Abstract
There are a number of single-profile impossibility theorems in social choice theory and welfare economics that demonstrate the incompatibility of unanimity/dominance criteria with various nonconsequentialist principles given some rationality restrictions on the rankings being considered. This article is concerned with examining what they have in common and how they differ. Groups of results are identified that have similar formal structures and are established using similar proof strategies.
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Notes
I owe this example to Paolo Piacquadio.
For this reason, Broome advocates eschewing the use of consequentialist terminology and instead distinguishing between teleological and nonteleological moral theories. A teleological theory is one in which there is a theory of the good that orders alternatives in terms of their relative goodness and rightness requires maximizing the good. Different theories of the good result in different teleological theories.
After my presentation in Montreal, I learned that Prasanta Pattanaik and Yongsheng Xu were exploring the structural unity of examples that exhibit a conflict between dominance, context-dependence (a form of nonconsequentialism), and a continuity (and possibly a rationality) condition imposed on the ranking being considered. I am grateful to them for providing me with a preliminary version of Pattanaik and Xu (2012).
The following notation is used for vector equalities and inequalities. For all \(x, y \in \mathbb {R}^m\), (i) \(x \ge y\) if and only if \(x_i \ge y_i\) for all i, (ii) \(x > y\) if and only if \(x \ge y\) and \(x \ne y\), (iii) \(x \gg y\) if and only if \(x_i > y_i\) for all i, and (iv) \(x = y\) if and only if \(x_i = y_i\) for all i. The origin in \(\mathbb {R}^k\) is denoted by \(\mathbf {0}_k\).
This property is known as Profile-Dependent Neutrality.
These two objections to Pareto Indifference are adapted from objections advanced by Sen (1977, 1979) to the strong neutrality principle of social choice theory. For further discussions of this issue, see Sen (1990) and Bossert and Weymark (2004). The second objection implicitly supposes that it is the individuals’ actual utilities that matter. If, as in Harsanyi (1977), utilities are “laundered” so as to correct factual errors or to mitigate anti-social attitudes, one could claim that a social ordering obtained using these laundered utilities can be welfarist. This is a controversial position on which I take no stand.
Kaplow and Shavell have in mind non-welfaristic principles that provide support for legal decisions. Misleadingly, they refer to any such principle as being a principle of fairness. For example, Kaplow and Shavell (2002, p. 39) say that “fairness” refers “to any principle that does not depend solely on the well-being of individuals.” This is a very different concept of fairness from those found in the theory of fair allocation, which are compatible with the Weak Pareto Principle. See Fleurbaey and Maniquet (2011) for a discussion of Paretian fairness criteria.
Kaplow and Shavell instead state their theorem in terms of real-valued representations of \(R_U\) and \(R^*_U\). However, their arguments do not require either \(R_U\) or \(R^*_U\) to be representable.
Note that the completeness of \(R_U\) is used at this point in the argument.
Campbell and Kelly (2002, pp. 80–81) claim that Kaplow and Shavell’s proof of their theorem does not make use of the transitivity of \(R_U\). This is not correct because transitivity is implicitly used in making their “Observation” (which is a restatement of Theorem 1 in terms of real-valued representations). Campbell and Kelly note that a somewhat weaker continuity condition suffices to establish Theorem 2.
The principles considered by Suppes (1966) and by Brun and Tungodden (2004) place restrictions on the ranking of the alternatives. By regarding pairs of alternatives for which a principle does not apply as being incomparable, each of these principles can alternatively be thought of as defining a quasiordering on the set of alternatives. As shown by Donaldson and Weymark (1998), every quasiordering is the intersection of orderings. Thus, these principles can be interpreted in terms of Sen’s intersection approach (see, e.g., Sen 1992, pp. 46–49). In this approach, the objective is to determine which pairwise rankings are shared by every ordering in some set of orderings. Of course, the same observation applies to any principle that can be interpreted in terms of a quasiordering. Brun and Tungodden regard their various dominance criteria as being applications of Sen’s intersection approach.
The results in this and the following section can be restated in terms of a single profile of individual preferences on X, but in order to relate these results to welfarism, it is more convenient to express them in terms of individual utilities.
