"They'd never admit it, so why should I"

This one has a grievance in it, and the grievance is often fair. The person you're arguing with holds too. You've seen them do it, on the roundabout and the figures and the film, and it seems to you that if you're the one who always climbs down then you're the one who's always wrong, and that isn't true and it isn't fair.

Two things about that. The first is that their holding is the same habit as yours, and it's not evidence that they think they're right. Karina Schumann and Carol Dweck ran four studies in 2014 on who accepts responsibility after doing something wrong, and found that the people who did it more readily were the ones who saw personality as something that changes rather than something fixed. Their reading is that accepting fault feels less threatening if you think of yourself as a person who can grow past it. The people who hold hardest, on that account, aren't the most certain. They're the ones for whom being wrong feels most like a verdict on who they are. If that's the person across from you, their holding tells you nothing about the roadworks.

The second is that "why should I" has an answer that's nothing to do with fairness. It's that your climb-down is the only move in the room that's yours to make. You can't make them admit anything. You can wait for them to go first, and you've been waiting for years, and the waiting has produced a lot of silent miles. Or you can say the five words, and then one of two things happens. Either they climb down too, which is much more common than you'd think, because your climb-down has taken the loss out of theirs. Or they don't, and you've still ended the argument and kept the evening, and their holding is now visibly theirs alone.

If it's always the same argument with the same person, in different clothes, that's the habit the earlier book You don't have to win this one is about, and it's worth reading alongside this one. This book is about the moment inside any argument, with anyone, when you know. What they know is their business. What you do at the turn is yours.