How should you decide what to do when you're uncertain about basic normative principles (e.g., Kantianism vs. utilitarianism)? A natural suggestion is to follow some "second-order" norm: e.g., "comply with the first-order norm you regard as most probable" or "maximize expected choiceworthiness". But what if you're uncertain about second-order norms too -- must you then invoke some third-order norm? If so, it seems that any norm-guided response to normative uncertainty is doomed to a vicious regress. In this paper, I aim (...) to rescue second-order norms from this threat of regress. I first elaborate and defend the suggestion some philosophers have entertained that the regress problem forces us to accept normative externalism, the view that at least one norm is incumbent on agents regardless of their beliefs or evidence concerning that norm. But, I then argue, we need not accept externalism about first-order (e.g., moral) norms, thus closing off any question of what an agent should do in light of her normative beliefs. Rather, it is more plausible to ascribe external force to a single, second-order rational norm: the enkratic principle, correctly formulated. This modest form of externalism, I argue, is both intrinsically well-motivated and sufficient to head off the threat of regress. (shrink)

In this paper, I argue for a new normative theory of rational choice under risk, namely expected comparative utility (ECU) theory. I first show that for any choice option, a, and for any state of the world, G, the measure of the choiceworthiness of a in G is the comparative utility (CU) of a in G—that is, the difference in utility, in G, between a and whichever alternative to a carries the greatest utility in G. On the basis of this (...) principle, I then argue that for any agent, S, faced with any decision under risk, S should rank his or her decision options (in terms of how choiceworthy they are) according to their comparative expected comparative utility (CECU) and should choose whichever option carries the greatest CECU. For any option, a, a’s CECU is the difference between its ECU and that of whichever alternative to a carries the greatest ECU, where a’s ECU is a probability‐weighted sum of a’s CUs across the various possible states of the world. I lastly demonstrate that in some ordinary decisions under risk, ECU theory delivers different verdicts from those of standard decision theory. (shrink)

In this paper, I argue for a new normative theory of rational choice under risk, namely expected comparative utility (ECU) theory. I first show that for any choice option, a, and for any state of the world, G, the measure of the choiceworthiness of a in G is the comparative utility (CU) of a in G—that is, the difference in utility, in G, between a and whichever alternative to a carries the greatest utility in G. On the basis of this (...) principle, I then argue that for any agent, S, faced with any decision under risk, S should rank his or her decision options (in terms of how choiceworthy they are) according to their comparative expected comparative utility (CECU) and should choose whichever option carries the greatest CECU. For any option, a, a's CECU is the difference between its ECU and that of whichever alternative to a carries the greatest ECU, where a's ECU is a probability-weighted sum of a's CUs across the various possible states of the world. I lastly demonstrate that in some ordinary decisions under risk, ECU theory delivers different verdicts from those of standard decision theory. (shrink)

Maximize Expected Choiceworthiness (MEC) is a theory of decision-making under moral uncertainty. It says that we ought to handle moral uncertainty in the way that Expected Value Theory (EVT) handles descriptive uncertainty. MEC inherits from EVT the problem of fanaticism. Roughly, a decision theory is fanatical when it requires our decision-making to be dominated by low-probability, high-payoff options. Proponents of MEC have offered two main lines of response. The first is that MEC should simply import whatever are the best solutions (...) to fanaticism on offer in decision theory. The second is to propose statistical normalization as a novel solution on behalf of MEC. This paper argues that the first response is open to serious doubt and that the second response fails. As a result, MEC appears significantly less plausible when compared to competing accounts of decision-making under moral uncertainty, which are not fanatical. (shrink)

