2. INTERPERSONAL COMPARISONS, EXTENDED PREFERENCES AND HARSANYI’S APPROACH

Formally, the problem of interpersonal comparisons can be expressed as follows. Let O be a set of outcomes, i.e. states of affairs, {x, y, z. . .}. Let N be a set of individuals, {1, 2, . . ., N}. N has a finite number of members N, each of whom exists in all of the outcomes (problems of variable and infinite populations are ignored here). A well-being comparison is not a ranking of outcomes, simpliciter, but rather a ranking of outcomes relativized to individuals. We say that one outcome is better for Jim than a second – and, if interpersonal comparisons are possible, that Sheila is better off in some outcome than Jim in some outcome. This relativization can be captured via the concept of an individual ‘history’. As I will define it, a history (x; i) is a pairing of an individual, i, and an outcome, x. The set H is the set of all histories. In other words, H is O × N. A well-being ranking, denoted ‘$\succurly{175}^{{\rm WB}}$’, is a quasiordering on ${\bf H}. `(x; i) \succurly{175}^{{\rm WB}} (y; j)$’ should be interpreted as ‘individual i in outcome x is at least as well off as individual j in outcome y’. Having strictly greater well-being ($\succurly{173}^{{\rm WB}}$) and being equally well off (~^WB) are derived from ≽^WB in the standard way.

A quasiordering, recall, is a binary relation which is transitive and reflexive but not necessarily complete.Footnote ⁶ For a given pair of histories (x; i) and (y; j), it is possible that neither $(x; i) \succurly{175}^{{\rm WB}} (y; j)$, nor $(y; j) \succurly{175}^{{\rm WB}} (x; i)$. We can therefore say that ≽^WB makes some interpersonal comparisons iff: there exists at least one pair of histories (x; i) and (y; j), with i and j distinct individuals (i ≠ j), such that $(x; i) \succurly{175}^{{\rm WB}} (y; j)$. Conversely, ≽^WB is only intrapersonally comparable iff, for every pair of histories such that $(x; i) \succurly{175}^{{\rm WB}} (y; j), i = j$.

While the formalization thus far expresses comparisons of well-being levels, we should also keep in view comparisons of well-being differences. Consider the set H × H, comprised of all pairs of histories. Then $\succurly{175}^{{\rm DIFF}}$ is a quasiordering on H × H.Footnote ⁷ ‘$((x; i), (y; j)) \succurly{175}^{{\rm DIFF}} ((z; l), (w; m))$’ should be interpreted as: the difference between the well-being of individual i in x and the well-being of individual j in y is at least as large as the difference between the well-being of individual l in z and individual m in w. $\succurly{175}^{{\rm DIFF}}$ makes some interpersonal difference comparisons iff there is at least one case in which $((x; i), (y; j)) \succurly{175}^{{\rm DIFF}} ((z; l), (w; m))$ and it is not the case that i = j = l = m.

The puzzle of interpersonal comparisons – thus – is to use information about individual preferences so as to construct a ≽^WB that makes some interpersonal level comparisons and/or a $\succurly{175}^{{\rm DIFF}}$ that makes some interpersonal difference comparisons.

What is an extended preference? We should distinguish between the general concept of an extended preference, and particular conceptions or versions of this general concept. A preference, on the part of some individual (Raj), is a ranking that connects to Raj's choices. While an outcome-preference is a choice-connected ranking of outcomes, an ‘extended preference’ is a choice-connected ranking of individual histories. Generically, I will use the term ‘spectator’ to refer to the holder of an extended preference, and the term ‘subject’ to refer to the individuals in histories. Phil and Jim are the ‘subjects’, respectively, of the histories (x; Phil) and (y; Jim). Raj, in holding extended preferences regarding these histories, is a ‘spectator’.

But what does it mean to rank histories? How are we to analyse the content of Raj's preference for (x; Phil) over (y; Jim)? Here, Harsanyi offers a particular conception of extended preferences. Raj is supposed to imagine ‘standing in Phil's shoes’ in outcome x. In other words, Raj is supposed to imagine what it would be like to possess the physical, social and other attributes that Phil has in outcome x, and also to have Phil's tastes. Raj is next supposed to imagine ‘standing in Jim's shoes’ in outcome y: possessing now the physical, social, etc. attributes of Jim in outcome y, along with Jim's tastes. And, finally, Raj is supposed to rank (x; Phil) versus (y; Jim) by deciding whether he prefers to have the first attribute bundle or the second.

As Harsanyi explains:

For short, I will term the particular conception of extended preferences which Harsanyi adopts the empathy-based conception, and I will express it as follows.

‘Empathy’ – a word that Harsanyi himself uses – is the capacity to assume someone else's perspective. On the ‘empathy’ based conception, the spectator develops extended preferences by exercising this capacity. In order to compare two histories, the spectator empathetically projects herself into the position of the first subject (considering both that subject's non-taste attributes in the first history, and his tastes), and then empathetically projects herself into the position of the second subject (considering now that subject's non-taste attributes in the second history, and his tastes). This empathetic projection by the spectator is nothing other than the thought exercise of taking on the properties (including the feelings and tastes) of each subject. Thus the ‘empathy-based’ conception of extended preferences might also be termed an ‘attribute-acquisition’ conception.

