The instant Hatsumomo sees Chiyo, she is cruel to her. She does this as a form of psychological warfare, to make Chiyo feel inferior so that Hatsumomo is in a better position to take advantage of her.
"Maybe you're just too pretty yourself to be able to see it elsewhere." (5.13)
A sumo says this to Hatsumomo. It's a good quote, but it's also wrong. Hatsumomo definitely sees that Chiyo is pretty, which positions Chiyo as Gion's Next Top Geisha, and Hatsumomo's number 1 rival.
"Everyone knows how you hate Mameha," Auntie told [Hatsumomo]. "You hate anyone more successful than you." (6.70)
In case you hadn't guessed by this point, Hatsumomo picks her rivals from people she is either threatened by or jealous of.
"She can't bear to have rivals," Mameha went on. "That's the reason she treats you as she does." (10.57)
Chiyo never could even imagine that she would be a rival to the great Hatsumomo, mainly because Hatsumomo makes her feel like garbage. She doesn't have enough self-worth to consider herself a rival.
Mameha must have made up her mind to use me in seeking her revenge on Hatsumomo. (10.86)
Mameha and Hatsumomo are like two queens on opposite side of the chessboard, and both use Chiyo/Sayuri as a pawn. To continue the chess metaphor, a pawn is always capable of becoming a queen herself, which Sayuri does.
"I had no idea what a fine day this would be," [Mother] said. "This morning when I woke up, two useless girls were living in the okiya. Now they'll be fighting it out…and with a couple of the most prominent geisha in Gion ushering them along!" (11.100)
Mother is thrilled because she knows that the two geisha will work extra hard to best one another, which means more money—and an even bigger cut for her.
"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.
[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.
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.
I'd earned more in the past six months than both Hatsumomo and Pumpkin combined. (27.1)
Looks like the competition pays off. Unfortunately, poor Pumpkin serves as collateral damage in the war between Hatsumomo and Mameha, which sets her up as a rival to Sayuri, too.