Published online by Cambridge University Press: 26 January 2009
The Interpersonal Addition Theorem, due to John Broome, states that, given certain seemingly innocuous assumptions, the overall utility of an uncertain prospect can be represented as the sum of its individual (expected) utilities. Given ‘Bernoulli's hypothesis’ according to which individual utility coincides with individual welfare, this result appears to be incompatible with the Priority View. On that view, due to Derek Parfit, the benefits to the worse off should count for more, in the overall evaluation, than the comparable benefits to the better off. Pace Broome, the paper argues that prioritarians should meet this challenge not by denying Bernoulli's hypothesis, but by rejecting one of the basic assumptions behind the addition theorem: that a prospect is better overall if it is better for everyone. This conclusion follows if one interprets the priority weights that are imposed by prioritarians as relevant only to moral, but not to prudential, evaluations of prospects.
1 This theorem is formally similar to the well-known aggregation theorem of Harsanyi, John (see his ‘Cardinal Welfare, Individualistic Ethics, and Interpersonal Compari-sons of Utility’, Journal of Political Economy, lxiii (1955)Google Scholar; repr., Essays on Ethics, Social Behavior and Scientific Explanation, Dordrecht, 1976)Google Scholar. However, while Harsanyi was concerned with aggregation of individual preferences, Broome aggregates individual betterness orderings. In addition to this philosophical difference, there is a technical difference as well: While Broome's betterness orderings range over uncertain prospects (= assignments of outcomes to states of nature), Harsanyi's preference orderings range over von Neumann-Morgenstern lotteries, i.e., over probability distributions on out-comes.
2 At least for those cases in which the set of individuals can be assumed to be fixed as one moves from one prospect or outcome to another. If different individuals are allowed to exist in different outcomes that are being compared with each other, the situation becomes much more complicated. Here we ignore this complication.
3 In fact, the three assumptions of the Interpersonal Addition Theorem imply that the probability functions pi that underlie the expectational utility measures Ui must coincide with the probability function p on which the overall measure u is based (cf. Broome, sect. 7. 1). But we cannot rely on this probability agreement in what follows if we want to question, as I am going to, one of the relevant assumptions of the theorem, The Principle of Personal Good. Instead, we must fix the probability assignment to states independently, in one way or another.
4 Jensen, K. K., ‘Measuring the Size of the Benefit and its Moral Weight’ Preference and Value: Preferentialism in Ethics, ed. Rabinowicz, W., Lund, 1996Google Scholar.
5 Note, however, that Broome has recently become much more sympathetic to the possibility of drawing such fine distinctions. He now admits that it makes good sense, conceptually, to distinguish between a person's good and its contribution to the overall evaluation (cf. , Broome, Valuing Lives, ts., 1999)Google Scholar. But he still thinks that this distinction, while conceptually motivated, does not make any real difference: Pace prioritarianism, a larger increment in individual welfare always makes a larger contribution to the overall goodness of a prospect.
6 This describes what Parfit, ‘Equality or Priority?’ calls the ‘moderate’ (teleological) version of the Priority View. He suggests that the extreme form of prioritarianism is Rawls's difference principle, which gives lexical priority to the improvements for the worse off, and not just a greater weight. For reasons to be explained below, in n. 15, 1 don't think this is quite right, but, in what follows, I shall concentrate on the moderate version. Also, I shall not consider deontological versions of the Priority View, according to which giving priority to the worse off is a normative requirement on action, which need not have any direct connection with the overall value of the resulting outcome.
7 Due to the non-linearity of the weight function w, the Priority View for Outcomes might appear to presuppose that the individual goodness of an outcome is measured on a common ratio scale, rather than just on a mere interval scale. For it is easy to see that, if w is non-linear, the comparisons between the sums of w-weighted individual goodness values are not invariant under transformations of the zero point of the scale. However, this argument is based on the questionable assumption that the shape of the weight function is fixed independently of our choice of the numerical representation for individual goodness. If the weight function instead is allowed to undergo appropriate compensatory adjustments as we move from one such representation to another, the need for an absolute zero for individual goodness is obviated. These adjustments in the shape of weight function will cancel out the effect of the scale transformation. (I owe this observation to Magnus Jiborn.)
8 In a more general formulation of prioritarianism, the overall goodness of an outcome is only required to be some increasing transform of the sum of its morally weighted individual goodness values. The transform in question must be linear, if prioritarianism is to avoid implausible implications. But if the transform is linear, then it may safely be ignored.
9 If the weighting function w is such that that the contribution of each person's welfare to the overall value of an outcome asymptotically approaches a fixed limit (the same for each individual) as that welfare increases, then the imposition of weights not only hampers impersonal compensations – it sometimes makes them impossible. For a fixed number of individuals, we will sometimes be unable to compensate a considerable loss to a person by any gains to others, however large. Example: Suppose the limit for individual contributions is, say, 100. That is, w(k) asymptotically approaches 100 as k increases to infinity. Suppose there are five individuals, 1,…, 5, such w(gi) = 90 for each i ≄ 1. Then the decrease in the welfare for individual 1 that diminishes his contribution by 50 units (i.e., diminishes w(g)1 by 50), cannot be compensated by any welfare gains, however large, for the remaining four individuals. For the total increase in the contributions made by these extra gains cannot ever be larger than 40 units.