Having articulated a conception of extended preferences, Harsanyi proceeds to arrive at interpersonal comparisons of well-being levels and differences by making two further assumptions. For short, call these the ‘Convergence’ premise and the ‘Bernoulli’ premise. The Convergence premise is that spectators have the very same extended preferences. If Raj prefers (x; Phil) to (y; Jim), then so does Sally. The ‘Bernoulli’ premise is that spectators’ extended preferences over history lotteries comply with the requirements of von-Neumann Morgenstern (vNM) expected utility theory, hence can be expectationally represented by vNM utility functions; and that a measure of well-being differences is derivable (in a linear fashion) from these utility functions.

The Convergence and Bernoulli premises, together, allow Harsanyi to define $\succurly{175}^{{\rm WB}}$ and $\succurly{175}^{{\rm DIFF}}$ as follows. Let v(.) be a vNM function expectationally representing the spectators’ common extended preferences over history lotteries. Then $(x; i) \succurly{175}^{{\rm WB}} (y; j)\, {\rm iff}\, v(x; i)\, \succurly{145} v(y; j).\, ((x; i),\break (y; j))\, \succurly{175}^{{\rm DIFF}}\, ((z; l), (w; m))$ iff $v(x; i) - v(y; j)\, \succurly{145}\, v(z; l) - v(w; m)$.

The critical literature on Harsanyi has discussed the Convergence and Bernoulli premises at length.Footnote ¹¹ The first seems untrue. There is no particular reason to believe that individuals will have identical extended preferences. Harsanyi offers an argument for Convergence, but the argument is flawed.

The Bernoulli premise remains open to reasonable debate, but is much more contestable than Harsanyi realized. Consider the simplest case, where Convergence happens to hold true and spectators do have identical extended preferences over histories and lotteries. Thus there is a common extended vNM utility function v(.). Why should it be true that $((x; i), (y; j)) \succurly{175}^{{\rm DIFF}} ((z; l), (w; m))$ iff $v(x; i) - v(y; j) \, \succurly{145}\, v(z; l) - v(w; m)$? Isn't it possible that spectators, in ranking lotteries, are risk-averse or risk-prone in well-being – in other words, that $\succurly{175}^{{\rm DIFF}}$ corresponds to differences in some convex or concave transformation of v(.)?

These criticisms of Harsanyi are familiar from the literature. In the next Part, I present a different and more novel criticism.Footnote ¹² This criticism is logically independent of the familiar critiques of Convergence and Bernoulli. And, in a sense, the criticism is deeper. I argue that Harsanyi's account of the content of extended preferences – the empathy-based conception – is problematic. In effect, Harsanyi stumbles from the get-go. An extended preference is some kind of preference or judgement on the part of the spectator; but to characterize it, specifically, as a preference to acquire subjects’ attributes and tastes faces serious difficulties. The critique of the empathy-based conception of extended preferences, which I develop in this paper, would also apply to a neo-Harsanyi view that relaxes Convergence and/or Bernoulli but retains the problematic analysis of extended preferences in terms of empathetic projection.

3. THE EMPATHY-BASED CONCEPTION OF EXTENDED PREFERENCES: A CRITIQUE

This Part describes two important flaws in the empathy-based conception of extended preferences – flaws that have received little attention in the literature. More precisely, these are flaws in any account of well-being comparisons (such as Harsanyi’s) that builds from this conception.Footnote ¹³ The first flaw is the ‘essential attribute’ problem; the second, the ‘wrong kind of preference’ problem.

4. A SYMPATHY-BASED CONCEPTION OF EXTENDED PREFERENCES

I propose a sympathy-based conception of extended preferences.Footnote ³³ The generic set-up remains the same one I used above, in discussing the empathy-based conception. O is the set of outcomes, N a finite set of individuals (each of whom exists in all outcomes), and H = O × N the set of all histories {(x; i)}. K is the set of spectators (for simplicity, assume that K = N). The outcomes in O are arbitrarily detailed specifications of possible worlds, but do not specify individuals’ preferences. Let R = (R ₁, R ₂, . . ., R_N) be a possible profile of ‘tastes’ (outcome and choice preferences) on the part of individuals 1, 2 . . . N. For any given R, each spectator k has extended preferences $\succurly{175}^{k}(R)$ (or, for short, $\succurly{175}^{k}$). Each ≽^k is a quasiordering of H.

The difference between the sympathy-based and empathy-based conception of extended preferences lies not in this set-up (which is quite general), but in the substantive content of ≽^k. In explicating the sympathy approach, I will distinguish between ‘intrapersonal’ and ‘interpersonal’ extended preferences. A spectator's extended preference as between two histories with the same subject is an ‘intrapersonal’ extended preference. A spectator's extended preference as between two histories with different subjects is an ‘interpersonal’ extended preference.

What exactly is the connection between extended preferences, in the sympathy-based sense, and ≽^WB? Part 5 will address this question for the general case, where Convergence may fail. At a minimum, however, if Convergence holds true (everyone has the same extended preferences, i.e. $\succurly{175}^{1} = \succurly{175}^{2} = \ldots = \succurly{175}^{N} = \succurly{175}$), then the sympathy-based conception says: $(x; i) \succurly{175}^{{\rm WB}} (y; j)$, the two outcomes and subjects the same or different, iff $(x; i) \succurly{175} (y; j)$.

