Play to keep the winnings. Also, more generally, especially in America, 'in deadly earnest; in such a way that the result will stand'.
The game of marbles is older than recorded history - marbles have been found in ancient Egyptian and Roman tombs. There is a version of the game that has been in vogue since at least the mid 19th century which is called keepies or for keeps. As the name suggests, whoever wins the game keeps all the marbles. This is cited in an 1861 edition of the journal The Ladies' Repository:
"See, mother, what a lot of marbles I've got!" said John.
"I want you to make me a great big bag to put them in."
"Why, where did you get so many, my son?" asked his mother.
"I won them from Peter Jones. See I got his glass taw too. I loaned him one of mine to play with while he put that in the ring. Isn't it pretty?"
"How much did you pay him for them?"
"Pay him! Nothing. He and I played for 'keeps' and I was the best player and won all his."
The US meaning of the term is recorded from almost the same date. In 1866, Shanks' Personal Recollections included this:
"'Fighting for keeps' is army slang and signifies fighting in deadly earnest."