Aggregating extended preferences

Abstract

An important objection to preference-satisfaction theories of well-being is that they cannot make sense of interpersonal comparisons. A tradition dating back to Harsanyi (J Political Econ 61(5):434, 1953) attempts to solve this problem by appeal to people’s so-called extended preferences. This paper presents a new problem for the extended preferences program, related to Arrow’s celebrated impossibility theorem. We consider three ways in which the extended-preference theorist might avoid this problem, and recommend that she pursue one: developing aggregation rules (for extended preferences) that violate Arrow’s Independence of Irrelevant Alternatives condition.

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Notes

  1. 1.

    In fact, the extended preferences program faces a number of challenges, and the program may not be a step in the right direction at all. Some important challenges concern (1) the precise nature and coherence of extended preferences, once we probe beyond the cursory sketch given above, and (2) the relationship between one individual’s ordinary preferences, (say) Agnes’s preference between eating meat and eating fish on the one hand, and, on the other, different individuals’ extended preferences over affairs which concern that individual, (say) Brandon’s preference between being Agnes and eating meat and being Agnes and eating fish. We discuss these issues in detail in a separate paper (Greaves and Lederman, forthcoming). In fact, we ourselves think that some of these other challenges are fatal to the extended preferences program. Solely by way of division of labour, the present paper focuses exclusively on the problem of aggregation.

  2. 2.

    In fact, it is not obvious that either individuals’ extended preference relations, or the objective ‘better off than’ relation, need to have the formal properties of an ordering—they may, for example, fail to be transitive and/or complete. We will return to this later.

  3. 3.

    This question, with the same motivation, has been raised by Adler (2014, 156; forthcoming, 26), who flags it as an important topic for future research. Voorhoeve (2014) discusses the incomparability problem that we focus on in Sect. 3. In Adler (2016), Adler presents an independent investigation of the problem of aggregation. We learned of this last paper after submitting the present paper but before it was published. There is significant overlap between the two papers: those interested in an alternative presentation of the material in sections 2–5 of the present paper, in particular, may wish to consult Adler’s paper.

  4. 4.

    For simplicity, we assume throughout the paper that both N and X are finite. As far as we know, nothing essential turns on the finiteness of the set of alternatives. But assuming that the population is finite is not entirely idle. As is well-known, Arrow’s theorem in its original form does not hold for infinite populations (Fishburn 1970); analogues of the ‘oligarchy theorems’ on which our Spinelessness Theorem is based can similarly fail in the setting of an infinite population. Essentially, the problem is this: Arrow’s conditions only imply that the set of ‘decisive’ groups (for a precise definition, see the Appendix) forms an ultrafilter in the powerset algebra of N, while the conditions of the oligarchy theorems only imply that the set of decisive groups forms a filter in this algebra. Since in the powerset algebra based on an infinite set there can be non-principal ultrafilters (and thus filters with infinitely descending chains under subset), Arrow’s theorem no longer implies the existence of an individual dictator, and the oligarchy theorems no longer imply the existence of an oligarchy. But analogous, equally troubling results can still be proven in the infinite setting: for example, if there is a well-behaved, \(\sigma\)-additive measure on the infinite population, it can be shown that under conditions analogous to Arrow’s, for any positive \(\epsilon\), no matter how small, then for some \(\delta <\epsilon\), there will be a group of measure \(\delta\) whose unanimous preferences are sufficient to determine the overall ordering (Kirman and Sondermann 1972). (An analogous modification of the oligarchy theorems can also be proven (see, e.g. Weymark 1984, Section 4).) The existence of such ‘invisible dictators’ is just as problematic as the dictator of Arrow’s original theorem (and mutatis mutandis for the oligarchy theorems). In short: while stating related results for infinite populations requires technical machinery we won’t introduce here, related conceptual points could be made in the infinite setting as well, so that the main line of argument does not depend on our simplifying assumption that the population is finite.

  5. 5.

