¹The simplest argument for this is that lower punishments are more likely to be payable as fines, which are simply transfers, rather than requiring costly punishments such as imprisonment or execution. This point is discussed at much greater length in Friedman, D. "Reflections on Optimal Punishment or Should the Rich Pay Higher Fines?," Research in Law and Economics, 1981.

²It will also depend on other characteristics of the two groups that effect how likely they are to commit the crime for a given level of expected punishment.

³If we made our model more complicated by allowing for the fact that different methods have different costs (making a voodoo doll might be cheaper than buying poison), the detailed quantitative conclusions would change. The division between those who chose the two methods would no longer be at p=.5; some people might use the method they thought less likely to work because it was cheaper. The logic of the argument, and the qualitative results, would remain the same.

⁴For simplicity, I ignore the cost of using either method.