Abstract
The principle that rational agents should maximize expected utility or choiceworthiness is intuitively plausible in many ordinary cases of decision-making under uncertainty. But it is less plausible in cases of extreme, low-probability risk (like Pascal’s Mugging), and intolerably paradoxical in cases like the St. Petersburg and Pasadena games. In this paper I show that, under certain conditions, stochastic dominance reasoning can capture most of the plausible implications of expectational reasoning while avoiding most of its pitfalls. Specifically, given sufficient background uncertainty about the choiceworthiness of one’s options, many expectation-maximizing gambles that do not stochastically dominate their alternatives ‘in a vacuum’ become stochastically dominant in virtue of that background uncertainty. But, even under these conditions, stochastic dominance will not require agents to accept options whose expectational superiority depends on sufficiently small probabilities of extreme payoffs. The sort of background uncertainty on which these results depend looks unavoidable for any agent who measures the choiceworthiness of her options in part by the total amount of value in the resulting world. At least for such agents, then, stochastic dominance offers a plausible general principle of choice under uncertainty that can explain more of the apparent rational constraints on such choices than has previously been recognized.
Introduction
Given our epistemic limitations, every choice you or I will ever make involves some degree of risk. Whatever we do, it might turn out that we would have done better to do something else. If our choices are to be more than mere leaps in the dark, therefore, we need principles that tell us how to evaluate and compare risky options.
The standard view in normative decision theory holds we should rank options by their expectations. That is, an agent should assign cardinal degrees of utility or choiceworthiness to each of her available options in each possible state of nature, assign probabilities to the states, and prefer one option to another just in case the probability-weighted sum (i.e., expectation) of its possible degrees of utility or choiceworthiness is greater. Call this view expectationalism.
Expectational reasoning provides seemingly indispensable practical guidance in many ordinary cases of decision-making under uncertainty. But it encounters serious difficulties in many cases involving extremely large finite or infinite payoffs, where it yields conclusions that are either implausible, unhelpful, or both. For instance, expectationalism implies that: (i) Any positive probability of an infinite positive or negative payoff, no matter how minuscule, takes precedence over all finitary considerations (Pascal, 1669). (ii) When two options carry positive probabilities of infinite payoffs of the same sign (i.e., both positive or both negative), and zero probability of infinite payoffs of the opposite sign, the two options are equivalent, even if one offers a much greater probability of that infinite payoff than the other (H´ajek, 2003). (iii) When an option carries any positive probabilities of both infinite positive and infinite negative payoffs, it is simply incomparable with any other option (Bostrom, 2011). (iv) Certain probability distributions over finite payoffs yield expectations that are infinite (as in the St. Petersburg game (Bernoulli, 1738)) or undefined (as in the Pasadena game (Nover and H´ajek, 2004)), so that options with these prospects are better than or incomparable with any guaranteed finite payoff. (v) Agents can be rationally required to prefer minuscule probabilities of astronomically large finite payoffs over certainty of a more modest payoff, in cases where that preference seems at best rationally optional (as in ‘Pascal’s Mugging’ (Bostrom, 2009)).
The last of these problem cases, though theoretically the most straightforward, has particular practical significance. Real-world agents who want to do the most good when they choose a career or donate to charity often face choices between options that are fairly likely to do a moderately large amount of good (e.g., supporting public health initiatives in the developing world or promoting farm animal welfare) and options that carry much smaller probabilities of doing much larger amounts of good (e.g., reducing existential risks to human civilization (Bostrom, 2013; Ord, 2020) or trying to bring about very longterm ‘trajectory changes’ (Beckstead, 2013)). Often, na¨ıve application of expectational reasoning suggests that we are rationally required to choose the latter sort of project, even if the probability of having any positive impact whatsoever is vanishingly small. For instance, based on an estimate that future Earth-originating civilization might support the equivalent of 10 to the fifty-second power of human lives, Nick Bostrom concludes that, ‘[e]ven if we give this allegedly lower bound...a mere 1 per cent chance of being correct, we find that the expected value of reducing existential risk by a mere one billionth of one billionth of one percentage point is worth a hundred billion times as much as a billion human lives’ (Bostrom, 2013, p. 19). This suggests that we should pass up opportunities to do enormous amounts of good in the present, to maximize the probability of an astronomically good future, even if the probability of having any effect at all is on the order of, say, 10 to the negative thirtieth power—meaning, for all intents and purposes, no matter how small the probability.