What is sympathy? While I empathize with you by projecting myself into your position, I sympathize with you by adopting an attitude of care and concern for you. Darwall has emphasized the distinction between empathy and sympathy, and the connection of sympathy (not empathy) to well-being.

The sympathy-based conception of extended preferences readily avoids the essential-attribute problem and the wrong-kind-of-preference problem. Consider first the intrapersonal case. In ranking (x; i) and (y; i), the spectator does not engage in the thought experiment of acquiring i's attributes in x and in y. Instead, his extended preference is reduced to a preference of the ordinary sort – an outcome preference – with an attitudinal restriction, namely a ranking of those outcomes while in the grip of an attitude of unreserved sympathy for the subject, i.

The fact that some of the subject's attributes are EL_k attributes in no way frustrates this exercise. For example, Sue Dean, with an attitude of concern for Cleopatra, can ask herself whether she prefers the outcome x' in which Cleopatra dies after Actium from a self-imposed snakebite, or instead the outcome x in which Cleopatra is imprisoned for life by the Romans. Because x' and x are not outcomes in which Sue Dean herself is supposed to be born in the first century bc, to be the lover of Marc Antony, etc., the outcomes are possible. And the fact that Sue Dean and Cleopatra are distinct individuals, each with some essential attributes that the other lacks, obviously does not prevent Sue from being sympathetic to Cleopatra. Sympathy often takes the form of being targeted upon some person distinct from the sympathizer.

The ‘wrong kind of preference problem’, remember, was that the sheer possession of a preference on someone's part for x over y does not guarantee that x is better than y (for the holder of the preference, or for anyone else), since preferences can be motivated by a wide range of considerations (moral, aesthetic, etc.). But the sympathy-based conception avoids this problem, in the intrapersonal case, by virtue of the connection between sympathy and well-being emphasized by Darwall – more precisely, the connection between having an attitude of sympathy, and being motivated to pursue what you believe lies in the interests of the sympathy target. Where spectator k holds an attitude of unreserved sympathy for subject i, it is necessarily the case that: if k believes x to be better for i's well-being than y, k prefers x to y.Footnote ³⁷

Thus k's weak preference for x over y, under a condition of unreserved sympathy with i, is evidence that $(x; i) \succurly{175}^{{\rm WB}} (y; i)$ – at least if k is suitably ‘idealized’ in the sense of having good information, deliberates, etc. And the fact that all spectators, under a condition of unreserved sympathy with i, weakly prefer x to y, is substantial evidence that $(x; i) \succurly{175}^{{\rm WB}} (y; i)$, if the spectators are suitably idealized – indeed, it arguably guarantees that $(x; i) \succurly{175}^{{\rm WB}} (y; i)$.Footnote ³⁸

Note that this linkage between the preferences held by spectators sympathetic to a particular subject, and the subject's well-being, becomes much weaker if we relax the requirement that spectators be unreservedly sympathetic to the subject. Imagine that some well-informed spectators are equally concerned about everyone in a group of subjects, including Jim, and under this condition prefer x to y. This doesn't tell us much about whether x is better for Jim. The spectators might believe that x is worse for Jim, but still prefer it on balance, despite their concern for him, because they are equally concerned about the others. Requiring the spectator's preferences to be ‘sympathetic’ constitutes a viable solution to the ‘wrong kind of preference’ problem in the intrapersonal case only if the sympathy involved is an attitude of care and concern exclusively (or at least largely)Footnote ³⁹ targeted at the subject.

One, interesting, variant of the intrapersonal case arises where the spectator is identical to the subject. Spectator k has an extended preference for (x; k) over (y; k). The sympathy-based conception analyses this as an outcome preference on k's part (a preference for x over y) under a condition of wholehearted self-sympathy. ‘Self-sympathy’ means an attitude of care and concern for the very same person who has the attitude. The person who is the target of the sympathizer's attitude is, now, the sympathizer himself. ‘Self-sympathy’ is, perhaps, an unfamiliar term. Here's a synonym: ‘self-interest’. In short, the spectator's extended preferences for the case where he is subject are nothing other than his self-interested outcome preferences.

The observation can be inverted. Wholehearted sympathy is a generalization of self-interest – an attitude of interest in some particular person, be it the holder of the attitude (self-interest), or someone else. The spectator's intrapersonal extended preferences are just rankings of outcomes with this generalized self-interest directed at the appropriate person, the subject.

Consider now the interpersonal case. It is tempting to propose: k has an extended preference for (x; i) over (y; j), i and j distinct, iff k prefers x over y under a condition of unreserved sympathy with ‘the subject’. However, this proposal fails. There is no one person to be the target of the spectator's sympathy in the interpersonal case. And holding simultaneous attitudes of unreserved sympathy with two different subjects is psychologically impossible. The spectator Robert cannot, at the same time, care only about Maurice and only about Jean.Footnote ⁴⁰ Thus, it will not work to analyse Robert's extended preference for (x; Maurice) over (y; Jean) as a preference for x over y while simultaneously being unreservedly sympathetic to both Maurice and Jean. Asking what Robert would prefer with this impossible combination of attitudes would be like asking what he prefers when simultaneously calm and panicky.

