I'll usually give one run-through of my side of the argument; if my position doesn't get any traction with the other party, then I'll drop it. I'm not going to spend a day typing messages to a stranger trying to tell him something he doesn't want to hear.
Arguing is a low priority with me. If anything, I need the other party to evince some obvious openmindedness and desire to hear more before I'll invest time and energy in an exchange.
Also, I don't feel bad backing off an argument. There's no loss of pride there. Since arguing is a low priority, my self-esteem isn't invested in the idea of winning or losing arguments.