A freelance film director recently described walking into a negotiation. She was ready: She had armed herself with stats and evidence and had practiced her pitch. But instead of diving into why she deserved the project—and the money that came along with it—she began with the following: “I just want to say up front that I’m going to negotiate, and the research shows that you’re going to like me less when I do.”

She could see the wheels turning in the minds of her colleagues. But she was right. When women ask for what they deserve,...