What prompts chief Bromden's descion to join in the fight? What does his participation reveal about the development of his character?

Chief tries to protect McMurphy in a few places. I'm not sure exactly where you are referring to but I know the Chief tries to protect McMurphy when he gets into a fight with the black orderlies.He also tries to protect McMurphy when they go to the Disturbed ward and EST therapy. Chief sees McMurrphy as outside the "Big Combine" system. In McMurphy there is hope for Chief, there is salvation.