11 In personal communication, Derek Parfit has confirmed that this is how he himself would interpret the Priority View. The prioritarian weights are moral, not prudential. They have according to Parfit no role to play in the determination of the individual goodness of prospects.
12 Roger Crisp has suggested an alternative treatment of this example (in private communication). Like me, and unlike Broome, Crisp takes prospect y to be better for each individual than prospect x, from the prioritarian point of view, but he suggests that prioritarians should keep the Principle of Personal Good for prospects intact. Therefore, they must conclude that y is overall better than x, even though y yields a worse outcome than x, overall, under each state of nature. This means that the prioritarians must reject dominance: A prospect is better even though its outcome is worse under each state. Needless to say, this is a very radical suggestion – too radical in my view.
13 See, however, the end of sect. I above for some qualifications having to do with the requirement of interpersonal comparability of the individual goodness values.
14 As we have assumed, w(10) – w(5) > w(16) – w(10). This implies that contribution Ci(y), which equals ½w(16) + ½w(5), must be smaller than ci(x), which equals ½w(10)
15 This shows, by the way, that it is incorrect to interpret Rawls's difference principle as the extreme, lexical form of the Priority View. Rawls's principle gives absolute priority to those people who are worse off than all others. The priority given to the worse off is not, on that principle, due to the fact that these people are ‘worse off than they might have been’. If their welfare level were arbitrarily increased but they would still be worse off than others, improving their lot would still have the same (absolute) priority.
16 Parfit, p. 23.
17 Qualification: One might think of a version of egalitarianism that would disregard the inequality in outcomes in its evaluation of prospects. As long as two prospects, as in our example, give the same expected welfare to each individual, they would be considered on this version of egalitarianism to be equally good overall, despite the fact that one of them is bound to result in an equal outcome while the other guarantees that the outcome will be unequal. The problem with such egalitarianism, however, is that it would violate the principle of dominance for overall betterness. To see that, recall the example from sect. Ill, with two equiprobable states, two individuals, i and j, and two prospects, x and y. Prospect x gives the same welfare benefits, equal to 10, to each individual under each state, while y gives 16 to i and 5 to j under one state, and under the other state it gives 5 to i and 16 to j. As we may assume, egalitarians will consider each outcome of y to be overall worse than the corresponding outcome of x, since the aggregated welfare is only slightly larger in the outcome of y but it is very unevenly distributed. At the same time, however, the presently considered version of egalitarianism for prospects would imply that prospect y is overall better than prospect x, since it promises the same expected welfare to both individuals and that welfare is somewhat higher than their expected welfare in x. Consequently, dominance is violated. Note that this is essentially the same problem as the one mentioned above, in n. 12, in connection with the Priority View.
18 Cf. Broome, ‘Equality versus Priority: A Useful Distinction’ http://aran. univ-pau. fr/ee/page3.html, 2001. See, however, at the same website, Marc Fleurbaey, ‘Equality versus Priority: How Relevant is this Distinction?'; 2001, for an interesting criticism of the idea that the Priority View is fundamentally opposed to egalitarianism. Note also that one might defend a weak version of egalitarianism that respects the separability requirement: If the aggregated expected welfare of a prospect is taken to be lexically prior to the equality considerations, then increasing the expected welfare of an individual, while keeping everything else constant, will always make the prospect overall better (even though the inequality of a prospect might thereby increase). Thus, in this sense, the individual contributions to the overall value of a prospect would be separable on this egalitarian view.
19 In personal communication, Derek Parfit has confirmed that this is how he himself would interpret the Priority View.
23 Cf. Schick, F., ‘Under Which Descriptions?’ Utilitarianism and Beyond, ed. Sen, A. and Williams, B., Cambridge, 1982Google Scholar.
24 Another worry in connection with ‘grand’ outcomes is whether the idea of such outcomes is conceptually coherent, to begin with. Here, I assume that the answer is, Tfes’ i.e., that it is meaningful to postulate such comprehensive possible ways for the world to be. But I am fully aware that this assumption itself is controversial.
25 Cf. Schick.
26 This essay is a short and relatively informal version of my ‘Prioritarianism and Uncertainty: On the Interpersonal Addition Theorem and the Priority View’ Exploring Practical Philosophy: From Action to Values, ed. Egonsson, D., Josefsson, J., Petersson, B. and Rønnow-Rasmussen, T., 2001Google Scholar. For comments and discussion I am indebted to Gustaf Arrhenius, John Broome, Johan Brannmark, Krister Bykvist, Erik Carlson, Roger Crisp, Sven Danielsson, Dan Egonsson, Marc Fleurbaey, Magnus Jiborn, Mats Johansson, Karsten Klint Jensen, Philippe Mongin, Derek Parfit, Erik Persson, Ingmar Persson, Toni Rønnow-Rasmussen, and Paul Weirich. Also, I wish to thank Peter Vallentyne for his very helpful and encouraging referee's report.