To be sure, Robert's attitudes of sympathy can change. He can, at one moment, feel unreservedly sympathetic to Maurice and, later, unreservedly sympathetic to Jean. But a preference for one possible outcome over a second is a synchronic relation between a preferrer and the two possibilities, each simultaneously presented to his mind via some mental representation. For Robert to stand in that synchronic relation to outcomes x and y, and at that one moment in time to be wholeheartedly sympathetic to two different people, is impossible.

Thus, in the interpersonal case, the account of extended preferences presented here asks the spectator to make a judgement about well-being. However, this analysis is a close cousin to the intrapersonal analysis. Although sympathy does not necessitate explicit well-being judgements – I can hold an attitude of sympathy for you without explicitly thinking about your well-being (or so it is plausible to believe) – sympathy is responsive to such judgements, as already discussed. If I am sympathetic to you, and come to believe/judge that some course of action benefits you, then I am motivated to pursue that course of action. The affective component of the intrapersonal analysis (wholehearted concern for the sympathy target) cannot be transposed to the interpersonal case, but the valuational component (the judgements about the target's well-being that motivate the sympathizer) can be. And that is what my account does. Indeed, the very same implicit or explicit views about the nature of well-being that motivate the sympathizer, in the intrapersonal case, will determine his well-being ranking when he makes explicit well-being judgements in the interpersonal case. For this reason, I describe the entire account of extended preferences, both (1) and (2), as the ‘sympathy-based conception’.Footnote ⁴¹

Note that the interpersonal analysis, like its intrapersonal cousin, avoids the ‘essential attribute’ and ‘wrong kind of preference’ objections. A spectator can take account of all of the subjects’ attributes, including their essential attributes, in arriving at his well-being judgements. A judgement made by one spectator to the effect that i in x is at least as well off as j in y is evidence that $(x; i) \succurly{175}^{{\rm WB}} (y; j)$, at least if the spectator is suitably idealized. And such a judgement by all such spectators is substantial evidence that $(x; i) \succurly{175}^{{\rm WB}} (y; j)$ – indeed, it arguably guarantees that $(x; i) \succurly{175}^{{\rm WB}} (y; j)$.

What exactly is the relation between ≽^k and the profile R? Remember that extended preferences (on both the empathy- and sympathy-based account) are potentially dependent on individual tastes. But what does such dependency involve, on the sympathy-based view?

First, there is a logical link between ≽^k and R_k. Given the definition of intrapersonal extended preferences, it follows that spectator k has a weak extended preference for (x; k) over (y; k) iff R_k is such that k has a self-interested weak preference for outcome x over outcome y. Second, there are empirical (not logical) links between ≽^k and each R_i, i ≠ k. In developing his intrapersonal extended preferences for histories belonging to subjects other than himself, and in making across-subject comparisons, the spectator k is permitted to take account of everyone else's tastes. Whether k does so depends upon his own tastes and values. For example, k might make the judgement that (x; i) is better for well-being than (y; j) given R_i and R_j, but not given R_i* and R_j*.

In the next Part, I will discuss how to derive ≽^WB from an array of extended preferences ($\succurly{175}^{1}, \ldots, \succurly{175}^{k}, \ldots,\succurly{175}^{N}$), where Convergence fails. But we should first consider three potential criticisms of the sympathy-based conception of extended preferences, as well as the question of representing ≽^k via utility functions.

One criticism is that the sympathy-based conception, used as an analysis of well-being, is circular. In the interpersonal case, the circularity is patent. In that case, the view says: Given Convergence, (x; i) ≽^WB (y; j) iff everyone judges that i in x is at least as well off as j in y. So well-being is ‘analysed’ in terms of well-being judgements – an obvious circularity. In the intrapersonal case, the circularity is more subtle, but still real. Even if sympathy does not require explicit thinking about the target's well-being, the attitude of sympathy has important connections to well-being, described in the previous paragraphs. But (so the critique goes) these connections are not happenstance. It is not as if sympathy is some independently specified attitude which, as it happens, motivates the holder to act in line with his judgements about the target's well-being, if he makes such judgements. Rather, this is a defining feature of the attitude: sympathy has conceptual, not just empirical, connections to well-being judgements and beliefs.Footnote ⁴² Insofar as the concept of well-being is used to identify the attitude of sympathy, which in turn is used – by the view tendered here – to analyse intrapersonal extended preferences and, thereby, intrapersonal well-being comparisons, the view is circular.

However, the circularities just described are not vicious circularities. Consider the interpersonal case (if the circularity is not vicious here, then a fortiori it is not in the intrapersonal case). To analyse a value relation between two items in terms of that very value relation is viciously circular. But the sympathy-based conception does not do that. Rather, it analyses a value relation in terms of individuals’ beliefs and judgements about that relation. What we have, here, is not troubling circularity, but a kind of self-reference. The value relation ≽^WB is instantiated just in case individuals have certain thoughts, thoughts about that very relation. This sort of self-reference is familiar from the literature on secondary properties.Footnote ⁴³ An object has the property of redness if it has surface reflectance characteristics which make it look red to normal observers. Note, here, how persons’ perceptions or judgements of redness are an ineliminable aspect of the property of redness.