    We note in passing that Harsanyi’s famed ‘aggregation theorem’ (Harsanyi 1955) describes one further structural constraint which emerges if we consider unit comparisons. Informally, the theorem says that if (1) each agent’s preferences are representable by a von Neumann–Morgenstern (vNM) utility function, (2) the output ordering is also representable by a vNM utility function, and (3) the output ordering satisfies an Ex Ante Strong Pareto condition, then the output ordering will be representable by a weighted sum of the individual vNM utility functions. Both the Strong Pareto condition and the claim that well-being should be representable by a vNM utility function are extremely plausible in the context of the extended preferences program. So Harsanyi’s theorem shows that any acceptable aggregation rule for extended preferences must have a particular functional form: it must be representable by a vector of weights on those individual utilities. This ‘single-profile’ version of Harsanyi’s theorem does not, as far as we are aware, have any implausible consequences; it simply exhibits a convenient way of expressing the family of functions to which the aggregation rule used in the extended preferences program must belong. ‘Multi-profile’ extensions of Harsanyi’s theorem (e.g. Mongin 1994), by contrast, appear more problematic for the extended preferences program. But they rely on a condition similar to Independence of Irrelevant Alternatives (IIA), a condition which, we will argue later, the theorist of extended preferences must reject even if she is to make sense of level-comparisons. More could be said here, but we won’t consider such problems further in the sequel, since given the solution we recommend they don’t pose a challenge to the theorist of extended preferences distinct from the one we will develop in Sects. 48.

  6. 6.

    Adler, for instance, sometimes suggests that the input to the aggregation rule should include all the extended preferences that any (actual) individual could have, or could have had, at any time (2012, 226–227). This is presumably extensionally equivalent to including all rationally permissible extended preferences. The resulting ‘possibilist’ version of the extended-preferences program would probably have to deviate from the formal framework as we have sketched it so far. In the first instance, it would have no special place for an assignment of preference relations to individuals, since individuals could have different relations in different possibilities. And if we did try to maintain some place for such an assignment in the possibilist setting, we would face problems arising from considerations of cardinality, which are likely to prevent there from being any surjective function from the set of individuals to the set of all rationally permissible extended-preference relations. To see this, recall, in particular, that we can identify the set of extended alternatives with the product \(N \times W\), where N is the set of individuals—so the cardinality of the set of extended alternatives is at least as great as the set of individuals—and that extended preference relations are binary relations on this set of extended alternatives; since binary relations are elements of the powerset of this product, Cantor’s theorem shows that there can be no surjection from the set of individuals onto the set of binary relations over extended alternatives. If rationally permissible preferences have the same cardinality as the whole powerset, then there can also be no surjection onto the set of rationally permissible preferences. In fact, a similar line of thought might be thought to show further that there can be no set of ‘all rational extended preference relations’ at all, and thus perhaps that this version of extended preference theory is itself incoherent: if (1) the specification of every possible world \(w \in W\) includes a specification of which extended preference relations are held by which individuals, (2) there is one extended alternative corresponding to each element of the product \(W \times N\), (3) there are more rationally permissible extended preference relations than extended alternatives (as is presumably the case), and (4) for every rationally permissible extended preference relation r and every individual i, there is some possible world in which i holds r, then the set of rationally permissible extended preference relations would have to have greater cardinality than itself, which is obviously impossible. However, this is not a special problem with (extended) preferences: a related argument can be used to show that if the objects of belief are sets of possible worlds, and belief states are represented by one set of possible worlds, then there is no set of all possible belief-states (Kaplan discovered this problem in the late 1970s, but it was not published until Kaplan (1995); a related puzzle is presented by Kripke (2011), who also discusses some of the history. A similar argument was independently discovered by Brandenburger (2003).) Since this mathematical fact presumably does not show that there is no interesting notion of rational belief, the corresponding argument does not show that there is no interesting notion of rational preference (extended or otherwise). In any event our argument will not rely on the assumption that one can make sense of the set of all rational preferences; see the next note.

