You do not have the right to kill the person. The state's power "is derived from the people." You are correct that an individual does not have the right to self-help in seeking redress from crimes committed against them. To allow that would result in anarchy and chaos as each person sought vengance for a percieved ill. But, that does not address the issue of the DP. Your mistake is that you have treated the action of the state as being one of acting on the behalf of the victim. Rather, the state acts on behalf of society. There are many ways you can look at the imposition of justice. It can be restorative or preventative. The DP acts under both models. It restores society to equilibruim by equating the punishment with the crime. Any lesser punishment is difficult to justify. For taking a life (especially pre-meditated), can a person ever repay society? Remember it is not the victim that must be repayd, it is society. As a preventative measure, a person who has committed a crime must be punished sufficiently to deter others from committing the same crime. The efficiency of the DP is debated, but I think that much of the reason that is has lost its efficacy is that it is not imposed properly. Trial costs often lead prosecutors to accept deals of life w/o parole when it is clear that a person is guilty. Additionally, exceptions are made for several groups of people, including youth, who have made the concious decision to kill. On top of that, I don't see the deterrance in knowing that even if you get the DP, you are going to live for another 25-30 years and have significant control over your time (the lengthy appeals process) and place (isolation means other inmates can't touch them) of death.