If you measure something, people will try to increase it. Just by presenting the number, people are encouraged to maximize it.
I tend to think that better discussion in a forum like this comes about with less back and forth. That's not to say that alternating comments on opposite sides of a viewpoint can't be enlightening, but I think it's stronger if those viewpoints come from multiple people on both sides.
I assume the goal is to encourage high-quality discussions (including extended discussions/back-and-forth), in which case average comment score will work against that.