  7. 7.

    Note that this closure condition on its own does not generate any of the cardinality difficulties mentioned in the previous note. Thus one can equally well state the ‘possibilist’ position by replacing ‘all rationally permissible preferences’ with ‘a very rich set of rational preferences, which includes non-actual ones’; our argument turns only on this set being closed under inverses.

  8. 8.

    Thus RF, T and C together are equivalent to the condition that \(\forall R \in D, f(R) \in {\mathcal {O}}\). We separate the conditions here because in later sections we consider weakening or dropping some of these conditions independently of others.

  9. 9.

    Binary relations which are not elements of \({\mathcal {O}}\) may also be such that it is rationally permissible to hold the associated extended preference-ordering: for example, it is at the very least arguable that rational preferences need not be complete (in other words, the inputs to the aggregation rule need not be complete). But assuming that the domain is as stated in the condition UD rather than some larger domain if anything makes it easier to find an acceptable aggregation rule: insofar as we can argue that there is no acceptable aggregation rule for a domain \(D \subseteq {\mathcal {O}}^N\), a fortiori there is no acceptable aggregation rule for a larger domain.

  10. 10.

    In the utility-function context, it is arguably natural, if the input to an aggregation rule is a profile of utility functions rather than merely orderings, for the output also to be a utility function (or a positive affine family of such functions) rather than merely an ordering. Any such output utility function, however, certainly induces an output ordering; thus an impossibility theorem formulated in terms of ‘utility aggregation rules’ in our sense (where the output relation is merely required to be some relation or other) applies a fortiori to these richer objects: we lose no generality in considering only UARs in our sense.

  11. 11.

    In our view, the distinction between Transitivity and Quasi-Transitivity is mainly of technical interest: we are not aware of any plausible reasons for thinking that rational preferences need not be transitive, but (at the same time) must be quasi-transitive. (Here is a purported reason that we regard as implausible. Consider three alternatives xyz that are arranged in close succession along some continuum: for example, shades of red, or amounts of sugar. It is sometimes claimed that such alternatives can have the property that both the difference between x and y and the difference between y and z are imperceptible, while (however) the difference between x and z is perceptible; further, that this might justify being indifferent between x and y, and being indifferent between y and z, while having a strict preference for x over z. This pattern of preferences satisfies Quasi-Transitivity, but not full Transitivity, since, here, strict preferences but not indifferences are transitive. We each reject this argument, but for different reasons. One of us thinks the argument goes wrong in its first step: there can be no such pattern of ‘imperceptible’ differences in the sense of ‘perceptible’ relevant to well-being. One of us thinks it goes wrong in its second step: granting the suggested pattern of perceptibility/imperceptibility exists, it does not justify this pattern of indifference and strict preference.) But in any case, even granting that the distinction is of more than technical interest, the main observation for present purposes is that our result would apply to quasi-transitive preferences as well.

  12. 12.

    Why not \(\pi (R) = (R_{\pi (1)}, \ldots , R_{\pi (|N|)})\)? Because we seek throughout, for notational convenience, to be defining left- rather than right-actions. That is, the action of permutations \(\pi\) on profiles R must be such that \(\left( \pi _2 \pi _1 \right) (R) = \pi _2 \left( \pi _1 (R) \right)\) (rather than that \(\left( \pi _2 \pi _1 \right) (R) = \pi _1 \left( \pi _2 (R) \right)\)). In the present case (and assuming the action of permutations \(\pi\) on N is itself a left-action), this condition is met by the definition given in the main text, but not by the alternative.

  13. 13.

    As stated this definition does depend on the details of the ‘more concrete’ framework introduced in Sect. 2. But all that we require here is an idea which can be stated independently of that framework, namely: that every permutation of individuals induces some corresponding permutation on the (abstract) set of alternatives. This induced permutation can then be used for the definitions which follow.

  14. 14.

