How we cite our quotes: (Chapter.Paragraph)
Quote #7
"You'll have to find yourself another little friend," [Hatsumomo] said to me. "After Pumpkin and I have had our talk, she'll know better than to speak a word to you in the future. Won't you, Pumpkin?" (12.52)
Just as Mameha uses Chiyo as a pawn against Hatsumomo, Hatusmomo uses Pumpkin as her own pawn against Chiyo.
Quote #8
[Mameha] was relishing the thought of seeing Hatsumomo destroyed. (15.34)
We later learn that Mameha takes on Chiyo at a request from the Chairman. But she definitely knows an opportunity to demolish her rival when she sees it, and she takes it.
Quote #9
For one thing, the bidding hadn't been a contest between Dr. Crab and Nobu at all. It had ended up a contest between Dr. Crab and the Baron. (24.3)
Men compete too, like these men trying to outbid each other for Sayuri's virginity. Men will later compete to be her danna too. Just as Mother knows the value of having two rivals fight for a goal, the geisha use the men's competitive nature to their advantage.