¹ See for example Mongin, Philippe, ‘Does Optimization Imply Rationality?’, Synthese 124 (2000), pp. 73–111CrossRefGoogle Scholar; Schick, Frederic, ‘Dutch Bookies and Money Pumps’, The Journal of Philosophy 83 (1986), pp. 112–19CrossRefGoogle Scholar. Two exceptions to the practical approach are John Broome and Wlodek Rabinowicz. Broome rejects a practical approach in general to questions of rationality, but he does not defend his position in specific relation to the VP. Rabinowicz argues that we should assess the VP by using both practical and non-practical arguments. Broome, John, ‘Are Intentions Reasons? And How Should We Cope with Incommensurable Values’, Practical Rationality and Preference: Essays for David Gauthier, ed. Morris, Christopher W. and Ripstein, Arthur (Cambridge, 2001), pp. 98–120CrossRefGoogle Scholar; Rabinowicz, Wlodek, ‘Money Pump with Foresight’, Imperceptible Harms and Benefits, ed. Almeida, Michael J. (Dordrecht, 2000), pp. 123–54CrossRefGoogle Scholar.

² Though I will not argue it here, I believe that much the same point applies for other practical arguments for requirements of rationality, such as the Dutch Book. For a discussion of the Dutch Book see Broome, John, Ethics out of Economics (Cambridge, 1999), pp. 157–61CrossRefGoogle Scholar. Velleman mounts a general critique of practical arguments in Velleman, J. David, ‘Deciding How to Decide’, Ethics and Practical Reason, ed. Gaut, Berys and Cullity, Garrett (Oxford, 1997), pp. 29–52Google Scholar.

³ I use the term ‘subjective value’ – as opposed to ‘expected value’ – to refer to value relative to the agent's epistemic position. As I explain below, I do this because I reject some of the axioms of expected value theory.

⁴ See Hare, Caspar, ‘Take the Sugar’, Analysis 70 (2010), pp. 237–47CrossRefGoogle Scholar. For a general account of the requirements of expected value theory, including negative transitivity see Peterson, Martin, An Introduction to Decision Theory (Cambridge, 2009)CrossRefGoogle Scholar, ch. 5.

⁵ See Peterson, An Introduction to Decision Theory, ch. 8.

⁶ Oddie and Menzies refer to ‘subjective value’ in Oddie, Graham and Menzies, Peter, ‘An Objectivist's Guide to Subjective Value’, Ethics 102 (1992), pp. 512–33CrossRefGoogle Scholar.

⁷ There is a difference between the evidence someone avails herself of and the evidence available to the person at the time. I find the latter more plausible as an account of what determines subjective value, though I will not argue for that here.

⁸ I believe things are perhaps more complicated than this because subjective betterness is a function of epistemic preferability and it has been plausibly argued that epistemic preferability is intransitive. See Sorensen, Roy, ‘Is Epistemic Preferability Transitive?’, Analysis 41 (1981), pp. 122–3CrossRefGoogle Scholar. This is not relevant to the fundamental argument of this article.

⁹ Non-standard betterness orderings are associated with non-standard or ‘deviant’ logics. Joseph Raz and John Broome have defended non-standard betterness orderings on the basis of the deviant logic of supervaluationism. Raz, Joseph, The Morality of Freedom (Oxford, 1986), p. 327Google Scholar; Broome, John, ‘Is Incommensurability Vagueness?’, Incommensurability, Incomparability, and Practical Reasoning, ed. Chang, Ruth (Cambridge, Mass. and London, 1997), pp. 67–89Google Scholar.

¹⁰ Davidson et al. initially formulated the VP as an argument about preferences. See Davidson, Donald, McKinsey, J. C. C., and Suppes, Patrick, ‘Outlines of a Formal Theory of Value, I’, Philosophy of Science 22 (1955), pp. 140–60CrossRefGoogle Scholar. I discuss the preference versions in more depth in section IV.