    The intuitive explanation just given suggests what might seem to be a different formal condition: for all R and \(\pi\), \(\pi \left( f (R)\right) = f \left( \pi (R)\right)\). But it’s easy to show that these are equivalent. Since the conditions hold for all permutations, they must hold for \(\pi ^{-1}\). But \(\pi ^{-1} \left( f(R)\right) = f \left( \pi ^{-1}(R)\right)\) iff \(\pi \left( \pi ^{-1} \left( f(R)\right) \right) = \pi \left( f (\pi ^{-1} R)\right)\). Moreover, \(\pi \left( \pi ^{-1} \left( f(R)\right) \right) = f(R)\) and, by definition, \(\pi \left( f \left( \pi ^{-1}(R)\right) \right) = \left( \pi (f)\right) (R)\). Since this holds for all R, the intuitive condition holds for all \(\pi\) iff the mathematically simpler one in the main text does.

  15. 15.

    Mathematically, there is nothing very original in this theorem: the key aspects of the proof are contained in the work of Sen (1970) and Weymark (1984).

  16. 16.

    More carefully: Whether there is in fact only one profile with this property (and Sufficient Diversity fails), or instead many (in such a way that Sufficient Diversity is met), depends on the apparently merely technical issue of whether the ‘individuals’ are in that setting individuated by their preference relation (so that there is exactly one profile to consider), or independently of that relation (so that permuting preference relations among individuals gives rise to a distinct profile).

  17. 17.

    In voting theory, it has been argued that aggregation rules that violate IIA are open to manipulation. However, no concept of manipulability is applicable in the extended preferences context: our question concerns how the facts about individuals’ extended preferences determine the facts about overall betterness, not how any choice should be based on individuals’ reports of their own preferences.

  18. 18.

    This particular proposal is problematic in a setting in which the data we start from is not merely a profile of ordinary preference orderings (one ordinary preference ordering for each individual), but rather a profile of ordinary utility functions. Any stipulation for fixing interpersonal unit comparisons of course automatically induces a standard of intrapersonal unit comparisons: the intrapersonal ratio (difference between x and y for i)/(difference between s and t for i), for example, must be equal to the product of the two interpersonal ratios (difference between x and y for i)/(difference between v and w for j), (difference between v and w for j)/(difference between s and t for i). The problem is that the intrapersonal unit comparisons that are induced by the above analogue of the Borda rule will not in general be consistent with the pre-existing intrapersonal unit comparisons already given in the profile of utility functions, since the Borda rule pays no attention to any features of individuals’ preferences or utilities that go beyond the induced ordinal ranking. The would-be structuralist therefore needs some other prescription, one that respects the existing cardinal information that is already present in individuals’ (ordinary) utility functions.

          There are various ways in which this can be done. The basic task is to select, from the positive affine family of utility functions that cardinally represent each individual’s ordinary preferences, one privileged representative utility function; the profile of representative utility functions across individuals then well-defines a standard of interpersonal comparisons. The best-known such selection rule, the ‘zero-one’ rule, is available in any situation in which every individual’s utility is bounded above and below: one can then select, for each individual, the utility function whose greatest lower bound is zero, and whose least upper bound is one. (This rule is employed, if not argued for, by Isbell (1959) and Schick (1971).) There are, of course, other possibilities: for example, one could equalise the greatest lower bound (setting this to zero for each individual) and the sum of the utilities of all other alternatives, or one could equalise the mean and the variance.

    These proposals suffice to recover consistency with existing interpersonal comparisons. Like the simpler model discussed in the main text, however, these proposals all require interpersonal comparisons to supervene on the profiles of individuals’ relative judgments on alternatives, and for that reason are open to conceptually similar objections. There are three main objections, which we record here for completeness. (We will state them using the zero-one rule just discussed.)

