Abstract
This paper discusses and evaluates some axiomatic approaches to weighted utilitarianism. It then offers a new and direct proof for weighted utilitarianism under full comparability which yields, as a corollary, U. Ebert’s result on rank-weighted utilitarianism.
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Notes
This seems fair because generalized utilitarianism can be rather naturally—and ordinally—obtained without recourse to informational requirements at all. This is essentially the force of Debreu’s theorem on additive separability (c.f. Fishburn 1970).
Here we are of course discarding such pathological examples as lexicographic preferences, but this is standard.
As per the language of d’Aspremont and Gevers (2002), we are implicitly making the assumption of domain universality on \(f\) here in our definition of a SWFL. A weaker condition to impose on the domain \(\mathcal {D}\subset \mathcal {U}\) of a SWFL is that of domain attainability: For all \(\vec {u}, \vec {v}, \vec {w} \in \mathbb {R}^{n}\) there exist \(x, y, z \in X\) and there exist \(u \in \mathcal {D}\) such that \(u(x, \cdot ) = \vec {u}\), \(u(y, \cdot ) = \vec {v}\) and \(u(z, \cdot ) = \vec {w}\).
In the presence of domain universality, XNE is equivalent to the more satisfying condition of formal welfarism that we may take to define \(R^{*}\) in our setting: For all \(v, w \in \mathbb {R}^n\), for all \(x,y \in X\) for all \(u\in U\): If \(u_i(x) = v_i\) and \(u_i(y) = w_i\) for all coordinates \(i\), then we have \(vR^{*}w \Leftrightarrow xRy.\)
The specified settings are in fact unnecessarily strong. A theorem of Sen (1977) shows, for example, that in light of domain universality, only Pareto indifference is required to obtain formal welfarism. This will be silently used in our survey of old results but is a relatively unimportant point.
To avoid clunkiness, we could have stated the properties far more intelligibly in terms of the associate SWO on \(\mathbb {R}^{n}\). But given our operation in the SWFL framework, we opt for consistency.
On the other hand, such statements give much more mileage as hypotheses in theorems than their more refined counterparts.
For example, \(u_{(1)}(x) = \min _i u_i(x)\) and \(u_{(n)}(x) = \max _i u_i(x)\).
The Weak Pareto axiom, WP, states that for all \(x, y \in X\), for all \(u \in \mathcal {U}\), if \(u_i(x) > u_i(y)\) for all \(i \in N\), then \(xPy\). It manifestly weakens SP.
The notation is very suggestive; \(f\) is invariant under such transformations of \(u \in \mathcal {U}\)—i.e. affine transformations with the same scale factor and varying additive ones. We will employ this useful shorthand henceforth.
The latter part of the proof can be carried out without TSM so long as \(f\) is continuous on \(\text {diag}(\mathbb {R}^n)\).
A goal specifically motivated by our justifications in the introduction.
The papers of Hammond (1976) and Maskin (1978) suggest that at one might have guessed at least SP, MIS, FC, A a priori. The fact that Deschamps and Gevers were able to dilute A into MIS and require no further conditions is rather remarkable and a testament to the strength of their result. In fact, MIS is even stronger than what they require in the paper, which is simply an undominated set of individuals of size at least 3.
The theorem itself was already known to U. Ebert, who gave a different, less transparent proof.
The latter problem can be avoided if we endow \(X\)—not terribly unreasonably—with a topology.
As we shall see, generalized utilitarianism “fits the bill” of an ordinal characterization if \(X\) is a topological space; this is a consequence of Debreu’s theorem.
More importantly, perhaps, it is most “focused.”
Particularly in the case of the generalized Gini family.
This step of the argument is also due to Ebert.
He bundles \(W\)’s continuity, Pareto property, symmetry (as a function \((\mathbb {R}^{n})^{+}\rightarrow \mathbb {R}\) which we see as anonymity), and “independence with respect to ordered vectors” (\(\mathcal {G}_n\)-separability) together as a baseline property \(W\) that all welfare orderings in the paper possess.
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Acknowledgments
The author would like to thank Amartya Sen and Eric Maskin for introducing him to the relevant literature and for useful commentary as this work progressed. The author thanks Akhil Mathew and Irving Dai for helpful conversations. Finally, the author thanks anonymous referees and the associate editor for their careful readings and thoughtful suggestions.
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Balasubramanian, A. On weighted utilitarianism and an application. Soc Choice Welf 44, 745–763 (2015). https://doi.org/10.1007/s00355-014-0865-0
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DOI: https://doi.org/10.1007/s00355-014-0865-0