Why do both sides in the death penalty debate reject the deterrence
argument? Please answer in paragraph form.

How many sides does a polygon have ?

Which debate ?

The debate of whether or not a just society should use the death
penalty as a punishment for committing a crime.

I think it is only common sense that the death penalty is at least a
slight deterant. Cruel and unusual punishment is also be at least a
slight deterant, but proof is doubtful and it can be argued that there
are side effects outweigh the deterant effect.
 frde is saying there are more than two sides, and a significant
number on most of the sides think the death penalty is a deterant and
would mention the effect of deterant.   Neil

I've read that most studies that support the deterrent effect of the
Death penalty are based on econometric modeling which as far as I can
tell is a very inconclusive method of predicting things like
deterrence.