    Firstly: the zero-one rule leads to arguably counterintuitive verdicts in particular cases, where intuition seems to hold that there might simply be more at stake for one person than there is for another. There are two ways of interpreting the dictates of the rule, which we will call ‘narrow’ and ‘broad’; these different interpretations of the rule are subject to different versions of the problem. A narrow interpretation of the rule is one according to which we select the most- and least-preferred alternatives, for the purpose of calibrating the utility functions of distinct agents, only from among the options in play in a given choice situation. The narrow interpretation is subject to obvious problems: clearly Kate and John can be such that Kate’s well-being is affected far more by choice of ice-cream flavour than John’s is; if ice-cream flavour choices are all that is relevant to the case at hand, the narrow interpretation of the rule is committed to denying this datum. This motivates moving to a broad interpretation of the rule, according to which we select the most- and least-preferred alternatives, for each agent, from among all conceivable options. Here the intuition is less clear: it is not obvious that there are pairs of people who exhibit differences in how much the realization of their most preferred and least preferred options matters to them. But insofar as there is some intuition that this could happen, that is an intuition that the proponent of the zero-one rule has to deny.

    Secondly: given the diversity of possible structuralist proposals, in the absence of any argument for one particular such proposal over the others, the postulation of any particular one would be unacceptably arbitrary. (This worry is pressed, in a discussion of interpersonal well-being comparisons, by Sen (1970, 98).) The worry obviously dissolves if it can be argued that one particular structuralist proposal is better than the others; for such an argument for the superiority of mean-variance normalisation, albeit in a different context, see Cotton-Barratt et al. (2014). (Cotton-Barratt et al. are actually arguing only for equalisation of variances, rather than of means and variances, since level comparisons are irrelevant in the context they focus on.)

    Thirdly: the verdicts that the zero-one rule yields on questions of interpersonal comparisons depend on some things that arguably they should not depend upon. Most obviously, on the ‘narrow’ interpretation, questions of interpersonal level comparisons regarding state of affairs x, or interpersonal unit comparisons regarding states of affairs x and y, can depend on whether or not some particular third state of affairs z is also included in the set \({\mathcal {S}}\) relative to which the zero-one rule is specified (the ‘set of states of affairs under consideration’). A natural response to this is to stipulate that \({\mathcal {S}}\) is to include all possible states of affairs—that the rule is to be interpreted broadly—but it is also unclear whether there is any privileged sense of ‘possible’ with boundaries that are sufficiently determinate for present purposes.

    For further discussion of the structuralist program, see e.g. Cotton-Barratt et al. (2014), Griffin (1986), Hammond (1991, 216), Hausman (1995), Jeffrey (1971, 655), Rawls (1999, 283–284).

  19. 19.

    Or relations; some prescription will be needed to deal with ties.

References

  1. Adler, M. (2014). Extended preferences and interpersonal comparisons: A new account. Economics and Philosophy, 30(2), 123–162.

    Article  Google Scholar 

  2. Adler, M. (2012). Well-being and fair distribution: Beyond cost-benefit analysis. Oxford: Oxford University Press.

    Google Scholar 

  3. Adler, M. (forthcoming). Extended preferences. In: M. Adler & M. Fleurbaey (Eds.), Oxford handbook of well-being and public policy. New York: Oxford University Press.

  4. Adler, M. D. (2015). Aggregating moral preferences. Economics and philosophy, 32(2), 283–321.

    Article  Google Scholar 

  5. Arrow, K. (1963). Social choice and individual values (2nd ed.). New York: Wiley.

    Google Scholar 

  6. Austen-Smith, D., & Banks, J. (2000). Positive political theory I. Michigan: University of Michigan Press.

    Google Scholar 

  7. Brandenburger, A. (2003). On the existence of a “complete” possibility structure. Cognitive Processes and Economic Behavior, 30–34.

  8. Brown, D. (1973). Acyclic choice. Cowles Foundation Discussion Papers 360. Cowles Foundation for Research in Economics, Yale University.

  9. Cotton-Barratt, O., MacAskill, W., & Ord, T. (2014). Normative uncertainty, intertheoretic comparisons, and variance normalisation. Unpublished manuscript.

