I see it all the time in mediations or facilitated conversations with teams - managers who the team know are bluffing, but they just won't admit they were wrong. Having worked for a couple of managers like that myself, and seen it in numerous mediations, I thought it may be useful to share some insights.
These issues often come up when I'm involved in team issues. One that springs to mind was a team of professionals who were so frustrated with their boss (the MD) that they submitted a collective grievance against him - he was fuming and didn't know what to do. Fortunately he had a good HR professional advising him, and she recommended either mediation or a facilitated conversation. We did a combination of the two. Due to a couple of potentially serious issues which couldn't be discussed with everybody, mediation between the MD and two of his team got them on track - but then we needed the full conversation with the team. This was the big one - they were all very angry with him!
When I facilitate these difficult conversations with teams there's always preparatory work to do, and in this case it included some pretty difficult private conversations with the MD. He was the boss and everybody knew it - and that's the way he wanted it to be. I helped him to reflect on what he was telling me, and he did so - he didn't really like what he saw in the mirror. Then he surprised me - he started to cry. He was sick and tired of the effort he was having to put in to defend himself against the team. I asked why he thought they were attacking him, and I helped him to realise it was because they didn't trust him - they felt he wasn't being honest with them. I explored with him whether there was any substance in why they would feel like that - and there was plenty of substance! He was worried about losing face and credibility though - he didn't want to admit he was wrong.
Those conversations were in the morning. I got him and the team together in the afternoon - we had four hours in the diary which everybody was dreading (except me!). It was a great discussion - people said how they were feeling and I facilitated the various discussions, capturing all the great actions they agreed as a team. It was a thoroughly enjoyable session. Why could something so potentially bad turn out so well? Because the MD showed humility - he displayed to everybody that he was a human being. With a little help from myself, he told people more about himself - he told them some personal stuff and he also said how much the issues were affecting him - he even described how his wife had told him to quit due to the stress of it all. Finally, he apologised to the whole team (without any prompt from me) - he said he had misled them and had got things wrong. He also explained why - because his ego was too big to admit he'd made mistakes.
Those who were angry with him then reassured him. They thanked him for being honest and explained how difficult they knew it had been to tell them what he just had. They told him what they liked about him and how they didn't want to raise the grievance. They even said it was nice to see he was human and had actual feelings. It was a real turning point. All I had to do then was facilitate a conversation between a happy team who got on well - it was easy and enjoyable!
It is hard to admit when you're wrong, but we should never under-estimate the power of humility!