This nesting approach with σ above also allows us to “fix” maximin/leximin under conditions of uncertainty to avoid Pascalian fanaticism, given a finite discretization of welfare levels or finite number of lexical thresholds. Let the welfare levels be t0>t1>⋯>tn, and define:
fk(x)=−∑iI(ui≤tk)
i.e.fk(x) is the number of individuals with welfare level at most tk, where uiis the welfare of individual i, and I(ui≤tk) is 1 if ui≤tk and 0 otherwise. Alternatively, we could use I(tk+1<ui≤tk).
In situations without uncertainty, this requires us to first choose among options that minimize the number of individuals with welfare at most tn, because fn takes priority over fk, for all k<n, and then, having done that, choose among those that minimize the number of individuals with welfare at most tn−1, since fn−1 takes priority over fk, for all k<n−1, and then choose among those that minimize the number of individuals with welfare at most tn−2, and so on, until t0.
This particular social welfare function assigns negative value to new existences when there are no impacts on others, which leximin/maximin need not do in general, although it typically does in practice, anyway.
This approach does not require welfare to be cardinal, i.e. adding and dividing welfare levels need not be defined. It also dodges representation theorems like this one (or the stronger one in Lemma 1 here, see the discussion here), because continuity is not satisfied (and welfare need not have any topological structure at all, let alone be real-valued). Yet, it still satisfies anonymity/symmetry/impartiality, monotonicity/Pareto, and separability/independence. Separability means that whether one outcome is better or worse than another does not depend on individuals unaffected by the choice between the two.
This nesting approach with σ above also allows us to “fix” maximin/leximin under conditions of uncertainty to avoid Pascalian fanaticism, given a finite discretization of welfare levels or finite number of lexical thresholds. Let the welfare levels be t0>t1>⋯>tn, and define:
i.e.fk(x) is the number of individuals with welfare level at most tk, where uiis the welfare of individual i, and I(ui≤tk) is 1 if ui≤tk and 0 otherwise. Alternatively, we could use I(tk+1<ui≤tk).
In situations without uncertainty, this requires us to first choose among options that minimize the number of individuals with welfare at most tn, because fn takes priority over fk, for all k<n, and then, having done that, choose among those that minimize the number of individuals with welfare at most tn−1, since fn−1 takes priority over fk, for all k<n−1, and then choose among those that minimize the number of individuals with welfare at most tn−2, and so on, until t0.
This particular social welfare function assigns negative value to new existences when there are no impacts on others, which leximin/maximin need not do in general, although it typically does in practice, anyway.
This approach does not require welfare to be cardinal, i.e. adding and dividing welfare levels need not be defined. It also dodges representation theorems like this one (or the stronger one in Lemma 1 here, see the discussion here), because continuity is not satisfied (and welfare need not have any topological structure at all, let alone be real-valued). Yet, it still satisfies anonymity/symmetry/impartiality, monotonicity/Pareto, and separability/independence. Separability means that whether one outcome is better or worse than another does not depend on individuals unaffected by the choice between the two.