  10. Fishburn, P. C. (1970). Arrow’s impossibility theorem: Concise proof and infinite voters. Journal of Economic Theory, 2(1), 103–106.

    Article  Google Scholar 

  11. Fleurbaey, M., & Maniquet, F. (2008a). Fair social orderings. Economic Theory, 34(1), 25–45.

    Article  Google Scholar 

  12. Fleurbaey, M. (2007). Social choice and just institutions: New perspectives. Economics and Philosophy, 23(1), 15–43.

    Article  Google Scholar 

  13. Fleurbaey, M., & Maniquet, F. (2008b). Utilitarianism versus fairness in welfare economics. In M. Fleurbaey, M. Salles, & J. Weymark (Eds.), Justice, political liberalism, and utilitarianism (pp. 263–280). Cambridge: Cambridge University Press.

    Chapter  Google Scholar 

  14. Greaves, H, & Lederman, H. Forthcoming. Extended preferences and interpersonal comparisons of well-being. Philosophy and Phenomenological Research.

  15. Griffin, J. (1986). Well-being: Its meaning, measurement, and moral importance. Oxford: Oxford University Press.

    Google Scholar 

  16. Hammond, P. (1991). Interpersonal comparisons of utility: Why and how they are and should be made. In J. Elster & J. E. Roemer (Eds.), Interpersonal comparisons of well-being (pp. 200–254). Cambridge: Cambridge University Press.

    Chapter  Google Scholar 

  17. Harsanyi, J. (1955). Cardinal welfare, individualistic ethics, and interpersonal comparisons of utility. Journal of Political Economy, 63(4), 309–321.

    Article  Google Scholar 

  18. Harsanyi, J. C. (1953). Cardinal utility in welfare economics and in the theory of risk-taking. Journal of Political Economy, 61(5), 434.

    Article  Google Scholar 

  19. Hausman, D. (1995). The impossibility of interpersonal utility comparisons. Mind, 104(415), 473–490.

    Article  Google Scholar 

  20. Isbell, J. R. (1959). Absolute games. In A. W. Tucker & R. D. Luce (Eds.), Contributions to the theory of games (Vol. 4, p. 357). Princeton: Princeton University Press.

    Google Scholar 

  21. Jeffrey, R. (1971). On interpersonal utility theory. Journal of Philosophy, 68(20), 647–656.

    Article  Google Scholar 

  22. Kaplan, D. (1995). A problem in possible-world semantics. Modality, Morality and Belief: Essays in Honor of Ruth Barcan Marcus, 41–52.

  23. Kirman, A. P., & Sondermann, D. (1972). Arrow’s theorem, many agents, and invisible dictators. Journal of Economic Theory, 5(2), 267–277.

    Article  Google Scholar 

  24. Kripke, S. A. (2011). A puzzle about time and thought. In Philosophical troubles. Oxford: Oxford University Press.

  25. Mongin, P. (1994). Harsanyi’s aggregation theorem: Multi-profile version and unsettled questions. Social Choice and Welfare, 11(4), 331–354.

    Article  Google Scholar 

  26. Rachels, S. (1998). Counterexamples to the transitivity of ‘better than'. Australasian Journal of Philosophy, 76(1), 71–83.

    Article  Google Scholar 

  27. Rawls, J. (1999). A theory of justice (revised ed.). Cambridge: Belknapp Press of Harvard University Press.

    Google Scholar 

  28. Schick, F. (1971). Beyond utilitarianism. The Journal of Philosophy, 68(20), 657–666.

    Article  Google Scholar 

  29. Sen, A. (1986). Social choice theory. In K. D. Arrow & M. D. Intriligator (Eds.), Handbook of mathematical economics (Vol. 3, pp. 1073–1181). Amsterdam: Elsevier.