Even hardened utilitarians who think we should normally do what maximizes expected welfare may find this conclusion troubling and counterintuitive. We intuit (or so I take it) not that the expectationally superior long-shot option is irrational, but simply that it is rationally optional: We are not rationally required to forego a high probability of doing a significant amount of good for a vanishingly small probability of doing astronomical amounts of good. And we would like decision theory to vindicate this judgment.
The aim of this paper is to set out an alternative to expectational decision theory that outperforms it in the various problem cases just described—but in particular, with respect to tiny probabilities of astronomical payoffs. Specifically, I will argue that under plausible epistemic conditions, stochastic dominance reasoning can capture most of the ordinary, attractive implications of expectational decision theory—far more than has previously been recognized—while avoiding its pitfalls in the problem cases described above, and in particular, while permitting us to decline expectationally superior options in extreme, ‘Pascalian’ choice situations.
Stochastic dominance is, on its face, an extremely modest principle of rational choice, simply formalizing the idea that one ought to prefer a given probability of a better payoff to the same probability of a worse payoff, all else being equal. The claim that we are rationally required to reject stochastically dominated options is therefore on strong a priori footing (considerably stronger, I will argue, than expectationalism). But precisely because it is so modest, stochastic dominance seems too weak to serve as a final principle of decision-making under uncertainty: It appears to place no constraints on an agent’s risk attitudes, allowing intuitively irrational extremes of risk-seeking and risk-aversion.
But in fact, stochastic dominance has a hidden capacity to effectively constrain risk attitudes: When an agent is in a state of sufficient ‘background uncertainty’ about the choiceworthiness of her options, expectationally superior options that would not otherwise stochastically dominate their alternatives can become stochastically dominant. Background uncertainty generates stochastic dominance much less readily, however, in situations where the balance of expectations is determined by minuscule probabilities of astronomical positive or negative payoffs. Stochastic dominance thereby draws a principled line between ‘ordinary’ and ‘Pascalian’ choice situations, and vindicates our intuition that we are often permitted to decline gambles like Pascal’s Mugging or the St. Petersburg game, even when they are expectationally best. Since it avoids these and other pitfalls of expectational reasoning, if stochastic dominance can also place plausible constraints on our risk attitudes and thereby recover the attractive implications of expectationalism, it may provide a more attractive criterion of rational choice under uncertainty.
I begin in §2 by saying more about standard expectational decision theory, as motivation and point of departure for my main line of argument. §3 introduces stochastic dominance. §4 gives a formal framework for describing decisions under background uncertainty. In §5, I establish two central results: (i) a sufficient condition for stochastic dominance which implies, among other things, that whenever Oi is expectationally superior to Oj , it will come to stochastically dominate Oj given sufficient background uncertainty; and (ii) a necessary condition for stochastic dominance which implies, among other things, that it is harder for expectationally superior options to become stochastically dominant under background uncertainty when their expectational superiority depends on small probabilities of extreme payoffs. In §6, I argue that the sort of background uncertainty on which these results depend is rationally appropriate at least for any agent who assigns normative weight to aggregative consequentialist considerations, i.e., who measures the choiceworthiness of her options at least in part by the total amount of value in the resulting world. §7 offers an intuitive defense of the initially implausible conclusion that an agent’s background uncertainty can make a difference to what she is rationally required to do. §8 describes two modest conclusions we might draw from the preceding arguments, short of embracing stochastic dominance as a sufficient criterion of rational choice. In §9, however, I survey several further advantages of stochastic dominance over expectational reasoning and argue that, insofar as stochastic dominance can recover the intuitively desirable implications of expectationalism, we have substantial reason to prefer it as a criterion of rational choice under uncertainty. §10 is the conclusion.