The characterization of interpersonal comparisons in terms of well-being judgements and beliefs is, indeed, quite useful. One way of gaining evidence about whether $(x; i) \succurly{175}^{{\rm WB}} (y; j)$ is by asking individuals (with good information, thinking clearly, etc.) whether they believe that i in x is better off than j in y. (By analogy, the process of constructing a successful device to detect red objects would involve asking observers which objects look red.) Turn this coin over: the very usefulness of this characterization underscores that it is not viciously circular. By contrast, the sham ‘analysis’ of $\succurly{175}^{{\rm WB}}$ in terms of well-being (rather than thoughts about well-being) offers no basis for determining whether i or j is indeed better off.

To be sure, the sympathy-based conception is not value-free. It links well-being to thoughts about well-being. Thus the analysis, although not viciously circular, is not as ‘nice’ as a characterization of well-being wholly in terms of non-value facts and non-evaluative thoughts. But philosophical efforts to produce a value-free analysis of well-being have been a failure. In particular, no value-free solution to the ‘wrong kind of preference’ problem has yet succeeded

A second critique of the sympathy-based conception is that it fails to satisfy what Harsanyi calls the ‘Principle of Acceptance’. This principle says that a spectator's extended preferences track the subject's outcome preferences in the intrapersonal case.

• Principle of Acceptance: $(x; i) \succurly{175}^{k} (y; i)$ iff R_i is such that i weakly prefers x to y.

The Principle of Acceptance was a central element of Harsanyi's account of extended preferences.Footnote ⁴⁴

However, the Principle of Acceptance is not a plausible component of the sympathy-based conception of extended preferences. The Principle, as just formulated, runs afoul of the ‘wrong kind of preference’ problem. Imagine that i prefers x to y on non-self-interested grounds. In such a case – given the connection between extended preferences and ≽^WB – it is problematic to require that k extendedly prefer (x; i) to (y; i).

A refinement to the Principle of Acceptance makes it more plausible.

• Modified Principle of Acceptance:$(x; i) \succurly{175}^{k} (y; i)$ iff R_i is such that i self-interestedly weakly prefers x to y.

However, this principle is far from compelling. I may care wholeheartedly about you, but refuse to take your views about your well-being as decisive. Sympathetic preferences may, in some cases, be paternalistic. This can occur, for example, where (1) the sympathizer gives more weight to some aspect of well-being than the target; and (2) the sympathizer believes that the realization of this aspect of well-being is sufficiently insensitive to the target's preferences. For example, the sympathizer may give greater emphasis to health; the target, to enjoyment. Moreover, the sympathizer believes (correctly) that a strong preference for health is not a precondition for its realization; people who care little for their health can still end up healthy. In one outcome, the target has a healthy but less enjoyable lifestyle; in a second, he is more indulgent but harms his health. The target self-interestedly prefers the second outcome; the sympathizer, caring wholeheartedly about the target, and knowing of his preference, believes that he undervalues health, and prefers the first outcome.

The Modified Principle of Acceptance precludes the spectator from ever having an intrapersonal history ranking that deviates from the subject’s. This is arguably too strong. By contrast, the sympathy-based conception (without that principle) permits the spectator to take account of the subject's preferences, without requiring the spectator to take those preferences as decisive. For example, the spectator believes that reading great literature or seeing great art makes a larger contribution to the target's well-being than watching sit-coms on TV, but only if the target himself has a taste for these more refined pursuits. The spectator thus extendedly prefers a history in which the target spends his free time experiencing art or literature, rather than sit-coms, iff the target self-interestedly prefers to spend his time this way. The sympathy-based conception of extended preferences (without the Modified Principle of Acceptance) permits this.

Still, some readers may wish to embrace the Modified Principle of Acceptance. They may argue: ‘It is true that a sympathizer k whose attitude of sympathy is targeted at some other person i may act paternalistically towards i. However, the best account of well-being is strongly non-paternalistic, in the following sense. If i is sufficiently idealized, and is ranking outcomes under a condition of self-sympathy (self-interest), then one outcome is at least as good for i as a second iff i weakly prefers the first – regardless of what others may prefer, and even if these others are also idealized, and also sympathetic towards i’.

Whether the most attractive account of well-being is strongly non-paternalistic, in the sense just described, is controversial. The sympathy-based model of extended preferences will yield a strongly non-paternalistic account of well-being if it is revised to incorporate the Modified Principle of Acceptance.

A third criticism of the sympathy-based conception (in both the original version and revised to incorporate the Modified Principle of Acceptance) is that nothing in my definitions of intra- and interpersonal extended preferences ensures that ≽^k is a quasiordering of H. For example, the spectator might have well-behaved intrapersonal rankings, and then make interpersonal judgements that yield an intransitivity.

The answer, here, is that k should treat his initial intra- and interpersonal rankings as provisional, and be willing to revise them to avoid intransitivity. Think of this as a rationality requirement on extended preferences. How k should engage in this revision process, so as to maximize consistency with his initial rankings, is a complicated topic that I will not attempt to address. It is clear, however, that if each of k's N intrapersonal rankings (his rankings of outcomes under a condition of sympathy with each subject) is itself a quasiordering, then these and his interpersonal rankings can always be revised in some fashion so that ≽^k is a quasiordering which never contradicts (either conforms to or ‘extends’) the initial intrapersonal rankings.Footnote ⁴⁵