    Google Scholar 

  30. Sen, A. (1970). Collective choice and social welfare. San Francisco: Holden-Day.

    Google Scholar 

  31. Temkin, L. (1987). Intransitivity and the mere addition paradox. Philosophy and Public Affairs, 16(2), 138–187.

    Google Scholar 

  32. Temkin, L. (2014). Rethinking the good: Moral ideals and the nature of practical reasoning. Oxford: Oxford University Press.

    Google Scholar 

  33. Voorhoeve, A. (2014). Review of Matthew D. Adler: Well-being and fair distribution: Beyond cost-benefit analysis. Social Choice and Welfare, 42(1), 245–54.

    Article  Google Scholar 

  34. Weymark, J. A. (1984). Arrow’s theorem with social quasi-orderings. Public Choice, 42(3), 235–246.

    Article  Google Scholar 

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Acknowledgments

We thank Christian List, Teruji Thomas and John Weymark for useful discussions and correspondence.

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Correspondence to Harvey Lederman.

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Hilary Greaves and Harvey Lederman have contributed equally to this paper.

Appendix: Proof of Theorem 2

Appendix: Proof of Theorem 2

The bulk of our proof is contained in the Field Expansion Lemma that forms the core of the proof of Arrow’s theorem, and in Weymark’s proof that this lemma in turn implies his Theorem 1 (Weymark 1984).

We use the following definitions. As before, let N be a finite set of individuals, and let X be a finite set of alternatives. Let \(G\subseteq N\) be an arbitrary set of individuals. Let \(x,y\in X\) be any alternatives. Let f be an arbitrary relation aggregation rule with domain \(D\subseteq \left( {\mathcal {P}}(X\times X) \right) ^N\). Then, relative to f,

  • G is semidecisive w.r.t. \(\left( x,y\right)\) iff \(\forall R\in D,\left(\left( \left( \forall i\in G \ xP_{i}y \right) \wedge \left( \forall i\notin G\ yP_{i}x\right) \right) \rightarrow x f^P(R) y\right).\)

  • G is decisive w.r.t. \(\left( x,y\right)\) iff \(\forall R\in D,\left(\forall i\in G \ xP_{i}y\rightarrow x f^P(R) y\right).\)

  • G is decisive iff G is decisive w.r.t. every pair of alternatives.

  • i has a veto w.r.t. \(\left( x,y\right)\) iff \(\forall R\in D,\left(yP_{i}x\rightarrow \lnot xf^P(R)y\right).\)

  • G is an oligarchy for \(Z \subseteq X\) iff for all \(x,y \in Z\), (i) G is decisive w.r.t. \(\left( x,y\right)\), and (ii) every member of G has a veto w.r.t. \(\left( x,y\right)\).

  • G is an oligarchy iff G is an oligarchy for X.

We recall also the following definitions, where \(\pi\) is a permutation of N, which induces a corresponding permutation of X:

Action of \(\pi\) on relations \(r \in \mathcal {R}\): \(\forall x,y \in X, \left(x \left( \pi (r) \right) y \leftrightarrow \left( \pi ^{-1}(x)\right) r \left( \pi ^{-1}(y) \right)\right) ;\)

Action of \(\pi\) on profiles \(R \in \mathcal {R}^N\): \(\pi (R) = \left( \pi \left( R_{\pi ^{-1}(1)}\right) , \ldots , \pi \left( R_{\pi ^{-1}(|N|)}\right) \right) ;\)

Action of \(\pi\) on aggregation rules f: \(\forall R \in \pi D, \left( \pi f \right) (R) = \pi \left( f \left( \pi ^{-1}R \right) \right)\).

A (Anonymity) : For all permutations \(\pi\) of N, \(f = \pi f\).

Our claim (recall) is

Theorem 2.

Let f be a RAR with domain \(D=\mathcal {R}_{rat}^{N}\). Suppose that f satisfies RF, WP, QT and IIA. Let \(Z\subseteq X\) be any set of extended alternatives with respect to which \(\left( \mathcal {R}_{rat}\right) ^{N}\) satisfies SD, and such that \(f|_{(\mathcal {R}_{rat}|_Z)^N}\) satisfies Anonymity. Then f is Spineless with respect to Z.