This paper aims to address the question of how one ought to choose when one is uncertain about what outcomes will result from one’s choices, but when one can nevertheless assign probabilities to the different possible outcomes. These choices are commonly referred to as choices (or decisions) under risk. I assume in this paper that one ought to make instrumentally rational choices—more precisely, one ought to adopt suitable means to one’s morally permissible ends. Expected utility (EU) theory is generally accepted (...) as a normative theory of rational choice under risk, or, more specifically, as a theory of instrumental rationality. According to EU theory, when faced with a decision under risk, one ought to rank one’s options (from least to most choiceworthy) according to their EU and one ought to choose whichever option carries the greatest EU (or one of them in the event that several alternatives are tied). The EU of an option is a probability-weighted sum of each of its possible utilities. In this paper, I argue that EU theory is not the correct theory of instrumental rationality. In its place, I argue for a new theory of instrumental rationality, namely expected comparative utility (ECU) theory. I first show that for any choice option, a, and for any state of the world, G, the measure of the choiceworthiness of a in G is the comparative utility of a in G—that is, the difference in utility, in G, between a and whichever alternative to a carries the greatest utility in G. On the basis of this principle, I then argue, roughly speaking, that for any agent, S, faced with any decision under risk, S ought to rank her options (in terms of how choiceworthy they are) according to their ECU and S ought to choose whichever option carries the greatest ECU (or one of them in the event that several alternatives are tied). For any option, a, a’s ECU is a probability‐weighted sum of a’s comparative utilities across the various possible states of the world. In this paper, I show that in some commonplace decisions under risk, ECU theory delivers different verdicts from those of EU theory. (shrink)

In response to cases that involve tiny probabilities of huge payoffs, some argue that we ought to discount small probabilities down to zero. However, this paper 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. This paper concludes that the money pump for Independence undermines the plausibility of discounting small probabilities. Much of the discussion (...) on Independence generalizes to other views that also violate Independence. (shrink)

Longtermists have recently argued that it is overwhelmingly important to do what we can to mitigate existential risks to humanity. I consider three mistakes that are often made in calculating the value of existential risk mitigation. I show how correcting these mistakes pushes the value of existential risk mitigation substantially below leading estimates, potentially low enough to threaten the normative case for existential risk mitigation. I use this discussion to draw four positive lessons for the study of existential risk. -/- (...) . (shrink)

Countenancing unbounded utility in ethics gives rise to deep puzzles in formal decision theory. Here, these puzzles are taken as an invitation to assess the most fundamental principles relating probability and value, with the aim of demonstrating that unbounded utility in ethics is compatible with a desirable decision theory. The resulting theory frames further discussion of Expected Utility Theory and of principles concerning symmetries of utility.

Which is better: a guarantee of a modest amount of moral value, or a tiny probability of arbitrarily large value? To prefer the latter seems fanatical. But, as I argue, avoiding such fanaticism brings severe problems. To do so, we must decline intuitively attractive trade-offs; rank structurally identical pairs of lotteries inconsistently, or else admit absurd sensitivity to tiny probability differences; have rankings depend on remote, unaffected events ; and often neglect to rank lotteries as we already know we would (...) if we learned more. Compared to these implications, fanaticism is highly plausible. (shrink)

For aggregative theories of moral value, it is a challenge to rank worlds that each contain infinitely many valuable events. And, although there are several existing proposals for doing so, few provide a cardinal measure of each world's value. This raises the even greater challenge of ranking lotteries over such worlds—without a cardinal value for each world, we cannot apply expected value theory. How then can we compare such lotteries? To date, we have just one method for doing so (proposed (...) separately by Arntzenius, Bostrom, and Meacham), which is to compare the prospects for value at each individual location, and to then represent and compare lotteries by their expected values at each of those locations. But, as I show here, this approach violates several key principles of decision theory and generates some implausible verdicts. I propose an alternative—one which delivers plausible rankings of lotteries, which is implied by a plausible collection of axioms, and which can be applied alongside almost any ranking of infinite worlds. (shrink)

ABSTRACT Average utilitarianism and several related axiologies, when paired with the standard expectational theory of decision-making under risk and with reasonable empirical credences, can find their practical prescriptions overwhelmingly determined by the minuscule probability that the agent assigns to solipsism—that is, to the hypothesis that there is only one welfare subject in the world, namely, herself. This either (i) constitutes a reductio of these axiologies, (ii) suggests that they require bespoke decision theories, or (iii) furnishes an unexpected argument for ethical (...) egoism. (shrink)

I have argued for a picture of decision theory centred on the principle of Rationally Negligible Probabilities. Isaacs argues against this picture on the grounds that it has an untenable implication. I first examine whether my view really has this implication; this involves a discussion of the legitimacy or otherwise of infinite decisions. I then examine whether the implication is really undesirable and conclude that it is not.

