Thesis

Tiny Probabilities of Vast Value

University of Oxford, 2022

The topic of this thesis is how we should treat tiny probabilities of vast value. This thesis consists of six independent papers. Chapter 1 This chapter shows that decision theories on which utilities are bounded, such as standard axiomatizations of Expected Utility Theory, violate Ex Ante Pareto if combined with an additive axiology, such as Total Utilitarianism. A series of impossibility theorems point toward Total Utilitarianism as the right account of axiology, while money-pump arguments put Expected Utility Theory in a favorable light. However, it is not clear how these two views can be reconciled. This question is particularly puzzling if utilities are bounded (as standard axiomatizations of Expected Utility Theory imply) because the total quantity of well-being might be infinite or arbitrarily large. Thus, there must be a non-linear transformation from the total quantity of well-being into utilities used in decision-making. This non-linear transformation is also required if one wishes to avoid Probability Fanaticism. However, such a transformation leads to violations of Ex Ante Pareto. So, the reconciliation of Expected Utility Theory and Total Utilitarianism prescribes prospects that are better for none and worse for some. discusses the idea that utilities are bounded. It shows that bounded decision theories prescribe prospects that are better for no one and worse for some if combined with an additive axiology. Chapter 2 At least at first glance, Expected Utility Theory tells us to be indifferent between two prospects when they are otherwise the same, except that one gives a better outcome than the other in a possible state of zero probability. But as some have suggested, Expected Utility Theory might be supplemented with dominance to get the verdict that the dominating prospect is better than the dominated one. However, I will show that if Expected Utility Theory is supplemented with dominance in this way, it will violate the Continuity axiom of Expected Utility Theory. , in turn, points out that standard axiomatizations of Expected Utility Theory violate dominance in cases that involve possible states of zero probability. Chapters 3-6 discuss the idea that we should ignore tiny probabilities in practical decision-making. Chapter 3 Nebel (2019) argues for the Repugnant Conclusion via the “Intrapersonal Repugnant Conclusion”, on which certainty of a mediocre life is better for individuals than a sufficiently small chance of an excellent life. In this chapter, I deny that accepting the Intrapersonal Repugnant Conclusion leads us to the Repugnant Conclusion. I point out that on many views which avoid the Repugnant Conclusion, we should discount small probabilities down to zero to avoid an implausibly reckless decision theory. But if we do, then Nebel’s crucial premise of Ex Ante Pareto fails because discounting at the individual level can fail to match up with discounting at the population level. argues that discounting small probabilities solves the ‘Intrapersonal Addition Paradox’ and thus helps avoid the Repugnant Conclusion. Chapter 4 Maximizing expected value leads to counterintuitive choices in cases that involve tiny probabilities of huge payoffs. In response to such cases, some have argued that we ought to discount very small probabilities down to zero. In this chapter, I discuss how exactly this view can be formulated. I begin by showing that less plausible versions of discounting small probabilities violate dominance. Then, I show that more plausible formulations of this view avoid these dominance violations, but instead, they violate the axiom of Independence—and in a particularly counterintuitive way. As a result of this violation, those who discount small probabilities can be exploited by a money pump. Lastly, I discuss one possible way of avoiding exploitation by this money pump. explores what the most plausible version of this view might look like and what problems the different versions have. Chapter 5 In response to cases that involve very small probabilities of huge payoffs, some argue that we ought to discount very small probabilities down to zero. However, this chapter shows that doing so violates Independence and Continuity, and as a result of these violations, those who discount small probabilities can be exploited by money pumps. Various possible ways of avoiding exploitation will be discussed. However, echoing the previous chapter, this chapter concludes that the money pump for Independence undermines the plausibility of discounting small probabilities. focuses on one type of problem, namely, money pumps. The Independence Money Pump, in particular, presents a difficult challenge for those who wish to discount small probabilities. Finally, Chapter 6 Morally speaking, what matters the most is the far future—at least according to Longtermism. The reason why the far future is of utmost importance is that our acts’ expected influence on the value of the world is mainly determined by their consequences in the far future. The case for Longtermism is straightforward: Given the enormous number of people who might exist in the far future, even a tiny probability of affecting how the far future goes outweighs the importance of our acts’ consequences in the near term. However, it seems that there is something wrong with a theory that lets very small probabilities of huge payoffs dictate one’s course of action. If, instead, we discount very small probabilities down to zero, we may have a response to Longtermism provided that its truth depends on tiny probabilities of vast value. Contrary to this, I will argue that discounting small probabilities does not undermine Longtermism. discusses the implications of discounting small probabilities for the value of the far future.

The topic of this thesis is how we should treat tiny probabilities of vast value. This thesis consists of six independent papers. Chapter 1 discusses the idea that utilities are bounded. It shows that bounded decision theories prescribe prospects that are better for no one and worse for some if combined with an additive axiology. Chapter 2, in turn, points out that standard axiomatizations of Expected Utility Theory violate dominance in cases that involve possible states of zero probability. Chapters 3-6 discuss the idea that we should ignore tiny probabilities in practical decision-making. Chapter 3 argues that discounting small probabilities solves the ‘Intrapersonal Addition Paradox’ and thus helps avoid the Repugnant Conclusion. Chapter 4 explores what the most plausible version of this view might look like and what problems the different versions have. Chapter 5 focuses on one type of problem, namely, money pumps. The Independence Money Pump, in particular, presents a difficult challenge for those who wish to discount small probabilities. Finally, Chapter 6 discusses the implications of discounting small probabilities for the value of the far future.

The topic of this thesis is how we should treat tiny probabilities of vast value. This thesis consists of six independent papers. Chapter 1 discusses the idea that utilities are bounded. It shows that bounded decision theories prescribe prospects that are better for no one and worse for some if combined with an additive axiology. Chapter 2, in turn, points out that standard axiomatizations of Expected Utility Theory violate dominance in cases that involve possible states of zero probability. Chapters 3-6 discuss the idea that we should ignore tiny probabilities in practical decision-making. Chapter 3 argues that discounting small probabilities solves the ‘Intrapersonal Addition Paradox’ and thus helps avoid the Repugnant Conclusion. Chapter 4 explores what the most plausible version of this view might look like and what problems the different versions have. Chapter 5 focuses on one type of problem, namely, money pumps. The Independence Money Pump, in particular, presents a difficult challenge for those who wish to discount small probabilities. Finally, Chapter 6 discusses the implications of discounting small probabilities for the value of the far future.