Why is it so important to be the one who says the last word? Is it because each statement wipes out the one before it? Or trumps it? Is the last word the tie-breaking victory goal? Each of these ideas backs up the conviction that there needs to be a winner in a discussion. It is so hard to lay that idea to rest.
Sometimes the last word is an attempt to get the other person to see that they are just plain wrong. It is not so much victory that is at stake, but truth. However, if you have already spoken the truth and it is contradicted, there is very little hope that a few more words will be just what is needed to convince. Usually every additional word only armors the hearer with more stubborn pride to resist. We don't like being shown that we are wrong.
One of the most emotionally charged discussions I ever got in was with my son, who in the end was infuriatingly correct. I finally admitted it, but said I really didn't like it! It involved a logic problem--something I put him on to as a kid by doing logic puzzles together. This discussion was humbling. I have since used this same problem in my logic classes, and experience a particular exhilarating glee when my students fall into the same trap I did.
There can be lots of reasons for wanting to have the last word. We should know what those reasons are, decide if they are justified or necessary or fruitful, and act accordingly. It may not give us the last word--but we need to realize it is always in our power to decide who gets it. It is usually a passive power. We have to shut up.