage is still not complete clear for me. I presumed that "stood at the head game" means "play first violin". Is it correct?
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