A final topic concerns the representation of extended preferences via utility numbers. Assume, first, that ≽^k is a complete quasiordering of the set H of histories. Standard results in utility theory show that, if H is finite or countable, ≽^k can be represented by a single utility function v^k(.) such that $(x; i) \succurly{175}^{k} (y; j)$ iff v^k(x; i) ⩾ v^k(y; j). Such a v^k(.) may (but need not) exist if ≽^k is complete and H is uncountable;Footnote ⁴⁶ and of course no single utility function can represent an incomplete quasiordering of H. However, every ≽^k – whether complete or incomplete, and regardless of the cardinality of H – can be represented by some set of utility functions V^k, such that (x; i) ≽^k (y; j) iff v^k(x; i) ≥ v^k(y; j) for all v^k(.) in V^k.Footnote ⁴⁷

5. CONSTRUCTING WELL-BEING COMPARISONS

Convergence can fail – even if the Modified Principle of Acceptance is adopted, and a fortiori if it is rejected. For example, two spectators can make different interpersonal judgements, because of a disagreement about the sources of well-being. We thus face the interesting question of constructing ≽^WB where spectators need not have the same extended preferences.Footnote ⁴⁸

Formally, we are looking for a rule that takes a profile P of extended preferences over set H, $P = (\succurly{175}^{1}, \succurly{175}^{2}, \succurly{175}^{3}, \ldots, \succurly{175}^{N})$, and maps this profile onto a well-being quasiordering of H, ≽^WB.Footnote ⁴⁹ I will use the symbol $\succurly{175}^{k}_{P}$ to mean the extended preferences of spectator k under profile P, and ≽^WB(P) to mean the well-being quasiordering associated with P – dropping the ‘P’ where the particular profile at issue is clear from context. A statement of identity between extended preferences, namely, $\succurly{175}^{k}_{P} = \succurly{175}^{l}_{Q}$, means that for all (x; i) and (y; j) in ${\bf H}, (x; i) \succurly{175}^{k}_{P} (y; j)$ iff $(x; i) \succurly{175}^{l}_{Q} (y; j)$. Similarly, $\succurly{175}^{{\rm WB}}(P) = \succurly{175}^{{\rm WB}}(Q)$ means that, for all (x; i) and (y; j) in ${\bf H}, (x; i) \succurly{175}^{{\rm WB}}(P) (y; j)$ iff $(x; i) \succurly{175}^{{\rm WB}}(Q)(y; j)$. Extended preferences can be heterogeneous (Convergence can fail), i.e. it is possible that $\succurly{175}^{k}_{P} \not= \succurly{175}^{l}_{P}$, with k and l different individuals.

This problem, obviously, has an Arrovian ‘feel’. But it differs in critical respects from the classic Arrow problem. First, the ‘inputs’ and ‘outputs’ need only be quasiorderings, not complete orderings. Second, what we are after is not a social ranking of outcomes, but a ranking of histories in light of well-being.

One plausible rule is the Pooling Rule: $(x; i) \succurly{175}^{{\rm WB}}(P) (y; j)$ iff $(x; i) \succurly{175}^{k}_{P} (y; j)$ for all spectators k. This Rule can also be expressed in terms of utility functions. The quasiordering $\succurly{175}^{{\rm WB}}$, however constructed, can always be represented by some set V of utility functions such that $(x; i) \succurly{175}^{{\rm WB}} (y; j)$ iff $v(x; i)\, \succurly{145}\, v(y; j)$ for all v(.) in V. In the case of the Pooling Rule, V is simply the union of V¹, V², . . ., V^N, where V^k represents $\succurly{175}^{k}$.

The Pooling Rule satisfies various plausible axioms: Unanimous Indifference and Weak Superiority, Independence, Spectator and Subject Anonymity, and either Strong or Weak Non-paternalism.

‘Unanimity’ axioms say that universal spectator judgements are decisive with respect to well-being. Well-being facts are accessible to the community of spectators (at least idealized ones). It is not possible for everyone to be wrong about well-being. Formally, these are analogous to the Pareto principles in social choice.

• Unanimous Indifference: (x; i) ~ ^k_P(y; j) for all k implies (x; i) ~ ^WB(P)(y; j).

• Unanimous Weak Superiority: $(x; i) \succurly{173}^{k}_{P} (y; j)$ for all k implies $(x; i) \succurly{173}^{{\rm WB}}(P) (y; j)$.

The Independence Axiom says that the well-being ranking of any pair of histories is wholly determined by spectators’ preferences between those histories.

• Independence: Let profiles P and Q and histories (x; i) and (y; j) be such that: for all $k, (x; i) \succurly{175}^{k}_{P} (y; j)$ iff $(x; i) \succurly{175}^{k}_{Q} (y; j)$. Then $(x; i) \succurly{175}^{{\rm WB}}(P) (y; j)$ iff $(x; i) \succurly{175}^{{\rm WB}}(Q) (y; j)$.

‘Independence’ is, of course, the analogue of Arrow's ‘Independence of Irrelevant Alternatives’. The former seems considerably more plausible than the latter. In determining whether some outcome x is socially preferred, dispreferred, or indifferent to some other outcome y (the Arrow problem), we may well want to take into consideration more information than the affected individuals’ ordinal rankings of the outcomes – in particular, information about the pattern of interpersonally comparable well-being in both outcomes. By contrast, the problem at hand is to arrive at a basis for ascriptions of interpersonally comparable well-being, as a function of spectators’ extended preferences. What relevant information about spectators’ extended preferences is excluded by Independence?

