I don't think having the last word means you won, because I almost never have the last word. If I can see that neither of us are going to change our minds/opinions, then I just shut up so the other person usually does have the last word. I just think it's ridiculous to keep arguing if it isn't going to get you anywhere.
Also, when others are around I assume they would rather not hear the bickering/arguing so I will curtail my comments; I think it makes me appear to have more class than the other person who would continue the bickering to the point of making the others present uncomfortable.
I know when I'm with a group of people, I hate to hear two out of the bunch arguing. It ruins the mood for everyone so I try not to do that.
And yes, the other person probably does "think" they won, but I know different so it doesn't bother me.