There are three conflicts in the short story, "The Rules of the Game."
First, there is the game, chess. Waverly learns chess and she wants to win. She actually excels. She beats her brothers and then gains national fame as a young player. Hence, if we look at chess as a contest, then chess is also a type of conflict. This is the external conflict of the story.
Second, there is also the conflict between Waverly and her mother. Will Waverly admit that her mother has given her everything within her power, or will Waverly assert her independence? At the end of the short story, she sees her relationship with her as a chess game. Here is how it ends:
Her black men advanced across the plane, slowly marching to each successive level as a single unit. My white pieces screamed as they scurried and fell off the board one by one. As her men drew closer to my edge, I felt myself growing light. I rose up into the air and flew out the window. Higher and higher, above the alley, over the tops of tiled roofs, where I was gathered up by the wind and pushed up toward the night sky until everything below me disappeared and I was alone.
Finally, related to this point, there is the internal conflict within herself. She is a girl - can she beat men? She is an immigrant, can she fit in outside of Chinatown? She is young, can she defeat her mother?