What is the most peaceful way to deal with other people's pride?

In people around me, pride seems to be one of the 2 things that are more challenging for me to deal with. The other one is stubbornness. I think both are caused by lack of exercise of the brain or mental-social blindness. Is just not doing anything about it the best way? Should the learning come from them? I hate to just confront them every time and seeing them leave. It makes me feel that i am not achieving my universal goal of catering to everyone's need. So far, i have managed with every other bad characteristic of people, except for pride and stubbornness. Pride/arrogance is the worst. I have severed partnerships and let people stop working with me and my endeavors because of this, but i have never asked people to go just because mainly of lack of skill. Their pride disgusts me...especially when it's way over spilling their brims. For some people who really have something to brag (laude in Harvard,self-made) although they shouldn't brag, i don't mind it much when they do.