Abstract
I discuss the trilemma that consists of the following three principles being inconsistent:
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1.
The Common Principle: if one distribution, A, necessarily brings a higher total sum of personal value that is distributed in a more egalitarian way than another distribution, B, A is more valuable than B.
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2.
(Weak) ex-ante Pareto: if one uncertain distribution, A, is more valuable than another uncertain distribution, B, for each patient, A is more valuable than B.
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3.
Pluralism about attitudes to risk (Pluralism): the personal value of a prospect is a weighted sum of the values of the prospect’s outcomes, but the weight each outcome gets might be different from the probability the prospect assigns to the outcome.
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Notes
I deliberately present the trilemma in an informal way. It can be formalized in different ways, but each unavoidably involves relying on some structural technical assumptions that would have to be explained at some length. As the aim of this paper is to explore the trilemma from a normative point of view, I chose not to commit myself to any specific formalization. The price of this choice is that, in some parts, the discussion involves some imprecise claims and uses some vague concepts. I explicitly highlight those places in the discussion where I believe this potentially hides some substantive normative issue. Throughout this paper I assume that personal value is cardinally measurable and interpersonally comparable and that the uncertainty (or risk) involved can be represented using a single probability distribution over a set of states. Although I did not find any formulation of this specific trilemma in the literature, the interested reader can find related results in the economic literature. For an excellent review see Mongin and Pivato (2015).
The term “in a more egalitarian way” is ambiguous, of course, but (as demonstrated below) the trilemma arises even in cases in which it is completely uncontroversial that the distribution with the higher total sum of personal value is also the more egalitarian one. Thus, there is no need for me to commit myself to a specific measure of inequality.
Rejecting the restrictive part of the principle (the one that demands that the value of a prospect is a weighted sum of the values of the prospects’ outcomes) cannot help one avoid the trilemma (and, in any case, as an anonymous referee commented, rejecting this part amounts to rejecting state-wise dominance on the level of personal value, which is highly unintuitive). The problematic part is the permissive one (the one that allows the weights to be different from the probabilities). Below I show that it is possible to build a distribution problem in which the conflict between The Common Principle and ex-ante Pareto arises in every case in which one of the patients is risk neutral (i.e. the value of a prospects for her is the prospect’s expected value) and the other patient assign to the states weights which are different from their probabilities, for any weights different from the probabilities. I do not even have to assume that the weights sum to 1. The trilemma, however, does not arise when the weight each outcome gets is equal to its probability. This will be discussed in Sect. 2. The probability in question can be understood either as objective probability or as the subjective probability of the distributer or in any other way according to which the same probability function is used when assessing the prospects from the point of view of all patients. I discuss this below.
I thank an anonymous referee for pointing this out to me.
At least some of the credit for this should be given to Parfit’s (1997) famous “levelling down objection”.
In fact, in Buchak’s theory (and in rank-dependent decision theories in general) the weights are not attached to the states, but rather to value levels, i.e. to the different possible levels of value the patient might gain from the prospect. However, when it comes to the simple example below and to its generalization (that follows) Pluralism captures these theories too. Since any plausible decision theory should cover such simple examples, the differences between rank-dependent decision theories and the theories characterized by Pluralism should not concern us.
And notice that had patient 2 used r(p) = p (i.e. had he been neutral to risk), Buchak’s theory would reduce to orthodox decision theory as 1.9 + 0.5(3) = 1.9 + 0.5(4.9 − 1.9) = 0.5 * 1.9 + 0.5 * 4.9. This is generally true in Buchak’s theory.
Alternatively, one can demand that, while the personal values of all patients might have a non-expectational form, this form must be identical for all patients. Interestingly, while taking this route cannot help one avoid the trilemma in cases in which all patients are risk-seeking, it might work when all patients are risk-avoidant (but this seems to depends on the formal framework used). Here is an example for the risk-seeking case.
0.5.
0.5.
K.
5, 14.
14,5.
L.
10,10.
10,10.
In case both patients obey Buchak’s theory and use r(p) = p0.5 the personal value from K is higher than that of L for both of them, but L brings a higher total sum of personal value, which is distributed in a more egalitarian way in both states.
