Mere Addition is equivalent to avoiding the Sadistic Conclusion in all plausible variable-population social orderings

Economic policy evaluations require social welfare functions for variable-size populations. Two important axioms in the population ethics literature are Mere Addition and avoidance of the Sadistic Conclusion, both of which focus on the sign of lifetime utility. The population ethics literature treats these axioms as closely related but distinct: one influential review calls avoidance of the Sadistic Conclusion "less controversial." Here, we provide weak, uncontroversial sufficient conditions for these two principles to be equivalent. Related results exist in prior literature, but these include only same-number utilitarian orderings and therefore exclude recent and theoretically important rank-dependent social evaluations that we include. [100 words].