Spectator anonymity expresses the idea that everyone's well-being views have equal weight. Subject anonymity expresses the idea that the well-being ranking of a given subject's histories does not depend upon the identity of the subject (were the spectators to have the same extended preferences with respect to someone else's histories, the well-being ranking would be the same).

Let π be any permutation mapping on the set of N subjects and spectators.

• Spectator Anonymity: If $\succurly{175}^{k}_{P} = \succurly{175}^{\pi(k)}_{Q}$ for all k, then $\succurly{175}^{{\rm WB}}(P) = \succurly{175}^{{\rm WB}}(Q)$.

• Subject Anonymity: For all k, let $\succurly{175}^{k}_P$ and $\succurly{175}^{k}_Q$ be such that, for all histories (x; i) and $(y; j), (x; i) \succurly{175}^{k}_{P} (y; j)$ iff $(x; \pi(i)) \succurly{175}^{k}_{Q} (y; \pi(j))$. Then $(x; i) \succurly{175}^{{\rm WB}}(P) (y; j)$ iff $(x; \pi(i)) \succurly{175}^{{\rm WB}}(Q) (y; \pi(j))$.

Non-paternalism axioms give a special role to each subject in ranking his own histories. Strong non-paternalism says that the well-being ranking of a given subject's histories is identical to the subject's extended preferences over those histories. If spectators are required to construct extended preferences in accordance with the Modified Principle of Acceptance, the Pooling Rule satisfies strong non-paternalism. If extended preferences can violate that principle, the Pooling Rule can violate strong non-paternalism, but it still satisfies weak non-paternalism – namely, that the well-being ranking of a subject's histories will never directly contradict the subject's own ranking.

• Strong Non-paternalism: $(x; i) \succurly{175}^{{\rm WB}}(P) (y; i)$ iff $(x; i) \succurly{175}^{i}_{P} (y; i)$.

• Weak Non-paternalism: If $(x; i) \succurly{173}^{i}_{P} (y; i)$, then not $(y; i) \succurly{175}^{{\rm WB}}(P) (x; i)$.

Notwithstanding its axiomatic virtues, the Pooling Rule might be criticized for producing ‘too much’ incompleteness in ≽^WB. Note, specifically, that two histories will be non-comparable if some spectator ranks them as non-comparable, or if two spectators have conflicting strict extended preferences over the histories.Footnote ⁵⁰

The force of this criticism is open to dispute. It might be argued that if facts about well-being just are facts about the convergent idealized extended preferences of a community of spectators, divergence in their views about two histories should yield well-being non-comparability between the two.Footnote ⁵¹ However, it is certainly important to investigate plausible rules for constructing ≽^WB other than the Pooling Rule. This is a topic for future research.

6. WELL-BEING DIFFERENCES

Extended preferences (as described thus far) provide ordinal information about spectators’ views regarding the histories in H. Whether (x; i) ≽^k (y; j) depends upon k's ordering of (x; i) and (y; j), more specifically whether he judges (x; i) to be at least as good as (y; j) if the two subjects are distinct, and whether he ranks x at least as good as y under a condition of sympathy with the subject if i = j. Similarly, $\succurly{175}^{{\rm WB}}$ is ordinal: $(x; i) \succurly{175}^{{\rm WB}} (y; j)$ means that the well-being level of i in x is at least as large as the well-being level of j in y.

The difference quasiordering ≽^DIFF is a natural way to represent cardinal well-being facts regarding the histories in H. ≽^DIFF is a quasiordering on ${\bf H} \times {\bf H}: ((x; i), (y; j)) \succurly{175}^{{\rm DIFF}} ((z; l), (w; m))$ is to be interpreted as: the difference in well-being between (x; i) and (y; j) is at least as large as the difference in well-being between (z; l) and (w; m). Moreover, so as to reflect truisms about well-being differences, and their connection to well-being levels, ≽^DIFF must satisfy certain additional ‘Difference Constraints’.Footnote ⁵²

If ≽^WB is a complete quasiordering on H, and ≽^DIFF a complete quasiordering on H x H that satisfies the Difference Constraints, then (depending on whether further technical conditions are satisfied) they may be jointly representable by a single utility function v(.), such that: $(x; i) \succurly{175}^{{\rm WB}} (y; j)$ iff $v(x; i) \succurly{145} v(y; j)$ and $((x; i), (y; j)) \succurly{175}^{{\rm DIFF}} ((z; l), (w; m))$ iff $v(x; i) - v(y; j) \succurly{145} v(z; l) - v(w; m)$. By extension, if ≽^WB is a possibly incomplete quasiordering on H, and ≽^DIFF a possibly incomplete quasiordering on H x H that satisfies the Difference Constraints, they may be jointly representable by a set of utility functions V such that: $(x; i) \succurly{175}^{{\rm WB}} (y; j)$ iff $v(x; i) \succurly{145} v(y; j)$ for all v(.) in V; and $((x; i), (y; j)) \succurly{175}^{{\rm DIFF}} ((z; l), (w; m))$ iff $v(x; i) - v(y; j) \succurly{145} v(z; l) - v(w; m)$ for all v(.) in V.Footnote ⁵³

