Okay, so there doesn't always have to be winner or a loser.  In fact, when you play this game in relationships?  Nobody is winner.  You either both win or you both lose.  

How do you both win?  Honest communication, that's how.  Did someone actually do something pretty bad to you?  You have two choices.  You can walk away.  This is the simplest.  This doesn't always mean "I can't forgive you" (though it can), but just that's not worth playing this game of winners and losers anymore.  Or, you can work it out.  Working it out means to actively work to get over your pain of what the other person did, rather than hating them and punishing them forever.  It means honest communication and finding forgiveness.  This can take years if the act against you is really bad (though forgiving is only worth it when the other person is actively working on doing better).  But it can also just mean completely understanding the person's motivations behind what they did and forgiving when you both really understand each other and make amends for what was done wrong (when the act is small).

You both lose if you walk away from a disagreement as the winner or the loser.  Because in a relationship there are no winners or losers.  There is only two people who are working hard to be good to the other person and sometimes failing (which is normal when it's just regular stuff).  Anything less than that is not a real relationship.  If one person is always being treated bad or you both are always treating each other bad, then that's not a relationship.  That's abusive and toxic.  

So, are you ready to really work on things with this person?  Are they worth it?  Are you capable of real change?  Are they?  Analyze what's going on and make a choice from there.