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 If we had perfect courts, I wouldn't have a problem with death penalty for murderers and rapist and child traffickers.  Of course, that is a big "IF".  

I don't trust the courts and therefore cannot support the death penalty for any crimes.  As badly as I want these crimes punished harshly, I'd let every one of them off if it prevents one innocent person from being put to death. 
 I am strongly in disagreement.

I think lifelong incarceration is inhumane and burdens victims & law-abiding people (eg via taxation).

I'm willing to bite the bullet on some nonzero error rate, and I would encourage you to realize that your alternative (eg life incarceration) will also have a non-zero error rate. 

It is sometimes argued that lifelong incarceration is cheaper than the death penalty, on account of the automatic appeals associated with the death penalty. Either these appeals do not tend to bring about justice, and they could be removed without an issue, or they do bring about justice and foregoing them for life incarceration would entail a higher error rate in life incarceration than in execution. 
 If someone is incarcerated for years, it harms their life, but it can be made partially right.  Once someone is killed, their eternity is settled.  That is not acceptable.  I don't like the cost associated with incarceration, but it is better than the murder of innocents. 
 There is no scenario where nobody innocent ever dies. It's just not on the table. 
 The primary issue with life incarceration or the death penalty is that it’s carried out by the monopolistic gang called the state. There’s no right answer in our system. 
The third way is to abolish the state and let the free market justice system work ( ie Bob Murphy, Bruce Benson and David Friedman) The closer the state can approximate these systems by getting out of the way (like allowing discrimination against criminals) it would be better than what we have. 
 I agree with that

However, I think in practice it would more closely resemble the ancien regime, with punishments like death and exile much more common. I think the "outlaw" system of the American West worked very well, with criminals denied the protection of the law (and thus subject to death or other injury). I also think in a sytem of private law there would be armed self-defense, and thus another avenue by which the death of criminals would become much more common.

Murphy seems to think that a private law system would become nearly pacifist, but I strongly disagree. The ability to be soft requires very strong security which in turn requires that somewhere along the line there is iron determination. We are seeing exactly that tension unravel in modern Europe and the Anglophone countries. 
 It’s a tricky task to predict a free market! I think different cultures would look different. Followed for your thoughtful comment.