The risk-avoidance case is non-trivial. When it comes to Buchak’s framework, when all patients use the same r(.) function and this function is convex, the trilemma does not arise. This is proven and some of its normative implications are discussed in a companion paper (under review). In any case, when it comes to the trilemma discussed here, demanding both that the functional forms of the personal values of all patients will be identical and that it will be a risk-avoidant one seems hard to justify as it seems at least permissible for a patient to be risk-neutral (i.e. to obey orthodox decision-theory).
This is a consequence of Harsanyi’s (1955) Utilitarian Theorem.
More generally, to accept Pluralism (which is a normative principle) is to accept that even when there is no disagreement on the epistemic level between different patients, they can assign different weights to the states. This blocks (for those who accept Pluralism) the general line of objection to ex-ante Pareto (suggested by Mongin 2016) according to which the principle is invalid when the agreement on the epistemic level is a result of deeper disagreements that cancel each other out. I thank an anonymous referee for pushing me on this point.
Which probability distribution is the correct one might, of course, be a matter of dispute in many cases, but the argument only needs the assumption that such a single normatively correct probability distribution exists in the relevant class of cases. Thus, philosophers who rejects “the uniqueness thesis” (see Kopec and Titelbaum 2016 for a review of the literature) might hesitate to follow this line of reasoning.
Indeed, this seems like a very natural assumption to make, at least in light of Buchak’s decision theory. However, see the discussion in the next section.
An anonymous referee pointed out that the same type of maneuver just discussed with respect to The Common Principle can be made also with respect to ex-ante Pareto, i.e. to accept The Common Principle without reservations, but ex-ante Pareto only when understood as referring to representations of value-rankings in terms of value functions that have an expectational form. I agree with the referee that this possibility naturally suggests itself, but I do not find it very promising. This is so because, unlike The Common Principle, ex-ante Pareto aggregates values across patients; not across states and so its validity seems to be independent of the representation used. The reason the maneuver seems promising in the case of The Common Principle is that while under the expected value representation, the strength of all reasons, including risk-related reasons, is represented by the values attached to the different outcomes, under non-expected value representations this is not the case. Under such representations some risk-related reasons are represented by the way the weights are assigned to the different states. Thus, when aggregating across states the type of representation used matters. However, no analogous move is available when it comes to ex-ante Pareto. The value of a prospect for a patient is independent of the way an outside observer represents it. I thank the referee for pressing me on this point.
References
Allais, M. (1953). Le comportement de l’homme rationnel devant le risque: Critique des postulats et axiomesde l’ecole Americaine. Econometrica, 21, 503–546.
Alon, S., & Gayer, G. (2016). Utilitarian preferences with multiple priors. Econometrica, 84(3), 1181–1201.
Blackorby, C., Donaldson, D., Mongin, P. (2004). Social aggregation without the expected utility hypothesis. Cahier no. 2004–020, Laboratoire d'économétrie, Ecole Polytechnique.
Bradley, R. (2005). Bayesian utilitarianism and probability homogeneity. Social Choice and Welfare, 24, 221–251.
Bradley, R. (2017). Decisions theory with a human face. Cambridge: Cambridge University Press.
Briggs, R. (2015). Cost of abandoning the sure-thing principle. Canadian Journal of Philosophy, 45(5–6), 827–840.
Broome, J. (1987). Utilitarianism and expected utility. The Journal of Philosophy, 84(8), 405–422.
Broome, J. (1990). Bolker-jeffrey expected utility theory and axiomatic utilitarianism. Review of Economic Studies, 57, 477–503.
Broome, J. (1991). Weighing goods. Cambridge, Mass.: Basil Blackwell.
Buchak, L. (2013). Risk and rationality. England: Oxford University Press.
Buchak, L. (2015). Revisiting risk and rationality: A reply to pettigrew and briggs. Canadian Journal of Philosophy, 45(5–6), 841–862.
Buchak, L. (2017). Taking risks behind the veil of ignorance. Ethics, 127(3), 610–644.
Ellsberg, D. (1961). Risk, ambiguity, and the savage axioms. Quarterly Journal of Economics, 75, 643–669.
Fleurbaey, M. (2010). Assessing risky social situations. Journal of Political Economics, 118(4), 649–678.