By choosing \(y^h\) so that \(y^h \gg x^h\) for all \(h \ne i, j\), the same proof strategy shows that Weak Pareto is inconsistent with Strong Dominance, Permutation Dominance, and the Suppes Grading Principle when U is a classical private goods profile with nonidentical preferences.
In order to avoid visual clutter, the budget lines are not shown in the diagram. The marginal rate of substitution is the same at \(x^i\), \(x^j\), \(y^i\), and \(y^j\). They are also equal at \(\bar{x}^i\), \(\bar{x}^j\), \(\bar{y}^i\), and \(\bar{y}^j\).
For a compact presentation of the Fleurbaey–Trannoy Theorem, see Fleurbaey (2006, Section 9.5).
For a formal proof of this claim, see Fleurbaey and Trannoy (2003, Lemma 1).
Type 2 dependence requires there to exist two situations with the same consequences that are not indifferent. When the consequences are utility vectors, type 2 dependence is equivalent to requiring Single-Profile Welfarism to be violated. Pattanaik and Xu (2012) have established a version of the Kaplow–Shavell Theorem in this more abstract setting and shown how Kaplow and Shavell’s impossibility result can be overturned by weakening their continuity condition.
Note that Nonidentical Preferences is a factual assumption about the profile U, whereas Minimal Relativism is a normative principle about the ordering \(\succeq \).
The terminology has been chosen to distinguish this dominance principle from the one considered in the next subsection.
Hare’s presentation of his theorem is somewhat imprecise and informal. I have modified his assumptions somewhat so as to simplify the presentation. I have also made explicit some of his implicit assumptions.
Strictly speaking, it is this implication of openness of the sets A and B that is needed, but for simplicity, I assume that A and B are open.
Because f assigns a unique consequence vector to each \(x \in X\), the binary relation R used in the Hare Theorem can be equivalently defined on \(C \times X\), just as \(\succeq \) is defined on \(N \times X\) in the Pattanaik–Xu Theorem. Pattanaik and Xu (2012) exploit this observation to draw out some of the connections between these two results. Note that in the Hare Theorem, it is X that provides the conditioning (i.e., contextual) variables, whereas in the Pattanaik–Xu Theorem, it is N.
For a discussion of associative duties, see Scheffler (2001).
For a formal statement of the multi-profile welfarism theorem, see Bossert and Weymark (2004, Theorem 2.2). Blackorby et al. (2005) have considered a richer multi-profile framework in which the consequence space includes non-welfare information in addition to the individual utilities. They have identified conditions, including a strong unrestricted domain condition, that precludes the non-welfare information from playing any role in the social ranking of the alternatives. Single-Profile Welfarism as defined here is concerned with a single profile of individual utility functions. If, instead, there is a single profile of individual preference orderings, welfarism presupposes that alternatives are treated neutrally. See Bossert and Weymark (2004, p. 1110).
Suzumura (2011, p. 675) describes the differing interpretations of Kaplow and Shavell’s claim as reflecting the conflict between those who support a single-profile approach to normative social evaluation and those who support a multi-profile approach.
These articles employ what Fleurbaey (2011) calls the equivalence approach. In this approach, two alternatives are ranked in the same way as the alternatives that they are indifferent to in some reference set. When the reference set is a monotonic path, it follows that vector dominance is only used to rank alternatives that lie on this path. The most fully developed exploration of the equivalence approach is provided by Fleurbaey and Maniquet (2011). Some complications that arise when preferences are incomplete are considered in Fleurbaey and Schokkaert (2013).
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This article is based on the first part of my Presidential Address to the Society for Social Choice and Welfare that I delivered on June 19, 2008 at Concordia University in Montreal. An expanded version of the second part has appeared as Weymark (2014). I am grateful for the comments received from Franz Dietrich, Marc Fleurbaey, Paolo Piacquadio, and an anonymous referee. I have also benefited from the discussion of my article in Montreal and at the Munich Center for Mathematical Philosophy Conference on Social Choice and Its Philosophical Applications held at Venice International University.
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Weymark, J.A. Conundrums for nonconsequentialists. Soc Choice Welf 48, 269–294 (2017). https://doi.org/10.1007/s00355-016-1021-9
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DOI: https://doi.org/10.1007/s00355-016-1021-9