"In a large house there are articles not only of gold and silver, but also of wood and clay; some are for special purposes and some for common use.  Those who cleanse themselves from the latter will be instruments for special purposes, made holy, useful to the Master and prepared to do any good work.  Flee the evil desires of youth and pursue righteousness, faith, love and peace, along with those who call on the Lord out of a pure heart.  Don’t have anything to do with foolish and stupid arguments, because you know they produce quarrels." ~ 2 Timothy 2:20-26   

My calling is in teaching Scripture so from that perspective I have come to understand that walking away from an argument hardly ever depicts a loser.  In fact the one who has wisdom enough to know when it is no longer fruitful (or able to produce a good result) is by far the one who is more wise.  This does not mean that you aren't to defend Scripture or correct or even rebuke because we are told in scripture there is a place and time for it.  Providing answers and correcting is required for believers and ministers, but the goal is to leave something corrected, dispel doubt with answers.  Even Jesus knew there was a time to dust His feet and walk away.  You have to discern spiritually when that moment has come.  The next time you feel a discussion is becoming argumentative in nature simply tell the other person that the discussion feels as if it is becoming argumentative and if the temperature of the discussion can't be brought down then you will have to step away.  My husband has had to do this and it, amazingly, causes the other person to quiet down and listen!  When people want answers they will listen.  When people want to be right, they will argue.  Even a fool figures out that after awhile he is the only one who is still talking and will be quiet.  The temptation to answer back in an argument or return evil for evil (get revenge) are the wood and clay of our flesh.  When we choose to walk away we cleanse ourselves of those common things and train ourselves to operate in purity of heart.  It is then that the Holy Spirit is making gold and silver out of us, something of value, purpose driven people, and a powerful tool for the Kingdom of God.  That should matter most, considering the Kingdom is eventually all that will remain.  Think wisely and take the high road.  That is the only road the Holy Spirit walks.  Stay with Him there.