Fleurbaey, M., & Mongin, P. (2016). The utilitarian relevance of the aggregation theorem. American Economic Journal: Microeconomics, 8(3), 289–306.
Fleurbaey, M., & Voorhoeve, A. (2013). Decide as you would with full information! an argument against ex ante pareto. In N. Eyal, S. Hurst, O. Norheim, & D. Wikler (Eds.), Inequalities in health: Concepts, measures and ethics (pp. 113–128). England: Oxford University Press.
Frick, J. (2015). Contractualism and social risk. Philosophy and Public Affairs, 43(3), 175–223.
Gilboa, I., Schmeidler, D., & Samet, D. (2004). Utilitarian aggregation of beliefs and tastes. Journal of Political Economy, 112, 932–938.
Goldschmidt, Z., & Nissan-Rozen, I. (2020). The intrinsic value of risky prospects. Synthese. https://doi.org/10.1007/s11229-020-02532-3
Harsanyi, J. C. (1955). Cardinal welfare, individualistic ethics and interpersonal comparisons of utility. Journal of Political Economy, 63, 309–321.
Jeffrey, R. (1965). The logic of decision. US: University of Chicago Press.
Kopec, M., & Titelbaum, G. M. (2016). The uniqueness thesis. Philosophy Compass, 11(4), 189–200.
Machina, M. J. (2008). Non-expected utility theory. In P. Macmillan (Ed.), The new palgrave dictionary of economics. London: Palgrave Macmillan.
Mongin, P. (1995). Consistent bayesian aggregation. Journal of Economic Theory, 66, 313–351.
Mongin, P. (2016). Spurious unanimity and the pareto principle. Economics and Philosophy, 32(3), 511–532.
Mongin, P., & Pivato, M. (2015). Social preferences and social welfare under risk and uncertainty. In M. Adler & M. Fleurbaey (Eds.), Oxford handbook of well-being and public policy. England: Oxford University Press.
Murray, D., & Buchak, L. (2019). Risk and motivation: When the will is required to determined what to do. Philosophers’ Imprint, 19(16), 1–16.
Nissan-Rozen, I. (2017). How to be an ex-post egalitarian and an ex-ante paretian. Analysis, 77(3), 550–558.
Parfit, D. (1984). Reasons and persons. England: Oxford University Press.
Parfit, D. (1997). Equality and priority. Ratio, 10(3), 202–221.
Pettigrew, R. (2015). Risk, rationality and expected utility theory. Canadian Journal of Philosophy, 45(5–6), 798–826.
Rabin, M. (2000). Risk aversion and expected utility theory: A calibration theorem. Econometrica, 68, 1281–1292.
Rowe, T. (2019). Risk and the unfairness of some being better off at the expense of others. Journal of Ethics and Social Philosophy, 16(1), 44–66.
Savage, L. (1954). The foundations of statistics. Wiley.
Stefánsson, H. O., & Bradley, R. (2015). How valuable are chances? Philosophy of Science, 82(4), 602–625.
Stefánsson, H. O., & Bradley, R. (2017). What is risk aversion? The British Journal for the Philosophy of Science, 70(1), 77–102.
Stefánsson, H. O. (2018). Is risk aversion irrational? examining the “fallacy” of large numbers. Synthese. https://doi.org/10.1007/s11229-018-01929-5
Sugden, R. (2004). The opportunity criterion: Consumer sovereignty without the assumption of coherent preferences. American Economics Review, 94(4), 1014–1033.
Thoma, J. (2019). Risk aversion and the long run. Ethics, 129(2), 230–253.
Thoma, J., & Weisberg, J. (2019). No escape from allias: Reply to buchak. Philosophical Studies. https://doi.org/10.1007/s11098-019-01322-z
von Neumann, J., & Morgenstern, O. (1947). Theory of games and economic behavior (2nd ed.). New Jersey: Princeton University Press.
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Nissan-Rozen, I. Weighing and aggregating reasons under uncertainty: a trilemma. Philos Stud 178, 2853–2871 (2021). https://doi.org/10.1007/s11098-020-01587-9
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DOI: https://doi.org/10.1007/s11098-020-01587-9