When Waverly challenges her mother in "Rules of the Game," she wants some degree of independence. Playing chess allows Waverly to have her own room, separate from her brothers, and she also gets out of eating all her rice if she doesn't feel like it. When her mother goes around Chinatown introducing Waverly as her daughter, Waverly tells her to stop doing so. It's clear that Waverly wants to claim all the credit for her chess prowess as her own so that she can become more independent from her mother.
In addition, Waverly wants to beat her mother at her own game. She says in the story's first line, "I was six when my mother taught me the art of invisible strength." Her mother teaches her to hide what she wants and that the "strongest wind cannot be seen." As Waverly improves her chess skills, she is developing this invisible strength to challenge her mother, just as her mother has taught her. She uses this strength to try to defeat her mother at her own game and to win some degree of freedom from her mother. Her mother also wants to use this invisible strength to control her daughter, and in this way they are similar.