I know it's hard admitting you're wrong. I find it hard too. For the longest time I would just get mad whenever I lost. It's a trait we all acquired because of society and also how we are all wired. Losing is one of the worst feelings we can feel and psychologically, hormones channel our losing memories to the deepest part of the brain so that we will never forget it. And believe it or not, it affects our entire body. Everything slows down and physically and emotionally.
So when I see someone in that kind of situation, where they just see they are wrong. If they back down and admit it, I suddenly gain a new found respect for them. It's hard to admit defeat. It's not only upsetting but it's also kind of embarrassing. But it's also something else - mature.