¹¹ Schick, ‘Dutch Bookies and Money Pumps’, pp. 117–18. We should be careful to distinguish this from Schick's claim that past choices should determine the value of the options at subsequent choice nodes. At p. 117 he denies the assumption that ‘if the agent knew of the arrangements he had already accepted, this would not affect the value he set on the arrangement just offered him’. See also Mongin, ‘Does Optimization Imply Rationality?’.

¹² McClennen, Edward F., Rationality and Dynamic Choice: Foundational Explorations (Cambridge, 1990)CrossRefGoogle Scholar.

¹³ Rabinowicz, ‘Money Pump with Foresight’.

¹⁴ Mongin grants that we can lose out in a value pump with only a few choice nodes. However, he says that foresight can allow us to avoid severe practical loss, like bankruptcy, which he equates with complete irrationality. Mongin, ‘Does Optimization Imply Rationality?’, pp. 84–6.

¹⁵ My formulation owes much to Peterson, Martin, ‘Parity, Clumpiness and Rational Choice’, Utilitas 19 (2007), pp. 505–13CrossRefGoogle Scholar.

¹⁶ For the alternatives see Rabinowicz, ‘Money Pump with Foresight’; and McClennen, Rationality and Dynamic Choice.

¹⁷ Duncan Macintosh defends the rationality of losing out in a value pump, but appeals solely to Jim's preferences, rather than to the value of the options his preferences are directed towards. Consequently, his argument is unpersuasive. MacIntosh, Duncan, ‘Intransitive Preferences, Vagueness, and the Structure of Procrastination’, The Thief of Time, ed. Andreou, Chrisoula and White, Mark D. (New York and Oxford, 2010), pp. 68–86CrossRefGoogle Scholar.

¹⁸ Philosophers disagree about the preference-attitudes we ought to have between incomparable options. I believe, along with John Broome, that we ought to be indifferent because neither option is better and our attitudes ought to respond to value. Most philosophers who have stated an opinion disagree. They believe that we ought to prefer neither of two incomparable options to the other, but that we also ought not to be indifferent between them. Nonetheless, they agree that it can be rational to have negatively intransitive preferences as a result of incomparability. See Broome, Ethics out of Economics, p. 155; Rabinowicz, Wlodek, ‘Value Relations’, Theoria 74 (2008), pp. 18–49CrossRefGoogle Scholar, at 25–30; Chang, Ruth, ‘The Possibility of Parity’, Ethics 112 (2002), pp. 659–88CrossRefGoogle Scholar, at 666; Gustafsson, Johan E. and Espinoza, Nicolas, ‘Conflicting Reasons in the Small-Improvement Argument’, The Philosophical Quarterly 60 (2010), pp. 754–63CrossRefGoogle Scholar; Raz, Joseph, The Morality of Freedom (Oxford, 1986), pp. 333–5Google Scholar; Boot, Martijn, ‘Parity, Incomparability and Rationally Justified Choice’, Philosophical Studies 146 (2009), pp. 75–92CrossRefGoogle Scholar.

¹⁹ When I refer to the VP simpliciter I am referring to both versions of that argument for the sake of simplicity.

²⁰ See for example Davidson et al., ‘Outlines of a Formal Theory of Value, I’; Peterson, ‘Parity, Clumpiness and Rational Choice’; Chang, Ruth, ‘Parity, Interval Value, and Choice’, Ethics 115 (2005), pp. 331–50CrossRefGoogle Scholar; Raiffa, Howard, Decision Analysis: Introductory Lectures on Choices under Uncertainty (Boston, Mass., 1970), pp. 75–80Google Scholar.

²¹ The same applies to the VP^u.

²² See Schoenfield, Miriam, ‘Chilling out on Epistemic Rationality’, Philosophical Studies 158 (2012), pp. 197–219CrossRefGoogle Scholar. Roy Sorensen presents another argument for intransitive correlative subjective value, which seems irrefutable. See Sorensen, Roy, ‘Subjective Probability and Indifference’, Analysis 43 (1983), p. 15CrossRefGoogle Scholar.

²³ This example is owed to Caspar Hare. See Hare, ‘Take the Sugar’, p. 238.