A de minimis risk is defined as a risk that is so small that it may be legitimately ignored when making a decision. While ignoring small risks is common in our day-to-day decision making, attempts to introduce the notion of a de minimis risk into the framework of decision theory have run up against a series of well-known difficulties. In this paper, I will develop an enriched decision theoretic framework that is capable of overcoming two major obstacles to the modelling (...) of de minimis risk. The key move is to introduce, into decision theory, a non-probabilistic conception of risk known as normic risk. (shrink)

Should we make significant sacrifices to ever-so-slightly lower the chance of extremely bad outcomes, or to ever-so-slightly raise the chance of extremely good outcomes? *Fanaticism* says yes: for every bad outcome, there is a tiny chance of extreme disaster that is even worse, and for every good outcome, there is a tiny chance of an enormous good that is even better. I consider two related recent arguments for Fanaticism: Beckstead and Thomas's argument from *strange dependence on space and time*, and (...) Wilkinson's *Indology* argument. While both arguments are instructive, neither is persuasive. In fact, the general principles that underwrite the arguments (a *separability* principle in the first case, and a *reflection* principle in the second) are *inconsistent* with Fanaticism. In both cases, though, it is possible to rehabilitate arguments for Fanaticism based on restricted versions of those principles. The situation is unstable: plausible general principles tell *against* Fanaticism, but restrictions of those same principles (with strengthened auxiliary assumptions) *support* Fanaticism. All of the consistent views that emerge are very strange. (shrink)

Standard decision theory has trouble handling cases involving acts without finite expected values. This paper has two aims. First, building on earlier work by Colyvan (2008), Easwaran (2014), and Lauwers and Vallentyne (2016), it develops a proposal for dealing with such cases, Difference Minimizing Theory. Difference Minimizing Theory provides satisfactory verdicts in a broader range of cases than its predecessors. And it vindicates two highly plausible principles of standard decision theory, Stochastic Equivalence and Stochastic Dominance. The second aim is to (...) assess some recent arguments against Stochastic Equivalence and Stochastic Dominance. If successful, these arguments refute Difference Minimizing Theory. This paper contends that these arguments are not successful. (shrink)

Dozens of articles have addressed the challenge that gambles having undefined expectation pose for decision theory. This paper makes two contributions. The first is incremental: we evolve Colyvan's ``Relative Expected Utility Theory'' into a more viable ``conservative extension of expected utility theory" by formulating and defending emendations to a version of this theory proposed by Colyvan and H\'ajek. The second is comparatively more surprising. We show that, so long as one assigns positive probability to the theory that there is a (...) uniform bound on the utility of possible gambles (and assuming a uniform bound on the amount of utility that can accrue in a fixed amount of time), standard principles of anthropic reasoning (as formulated by Brandon Carter) place lower and upper bounds on the expected values of gambles advertised as having no expectation--even assuming that with positive probability, all gambles advertised as having infinite expected utility are administered faithfully. Should one accept the uniform bound premises, this reasoning thus dissolves (or nearly dissolves, in some cases) several puzzles in infinite decision theory. (shrink)

Many moral theories are committed to the idea that some kinds of moral considerations should be respected, whatever the cost to ‘lesser’ types of considerations. A person's life, for instance, should not be sacrificed for the trivial pleasures of others, no matter how many would benefit. However, according to the decision-theoretic critique of lexical priority theories, accepting lexical priorities inevitably leads us to make unacceptable decisions in risky situations. It seems that to operate in a risky world, we must reject (...) lexical priorities altogether. This paper argues that lexical priority theories can, in fact, offer satisfactory guidance in risky situations. It does so by equipping lexical priority theories with overlooked resources from decision theory. (shrink)

The existence of a transitive, complete, and weakly independent relation on the full set of gambles implies the existence of a non-Ramsey set. Therefore, each transitive and weakly independent relation on the set of gambles either is incomplete or does not have an explicit description. Whatever tools decision theory makes available, there will always be decision problems where these tools fail us. In this sense, decision theory remains incomplete.

