It takes two to tangle, so the adage goes.
Two individuals communicating their thoughts and ideas through arguments can never achieve effective communication.
Dealing with argumentative people requires wisdom. It is wise to maintain a lead by being quiet and maintaining a peaceful attitude. Engaging with difficult people through deep listening 
is a big bonus.
I personally, always prefer to listen to the wisdom in the other person’s arguments. Patiently, with rapt, undivided attention, I am able to get the additional benefit of the argumentative person’s wisdom, albeit small. In the process, I add to my own little wisdom too.
In maintaining my peace, I succeed in making the other argumentative person feel happier; just by making him/her feel listened to, and heard.
It is unfortunate though that argumentative people ultimately lose out on an all-rounded, valuable wisdom.
I love .. 
.. to remind myself, always, that the Almighty Creator founded this earth .. 
.. by wisdom; and by understanding created the heavens we all live under. So doing all I could to add to my wisdom base is a good strategy.
However, if I may add that the best wisdom is this one that comes from above. It is therefore worth spending my entire lifetime, with all my energy, seeking and pursuing this pure wisdom.
-Annie