¹ McCarthy, C., No Country for Old Men (London, 2005)Google Scholar.

² Anton's game has some superficial affinities with infinite decision problems in decision theory, such as the St Petersburg Game, and Satan's Apple. However, besides the fact that those are puzzles for rationality not morality, the St Petersburg Game involves a divergent infinite series (the gamble has infinite expected utility), whereas Anton's game involves a convergent series (one expected death rather than an infinity of them); and Satan's Apple uses infinity to reveal a tension between choice-by-choice rationality, and the rationality of those choices’ cumulative effects. See e.g. Nover, H. and Hájek, A., ‘Vexing Expectations’, Mind 113 (2004), pp. 237–49CrossRefGoogle Scholar; Arntzenius, F., Elga, A. and Hawthorne, J., ‘Bayesianism, Infinite Decisions, and Binding’, Mind 113 (2004), pp. 251–83CrossRefGoogle Scholar. For an intriguing take on ethical versions of these and other problems of infinite decision theory, see Bostrom, N., ‘Infinite Ethics’, Analysis and Metaphysics 10 (2011), pp. 9–59 Google Scholar. Notice that, since Anton's game is a supertask, we need not address the otherwise very interesting question of time discounting.

³ I mean ‘ought’ in the belief- or perhaps evidence-relative sense; not, in any case, the fact-relative one.

⁴ Anton's game has something in common with trolley problems. We might be inclined, therefore, to think that n = 5, as people typically use that ratio when working out when it is permissible to turn a trolley. I am not sure if the analogy is apt. Although Anton has said that he will kill Victim if you refuse to flip, by flipping you are not really redirecting an existing threat. Anton can change his mind; he has not yet decided whom to kill. I think your causal involvement in either death is about the same. But, as already noted, nothing depends on this view: if you think the analogy to trolley cases is right, then just assume throughout that there are five Victims.

⁵ It is important to remember here that the number of victims should correspond to whatever number you thought would warrant indifference between flipping and not flipping in the certain version of the game described above.

⁶ See Fabre's and Moellendorf's arguments against totally discounting sunk costs in the ethics of war, in Fabre, C., ‘War Exit’, Ethics 125 (2015), pp. 631–52CrossRefGoogle Scholar; Moellendorf, D., ‘Two Doctrines of Jus Ex Bello’, Ethics 125 (2015), pp. 653–73CrossRefGoogle Scholar.

⁷ For the accusation, see, for example, Fried, B. H., ‘What Does Matter? The Case for Killing the Trolley Problem (or Letting It Die)’, Philosophical Quarterly 62 (2012), pp. 505–29CrossRefGoogle Scholar.

⁸ Lara Buchak, for example, aims to show that a range of attitudes to risk can be rationally permissible, not that any of them are rationally required. See Buchak, L., Risk and Rationality (Oxford, 2013)CrossRefGoogle Scholar.

⁹ Even if one cannot simply choose to believe p, there are surely things one can do to bring it about that one believes p.

¹⁰ Altham, J. E. J., ‘Ethics of Risk’, Proceedings of the Aristotelian Society 84 (1983), pp. 15–29 CrossRefGoogle Scholar.

¹¹ Buchak, Risk and Rationality, p. 233.

¹² For a similar point, criticising ex post forms of contractualism, see Frick, J., ‘Contractualism and Social Risk’, Philosophy and Public Affairs 43 (2015), pp. 175–223 CrossRefGoogle Scholar. Also relevant are Otsuka, M. and Voorhoeve, A., ‘Why It Matters That Some Are Worse Off Than Others: An Argument against the Priority View’, Philosophy and Public Affairs 37 (2009), pp. 171–99CrossRefGoogle Scholar.

¹³ For the idea that we have independent reasons to care about equality, over and above how it contributes to the well-being of those affected, see Temkin, L. S., Inequality (Oxford, 1993)Google Scholar.

¹⁴ Chang, R., Incommensurability, Incomparability, and Practical Reason (London, 1997)Google Scholar.

¹⁵ Chang, R., ‘The Possibility of Parity’, Ethics 112 (2002), pp. 659–88CrossRefGoogle Scholar.

¹⁶ Broome, J., ‘Is Incommensurability Vagueness?’, Incommensurability, Incomparability, and Practical Reason, ed. Chang, Ruth (Oxford, 1997), pp. 67–89 Google Scholar. Chang rejects the view that what she calls parity can be understood as vagueness.

¹⁷ In support of the idea that moral vagueness is ontic vagueness, see Schoenfield, M., ‘Moral Vagueness Is Ontic Vagueness’, Ethics 126 (2015), pp. 257–82CrossRefGoogle Scholar. On vague value generally: Dougherty, T., ‘Vague Value', Philosophy and Phenomenological Research 89 (2013), pp. 352–72CrossRefGoogle Scholar.

¹⁸ For a similar point about formal semantics and vagueness in language, see Cook, R. T., ‘Vagueness and Mathematical Precision’, Mind 111 (2002), pp. 225–47CrossRefGoogle Scholar. What he says about degree-theoretic semantics and vague language is apt here: ‘Degree-theoretic semantics provides a good model of how vague language behaves. On this view, the formalism is not a description of what is really occurring but is instead a fruitful way to represent the phenomenon, that is, it is merely one tool among many that can further our understanding of the dissatisfied in question. In particular, not every aspect of the model need correspond to actual aspects of the phenomenon being modeled’ (p. 234).