How are we to construct ≽^DIFF, as a function of spectators’ preferences and well-being judgements? This is a large and complicated topic, which I lack space to discuss in detail here. However, two basic possibilities should be noted. The first strategy is to invite spectators to make their own judgements about well-being differences.Footnote ⁵⁴ For any given four-tuple of histories, spectator k asks herself: do I believe the well-being difference between the first two to be at least as large as the well-being difference between the second two? If spectator k's difference judgements are sufficiently coherent, they will take the form of a personal difference quasiordering ≽^{k -Diff}: a quasiordering on H × H which satisfies the Difference Constraints vis-à-vis ≽^k. We can then formulate some rule (e.g. a Pooling Rule) for mapping a profile of such judgements $(\succurly{175}^{1{\hbox{-}}{\rm Diff}}, \succurly{175}^{2{\hbox{-}}{\rm Diff}}, \ldots, \succurly{175}^{N{\hbox{-}}{\rm Diff}})$ onto $\succurly{175}^{{\rm DIFF}}$.

Note that this strategy does not create a direct connection between spectator k's personal difference quasiordering, ≽^{k -Diff}, and her preferences when sympathetic to various subjects. Still, there remains a substantial, indirect connection – since ≽^{k -Diff} must satisfy Difference Constraints relative to ≽^k, and ≽^k does have a direct nexus to the spectator's sympathetic preferences. Recall that (x; i) ≽^k (y; i) iff spectator k weakly prefers x to y when unreservedly sympathetic to subject i. These sympathetic preferences will, in turn, help structure ≽^{k -Diff}. For example, if k under a condition of unreserved sympathy with i weakly prefers outcome x to y to z, i.e. $(x; i) \succurly{175}^{k} (y; i) \succurly{175}^{k} (z; i)$, then $\succurly{175}^{k{\hbox{-}}{\rm Diff}}$ must assign a difference between (x; i) and (z; i) at least as large as the difference between (y; i) and (z; i).

A second strategy (call it the Bernoulli strategy) is to involve sympathy more directly in $\succurly{175}^{k{\hbox{-}}{\rm Diff}}$.Footnote ⁵⁵ Let u^k_i(.) be a vNM utility function that expectationally represents spectator k's preferences over outcome lotteries under a condition of unreserved sympathy with subject i. Then the Bernoulli strategy uses u^k_i(.) to define $\succurly{175}^{k{\hbox{-}}{\rm Diff}}$ in the intrapersonal case, with respect to subject i's histories: $((x; i), (y; i)) \succurly{175}^{k{\hbox{-}}{\rm Diff}} ((z; i), (w; i))$ iff $u^{k}_{i}(x) - u^{k}_{i}(y) \succurly{145} u^{k}_{i}(z) - u^{k}_{i}(w)$.Footnote ⁵⁶ As with the first strategy, ≽^DIFF is constructed from ($\succurly{175}^{1{\hbox{-}}{\rm Diff}}, \succurly{175}^{2{\hbox{-}}{\rm Diff}}, \ldots, \succurly{175}^{N{\hbox{-}}{\rm Diff}}$) using a Pooling Rule or some other rule.

While Harsanyi's derivation of difference comparisons from vNM utility functions was empathy-based, the Bernoulli strategy here is sympathy-based. Harsanyi asked spectators to rank lotteries over histories, with the spectator meant to think about the probability assigned by a lottery to a history (x; i) as the probability that the spectator will ‘stand in the shoes’ (acquire the attributes) of subject i in outcome x. By contrast, the Bernoulli strategy under discussion now asks each spectator to rank, in turn, N subsets of lotteries over histories – the subset consisting of lotteries over histories that have individual 1 as subject and thus take the form (x; 1), the subset consisting of lotteries over histories of the form (x; 2), . . ., the subset consisting of lotteries over histories of the form (x; N) – with each such ranking understood as the spectator's ranking of lotteries over outcomes when unreservedly sympathetic with the given subject. The spectator is never asked to imagine acquiring someone else's identity.

Still, the Bernoulli strategy will be controversial.Footnote ⁵⁷ It is a truism about sympathy that my judgements about your welfare levels in various outcomes are mirrored by my preferences over those outcomes when unreservedly sympathetic to you. It may be true, but is hardly a truism, that my judgements about differences between your welfare levels in various outcomes are mirrored by differences in the values of a vNM utility function expectationally representing my preferences for lotteries over those outcomes when unreservedly sympathetic to you. Imagine that k, under a condition of sympathy with subject i, strictly prefers outcome z to y to x; and k makes the judgement that the difference between i's well-being in z and y is exactly equal to the difference in his well-being between y and x. Why are we justified in insisting (as the Bernoulli strategy does) that spectator k, under a condition of sympathy with i, must now be indifferent between y and a lottery giving a 50% probability of z and a 50% probability of x? Why would it be incoherent or problematic for k to rank y above or below this lottery?

A full analysis of the pros and cons of the two strategies for defining ≽^{k -Diff} must be left to another day.Footnote ⁵⁸ What should be emphasized, here, is that each represents an intellectual extension of the sympathy-based conception of extended preferences. The first strategy asks spectator k to make well-being judgements, regarding differences, just as the construction of ≽^k, in the interpersonal case, asks spectator k to make judgements of well-being levels. The second, Bernoulli, strategy brings sympathy into the picture more directly. Moreover, as already noted, the first strategy is indirectly constrained by the use of sympathy in defining ≽^k. Neither strategy employs the problematic device of empathetic projection.