:We address the question, in decision theory, of how the value of risky options should be assessed when they have no finite standard expected value, that is, where the sum of the probability-weighted payoffs is infinite or not well defined. We endorse, combine and extend the proposal of Easwaran to evaluate options on the basis of their weak expected value, and the proposal of Colyvan to rank options on the basis of their relative expected value.

This paper explores the consequences of applying two natural ideas from epistemology to decision theory: (1) that knowledge should guide our actions, and (2) that we know a lot of non-trivial things. In particular, we explore the consequences of these ideas as they are applied to standard decision theoretic puzzles such as the St. Petersburg Paradox. In doing so, we develop a “knowledge-first” decision theory and we will see how it can help us avoid fanaticism with regard to the St. (...) Petersburg puzzle and related puzzles. The result will be a decision theory that gives a novel, but well-motivated, reason for discounting small probabilities when making decisions. We examine the merits and demerits of such a decision theory. (shrink)

A decade ago, Harris Nover and I introduced the Pasadena game, which we argued gives rise to a new paradox in decision theory even more troubling than the St Petersburg paradox. Gwiazda's and Smith's articles in this volume both offer revisionist solutions. I critically engage with both articles. They invite reflections on a number of deep issues in the foundations of decision theory, which I hope to bring out. These issues include: some ways in which orthodox decision theory might be (...) supplemented; the role of simulations of such infinite games; the role of small probabilities, and of idealization, in decision theory; tolerance about practical norms; and alternative ways of understanding decision theory's job description. (shrink)

In this article, we re-examine Pascal’s Mugging, and argue that it is a deeper problem than the St. Petersburg paradox. We offer a way out that is consistent with classical decision theory. Specifically, we propose a “many muggers” response analogous to the “many gods” objection to Pascal’s Wager. When a very tiny probability of a great reward becomes a salient outcome of a choice, such as in the offer of the mugger, it can be discounted on the condition that there (...) are other symmetric, non-salient rewards that one may receive if one chooses otherwise. (shrink)

Orthodox decision theory is fanatical in the way it treats small probabilities of enormous value, if unbounded utility functions are allowed. Some have suggested a fix, Nicolausian discounting, according to which outcomes with small enough probabilities should be ignored when making decisions. However, there are lotteries involving only small-probability outcomes, none of which should intuitively be ignored. So the Nicolausian discounter needs a procedure for distinguishing the problematic cases of small-probability outcomes from the unproblematic ones. In this paper, I present (...) a dilemma for Nicolausian discounting: the view must be coupled either with a procedure that delivers fanatical verdicts of its own, as bad as those of orthodox decision theory, or with one that entails intransitive cyclic preferences. (shrink)

We begin by showing that every theory of the value of uncertain prospects must have one of three unpalatable properties. _Reckless_ theories recommend giving up a sure thing, no matter how good, for an arbitrarily tiny chance of enormous gain; _timid_ theories permit passing up an arbitrarily large potential gain to prevent a tiny increase in risk; _non-transitive_ theories deny the principle that, if A is better than B and B is better than C, then A must be better than (...) C. Having set up this trilemma, we study its horns. Non-transitivity has been much discussed; we focus on drawing out the costs and benefits of recklessness and timidity when it comes to axiology, decision theory, and normative uncertainty. (shrink)

Many population axiologies avoid the Repugnant Conclusion by endorsing Superiority: some number of great lives is better than any number of mediocre lives. But as Nebel shows, RC follows from the Intrapersonal Repugnant Conclusion: a guaranteed mediocre life is better than a sufficiently small probability of a great life. This result is concerning because IRC is plausible. Recently, Kosonen has argued that IRC can be true while RC is false if small probabilities are discounted to zero. This article details the (...) unique problems created by combining Superiority with discounting. The resultant view, Superiority Discounting, avoids the Repugnant Conclusion only at the cost of the Preposterous Conclusion: near-certain hell for arbitrarily many people is better than near-certain heaven for arbitrarily many people. (shrink)