¹⁹ The right response here might be to develop the ‘irrelevant utility’ idea to accommodate these kinds of cases, and argue that what matters here is not the probability, for each person, that she will suffer this harm, but the probability that someone will suffer this harm. Thanks to a referee for pointing this out. For more on this idea, see in particular Otsuka, M., ‘Risking Life and Limb: How to Discount Harms by Their Probability’, Identified Versus Statistical Lives: An Interdisciplinary Perspective, ed. Eyal, Nir, Cohen, I. Glenn and Daniels, Norman (Oxford, 2015), pp. 77–93 CrossRefGoogle Scholar.

²⁰ This is one view among many others about the moral significance of risk, and since I cannot hope to survey and address them all, I focus on summarizing the heart of the argument I find most plausible. But for further discussion, see Oberdiek, J., ‘The Moral Significance of Risking’, Legal Theory 18 (2012), pp. 339–56CrossRefGoogle Scholar; Perry, S., ‘Risk, Harm, Interests, and Rights’, Risk: Philosophical Perspectives, ed. Lewens, Tim (New York, 2007), pp. 190–210 Google Scholar; Hayenhjelm, M. and Wolff, J., ‘The Moral Problem of Risk Impositions: A Survey’, European Journal of Philosophy 20 (2012), pp. 26–51 CrossRefGoogle Scholar; Zimmerman, M. J., ‘Risk, Rights, and Restitution’, Philosophical Studies 128 (2006), pp. 285–311 CrossRefGoogle Scholar. The most sustained individual treatment of the ethics of risk is Hansson, S. O., The Ethics of Risk (London, 2013)CrossRefGoogle Scholar.

²¹ Lazar, S., ‘Risky Killing and the Ethics of War’, Ethics 126 (2015), pp. 91–117 CrossRefGoogle Scholar.

²² This argument has some affinities with Philip Pettit's account of the robustly demanding good of respect, in Pettit, P., The Robust Demands of the Good (Oxford, 2015)CrossRefGoogle Scholar.

²³ Others would agree. See Fabre, ‘War Exit’, p. 637; Moellendorf, ‘Two Doctrines’, p. 666.

²⁴ Jeff McMahan tentatively defends a version of the Died in Vain variant (he calls it the Redemption Thesis). In general, though, he is in favour of an absolute discount for sunk costs: permissibility in these cases is determined only by forward-looking considerations. David Rodin has a subtly different but deontically equivalent view. See McMahan, J., ‘Proportionality and Time’, Ethics 125 (2015), pp. 696–719 CrossRefGoogle Scholar; Rodin, D., ‘The War Trap: Dilemmas of Jus Terminatio’, Ethics 125 (2015), pp. 674–95CrossRefGoogle Scholar.

²⁵ This is contrary to the views of Fabre, ‘War Exit’ and Moellendorf, ‘Two Doctrines’, who think that we have a ‘proportionality budget’ which is set at T₁, such that once it is expended, no further sacrifices are permissible. In our case, that would mean if you get tails on the first flip, you must not flip again. This is surely too severe.

²⁶ Thanks here to Garrett Cullity.

²⁷ This kind of moderate deontology has been the norm at least since it was first characterized by Kagan, S., The Limits of Morality (Oxford, 1989)Google Scholar. See Quinn, W. S., ‘Actions, Intentions, and Consequences: The Doctrine of Doing and Allowing’, Philosophical Review 89 (1989), pp. 287–312 CrossRefGoogle Scholar; McMahan, J., The Ethics of Killing: Problems at the Margins of Life (New York, 2002)CrossRefGoogle Scholar; Frowe, H., Defensive Killing (Oxford, 2014)CrossRefGoogle Scholar.

²⁸ Thanks to Johann Frick for this suggestion.

²⁹ See Daniels, N., ‘Can There Be Moral Force to Favoring an Identified over a Statistical Life?’, Identified Versus Statistical Lives: An Interdisciplinary Perspective, ed. Cohen, I. Glenn, Daniels, Norman and Eyal, Nir (Oxford, 2015), pp. 110–23CrossRefGoogle Scholar.

³⁰ For discussion along these lines, see Hedden, B., ‘Options and Diachronic Tragedy’, Philosophy and Phenomenological Research 87 (2013), pp. 1–35 Google Scholar; Tenenbaum, S. and Raffman, D., ‘Vague Projects and the Puzzle of the Self-Torturer’, Ethics 123 (2012), pp. 86–112 CrossRefGoogle Scholar. For discussion among decision theorists about whether rationality can require you to bind yourself to a sequence of actions, see, for example, Arntzenius et al., ‘Infinite Decisions’; Elga, A., ‘Subjective Probabilities Should Be Sharp’, Philosophers' Imprint 10 (2010), pp. 1–11 Google Scholar.

³¹ These questions raise a nest of problems to do with the actualism/possibilism debate in ethical theory, to which I cannot do justice here. For some highlights of that debate, see Jackson, F. and Pargetter, R., ‘Oughts, Options, and Actualism’, Philosophical Review 95 (1986), pp. 233–55CrossRefGoogle Scholar; Ross, J., ‘Actualism, Possibilism, and Beyond’, Oxford Studies in Normative Ethics 2 (2012), pp. 74–96 CrossRefGoogle Scholar.

³² For helpful discussion of the ideas that this article is based on, I thank Rachael Briggs, David Grewal, Alan Hájek and Brendan O’Grady. I presented earlier versions of this article, and another related to it, at the Australasian Association of Philosophy, ANU's CRNAP workshop, and in departmental seminars at Roskilde, Oxford, St Andrews, Wellington, Stirling, Auckland and Otago. Thanks to the conveners of those seminars, and the participants, for their helpful comments and discussion. Thanks for comments on the draft to Ed Elliott, Cécile Fabre, Johann Frick, Al Hájek, Theron Pummer, Victor Tadros, Frej Klem Thomsen, David Wiens and Robbie Williams. Lastly, thanks to a reviewer for this journal.