When you flip a coin, it's customary to ask the other person, "heads or tails?"
That choice is meaningless.
You picking "tails" doesn't change the probability of you getting what you want. It could just as easily have been assigned to you by someone else. But what it does do, is establish buy-in.
From the moment you call "tails", you've bound yourself to abide by the result of the flip. And that's really the point - nobody cares whether you're heads or tails, they just care that you agree to it.