What is the author's intention in The Rules of the Game?
I am sure that there will be differences of opinion.
Since the author does not explicitly say why she wrote the story, we cannot say for certain what the author's intention was. However, there are a few clues in the work to suggest why she might have written it.
First, the story here is to introduce Waverly and her upbringing and life. It shows that she is from an immigrant family in San Francisco. It also shows her family dynamics, and most importantly it shows what her relationship to her mother is like.
Secondly, the story also shows what kind of person Waverly is. She is smart and capable. She is also a fighter. She learns the "art of invisible strength" from her mother, and she is willing to use it on her mother.
From this perspective, we can say that the author's purpose is to write about an interesting character and her relationship with her mother, as she grows up.