And it turns out that the utilitarian approach of adding up utilities is *not* a bargaining solution, because it violates Pareto-optimality in some cases. Does that “disprove” total utilitarianism?
I’m not sure this is right. As soon as you maximize a weighted sum with non-negative coefficients your solution will be weakly Pareto optimal. As soon as all coefficients are strictly positive, it will be strongly Pareto optimal. The axioms mentioned above don’t imply non-negative coefficients, so theoretically they are also satisfied by “anti-utilitarianism” which counts everyone’s utility negatively. But one can add stronger Pareto axioms to force all coefficients to be strictly positive.
The problem with the utilitarian Bargaining solution is that it is not independent of affine transformations of utility functions. Just summing up utility functions is underspecified, one also needs to choose a scaling for the utility functions. A second criterion that might not be satisfied by the utilitarian solution (depending on the scaling chosen) is individual rationality, which means that everyone will be better off given the bargaining solution than some disagreement outcome.
I’m not sure this is right. As soon as you maximize a weighted sum with non-negative coefficients your solution will be weakly Pareto optimal. As soon as all coefficients are strictly positive, it will be strongly Pareto optimal. The axioms mentioned above don’t imply non-negative coefficients, so theoretically they are also satisfied by “anti-utilitarianism” which counts everyone’s utility negatively. But one can add stronger Pareto axioms to force all coefficients to be strictly positive.
The problem with the utilitarian Bargaining solution is that it is not independent of affine transformations of utility functions. Just summing up utility functions is underspecified, one also needs to choose a scaling for the utility functions. A second criterion that might not be satisfied by the utilitarian solution (depending on the scaling chosen) is individual rationality, which means that everyone will be better off given the bargaining solution than some disagreement outcome.
You’re right; I meant to refer to the violation of individual rationality. Thanks!