²⁴ Ruth Chang has argued that SIAs do not exploit borderline cases of vague predicates. I do not find her argument persuasive, but even if it is, Chang's argument still implies that options can be ordered negatively intransitively in terms of their subjective value. See Chang, ‘The Possibility of Parity’.

²⁵ A complete account of epistemicist comparabilism has yet to be provided in the literature. Broome discusses epistemicist accounts at Broome, Ethics out of Economics, p. 152. For defences of epistemicism see Williamson, Timothy, Vagueness (London, 1994)Google Scholar; Sorensen, Roy A., ‘Vagueness, Measurement, and Blurriness’, Synthese 75 (1988), pp. 45–82CrossRefGoogle Scholar.

²⁶ Peterson, ‘Parity, Clumpiness and Rational Choice’, pp. 511–12.

²⁷ For a discussion of an alternative see Hare, ‘Take the Sugar’.

²⁸ Compare Peterson, ‘Parity, Clumpiness and Rational Choice’.

²⁹ As I will show below, it is not clear which of these two approaches Chang favours given what she says about the VP. See Chang, ‘Parity, Interval Value, and Choice’, p. 347; Chang, Ruth, ‘Voluntarist Reasons and the Sources of Normativity’, Reasons for Action, ed. Sobel, David and Wall, Steven (Cambridge, 2009), pp. 264–5Google Scholar.

³⁰ Chang, ‘Voluntarist Reasons and the Sources of Normativity’, p. 264.

³¹ Chang, Ruth, ‘Are Hard Choices Cases of Incomparability?’, Philosophical Issues 22 (2012), pp. 106–26CrossRefGoogle Scholar, at 115.

³² In her discussions of the VP, Chang says that she believes PC solely on the basis of the VP. The reasons created in a choice series are not the ‘voluntarist reasons’ she defends in other places. See Chang, ‘Voluntarist Reasons and the Sources of Normativity’, p. 264. See also Chang, Ruth, ‘Grounding Practical Normativity: Going Hybrid’, Philosophical Studies 164 (2013), pp. 163–87CrossRefGoogle Scholar.

³³ Chang says that the VP is a problem for incomparability, but not for parity at Chang, ‘Grounding Practical Normativity’, p. 124 n. 11. But it is not clear what precludes an incomparabilist from appealing to exactly the same considerations about the normative importance of past choices as the proponent of parity.

³⁴ Bratman, Michael E., Intention, Plans, and Practical Reason (Cambridge, Mass. and London, 1999), pp. 23–7Google Scholar.

³⁵ Chang argues that her account of hybrid voluntarism, which is very similar to PC, avoids the bootstrapping problem. Chang, ‘Grounding Practical Normativity’, pp. 183–4. However, she does not consider this modified form of the bootstrapping problem. It is also not clear that Chang avoids the original bootstrapping problem either.

³⁶ The equivalent part in the VP^u is 1^u(b).

³⁷ Chang, ‘Parity, Interval Value, and Choice’, p. 347.

³⁸ The contrast with John Broome's defence of the normative importance of past choices is instructive. Broome defends that claim on the basis of persuasive non-practical argument and argues that a happy benefit of the claim is that it provides (admittedly extremely fragile) protection from the value pump. Broome argues that unrepudiated intentions place us under normative requirements, but not reasons, to carry out the means to the options chosen earlier in the choice series. See Broome, ‘Are Intentions Reasons?’.

³⁹ Chang, ‘Parity, Interval Value, and Choice’, p. 347.

⁴⁰ This is different from Martin Peterson's critique of the argument. Peterson contends that Chang's conclusion follows from her premises, but that the conclusion ought to be rejected. Peterson, ‘Parity, Clumpiness and Rational Choice’, p. 512. I am saying that Chang's conclusion does not follow from her premises.

⁴¹ An earlier version of this article was presented to the Oxford Philosophy Department Ockham Society in 2013. I am thankful to all three people present for their helpful feedback. I am also thankful to Roger Crisp and Brian Hedden for helpful comments and suggestions. My biggest debt of gratitude is to the anonymous reviewer for Utilitas whose penetrating criticisms have made this article significantly better.