The proof uses the following lemmas.

Lemma 2

(Field Expansion Lemma). Let f be an RAR that satisfies QT, WP and IIA, and whose domain D satisfies SD w.r.t. X. If a subpopulation \(G\subseteq N\) is semidecisive over any pair of alternatives, then G is decisive.

Proof.

See e.g. Arrow (1963, 98-100), Sen (1986, 1080). (Arrow and Sen officially assume Universal Domain, but in fact their proofs of this Lemma only require the far weaker condition SD.)\(\square\)

Lemma 3.

Let f be an RAR whose domain D satisfies SD w.r.t. X. Then there is at most one oligarchy relative to f.

Proof.

See Weymark (1984, Lemma 2); again, the proof only requires SD rather than the UD condition officially assumed by Weymark. \(\square\)

Given Lemmas 2 and 3, we can establish the following:

Lemma 4

(Weymark’s oligarchy theorem). Let f be an RAR that satisfies RF, QT, WP and IIA, and whose domain D satisfies SD w.r.t. X. Then there exists a unique oligarchy relative to f.

Proof.

Weymark (1984), Theorem 1. \(\square\)

The proof of our theorem is then as follows.

Proof.

If f satisfies IIA, then f naturally induces a RAR \(f|_{\left( \mathcal {R}_{rat}|_Z \right) ^N}\) for the aggregation of preferences over \(Z \subseteq X\), by restriction. Given that \(\mathcal {R}_{rat}^N\) satisfies SD w.r.t. Z, so does \(\left( \mathcal {R}_{rat}|_Z\right) ^N\). Thus, applying Lemma 4 to the RAR \(f|_{\left( \mathcal {R}_{rat}|_{Z}\right) ^{N}}\) establishes that there exists a unique oligarchy G (for Z) with respect to \(f|_{\left( \mathcal {R}_{rat}|_{Z}\right) ^{N}}\).

We next show that \(G=N\). Suppose, for contradiction, that \(G\subsetneq N\). Let \(\pi\) be any permutation of N that maps one or more members of \(N \setminus G\) to members of G (since \(G\subsetneq N\), such a permutation exists). It is straightforward to check that if G is an oligarchy (for Z) relative to \(f|_{\left( \mathcal {R}_{rat}|_Z \right) ^N}\), then, for any permutation \(\pi\) of N such that \(\left( \mathcal {R}_{rat}|_Z \right) ^N\) is closed under \(\pi\), \(\pi G\) is an oligarchy (for \(\pi (Z)\), and thus since \(\pi (Z)= Z\), for Z) relative to \(\pi f|_{\left( \mathcal {R}_{rat}|_Z \right) ^N}\). Since (by assumption) \(f|_{\left( \mathcal {R}_{rat}|_Z \right) ^N}\) satisfies Anonymity, however, we have \(f_{\left( \mathcal {R}_{rat}|_Z \right) ^N} = \pi f_{\left( \mathcal {R}_{rat}|_Z \right) ^N}\). Since we have chosen \(\pi\) such that \(\pi G\ne G\), this contradicts Lemma 3.

In case \(G=N\), every individual has a veto for every pair of alternatives in Z, relative to \(f|_{\left( \mathcal {R}_{rat}|_{Z}\right) ^{N}}\). But if this is true relative to \(f|_{\left( \mathcal {R}_{rat}|_{Z}\right) ^{N}}\), then by IIA it is also true relative to f. That is, f is Spineless with respect to Z, as claimed. \(\square\)

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Greaves, H., Lederman, H. Aggregating extended preferences. Philos Stud 174, 1163–1190 (2017). https://doi.org/10.1007/s11098-016-0748-6

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Keywords

  • Interpersonal well-being comparisons
  • Extended preferences
  • Preference-satisfaction theory
  • Theories of well-being