Abstract
In the context of uncertainty, belief-weighted relative utilitarianism consists in comparing acts according to a weighted sum of the (0, 1) -normalized subjective expected utilities they yield. The weights may change with the profile of beliefs but do not depend upon the profile of individual utilities for the outcomes. This class of social welfare functions is characterized by the Pareto principle, the sure-thing principle, a continuity condition, and an independence condition requiring that the social ranking of two acts is unaffected by the addition of an outcome that leaves everyone’s best and worst outcomes unchanged. The weights must be constant if the social ranking of constant acts is independent of individual beliefs. Anonymity then pins down plain relative utilitarianism: acts are compared according the sum of (0, 1)-normalized subjective expected utilities they generate.
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Notes
Postulate P3 guarantees the existence of a state-independent ordering of the outcomes interpretable as the “taste” component of the preference relation over acts. Postulate P4 guarantees the existence of a well-defined subjective likelihood relation on events intepretable as the “belief” component of the preference relation.
The Pareto principle is controversial in the current context: see Gilboa et al. (2004) and Gilboa et al. (2014) for a criticism and two proposals. The literature is divided into (i) a strand that maintains the SEU requirement on society’s preference but weakens the Pareto principle and (ii) a strand that keeps the Pareto principle but weakens the SEU axiom. Our contribution belongs to the latter strand.
In Sprumont (2018), we argue that a stronger independence condition is often justified and leads to comparing acts on the basis of some weighted product of the normalized utilities they generate. The social preference thus obtained violates the sure-thing principle.
The bulk of the literature on state-dependent subjective expected utility assumes a finite state space. See however Wakker and Zank (1999) for a formulation using a Savage state space.
The literature often refers to what we call the Pareto Principle as the Ex-ante Pareto Principle in order to distinguish it from the (weaker) Ex-post Pareto Principle. The latter says that the social preference over outcomes (if well-defined) should respect the individual preferences over outcomes.
In contrast, three of Dhillon and Mertens’ (1999) axioms—Continuity, Independence of Redundant Alternatives, and Monotonicity—impose restrictions across preference profiles.
Because the beliefs \(p^{*}(R_{1}),\ldots ,p^{*}(R_{n})\) are non-atomic measures, proviso (iii) in this axiom can be rewritten as follows: for all \( x^{\prime }\in X^{\prime }\) and \(i\in N\) there exists \(a\in A(X)\) such that \( x^{\prime }I_{i}a.\)
Although widely accepted and much weaker than Independence of Inessential Expansions, Independence of Redundant Outcomes is not innocuous. Imagine two colleagues 1 and 2 who regularly have lunch together. If there are two nearby restaurants a and b, 1 prefers a over b and 2 prefers b over a, having lunch at b half of the time seems fair. If there are ten restaurants, 1 finds \(a_{1},\ldots ,a_{9}\) equally good but dislikes b, and 2 likes b but finds \(a_{1},\ldots ,a_{9}\) equally bad, is it still fair for the two colleagues to have lunch at b half of the time?
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This paper results from research supported by a grant from the FRQSC. I thank two reviewers and an associate editor for very hepful remarks and suggestions.
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Sprumont, Y. Relative utilitarianism under uncertainty. Soc Choice Welf 53, 621–639 (2019). https://doi.org/10.1007/s00355-019-01202-9
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DOI: https://doi.org/10.1007/s00355-019-01202-9