I would support the death penalty, in theory, in certain extremely limited cases, if it could be applied justly and equitably, and if it could always be applied with absolutely no execution of persons who are, in reality, not guilty.
But Tim, what do you mean by "guilty" above. My bet is in the traditional sense based on Libertarian free will, in which this person isn't just an unfortunate loser of the lottery but can deserve it.
I am talking about those "unfortunate" lottery losers who could warrant execution (for sake of the greater society) by doing things to make life even more unfortunate for others, e.g. those who have performed, and will likely continue to perform (given the chance) the most atrocious acts of violence against others, including murder.