Should you be willing to forego any sure good for a tiny probability of a vastly greater good? Fanatics say you should, anti-fanatics say you should not. Anti-fanaticism has great intuitive appeal. But, I argue, these intuitions are untenable, because satisfying them in their full generality is incompatible with three very plausible principles: acyclicity, a minimal dominance principle, and the principle that any outcome can be made better or worse. This argument against anti-fanaticism can be turned into a positive argument (...) for a weak version of fanaticism, but only from significantly more contentious premises. In combination, these facts suggest that those who find fanaticism counterintuitive should favor not anti-fanaticism, but an intermediate position that permits agents to have incomplete preferences that are neither fanatical nor anti-fanatical. (shrink)

Fanatics judge a lottery with a tiny probability of arbitrarily high value as better than the certainty of some modest value, and they are prone to getting swindled. You need only make the lie “big enough” to get one over on them. I put forward an elegant solution to the fanatic’s problem. When coming to a fully rational decision, agents may ignore outlandish possibilities.

The lesson to be learned from the paradoxical St. Petersburg game and Pascal’s Mugging is that there are situations where expected utility maximizers will needlessly end up poor and on death’s door, and hence we should not be expected utility maximizers. Instead, when it comes to decision-making, for possibilities that have very small probabilities of occurring, we should discount those probabilities down to zero, regardless of the utilities associated with those possibilities.

Decision theory aims to provide mathematical analysis of which choice one should rationally make in a given situation. Our current decision theory norms have been very successful, however, several problems have proven vexing for standard decision theory. In this paper, I show that these problems all share a similar structure and identify a class of problems which decision theory overvalues. I demonstrate that agents who follow current standard decision theory can be exploited and have their preferences reordered if offered decision (...) problems of this class. I show that preference reordering is a serious problem, which motivates my search for a decision theory which is immune to exploitation. I find Dr. Nick Smith’s theory of Rationally Negligible Probabilities cannot be exploited in this way and discuss why agents should adopt it. (shrink)

How should an agent decide what to do when she is uncertain not just about morally relevant empirical matters, like the consequences of some course of action, but about the basic principles of morality itself? This question has only recently been taken up in a systematic way by philosophers. Advocates of moral hedging claim that an agent should weigh the reasons put forward by each moral theory in which she has positive credence, considering both the likelihood that that theory is (...) true and the strength of the reasons it posits. The view that it is sometimes rational to hedge for one's moral uncertainties, however, has recently come under attack both from those who believe that an agent should always be guided by the dictates of the single moral theory she deems most probable and from those who believe that an agent's moral beliefs are simply irrelevant to what she ought to do. Among the many objections to hedging that have been pressed in the recent literature is the worry that there is no non-arbitrary way of making the intertheoretic comparisons of moral value necessary to aggregate the value assignments of rival moral theories into a single ranking of an agent's options. -/- This dissertation has two principal objectives: First, I argue that, contra these recent objections, an agent's moral beliefs and uncertainties are relevant to what she rationally ought to do, and more particularly, that agents are at least sometimes rationally required to hedge for their moral uncertainties. My principal argument for these claims appeals to the enkratic conception of rationality, according to which the requirements of practical rationality derive from an agent's beliefs about the objective, desire-independent value or choiceworthiness of her options. Second, I outline a new general theory of rational choice under moral uncertainty. Central to this theory is the idea of content-based aggregation, that the principles according to which an agent should compare and aggregate rival moral theories are grounded in the content of those theories themselves, including not only their value assignments but also the metaethical and other non-surface-level propositions that underlie, justify, or explain those value assignments. (shrink)

This paper is about compositionality, belief reports, and related issues. I begin by introducing Putnam’s proposal for understanding compositionality, namely that the sense of a sentence is a function of the sense of its parts and of its logical structure (section 1). Both Church and Sellars think that Putnam’s move is superfluous or unnecessary since there is no relevant puzzle to begin with (section 2). I will urge that Putnam is right in thinking that there is indeed a puzzle with (...) a discussion of translation and belief individuation (section 3). Later Salmon (2001/ 2007) reinforces Church’s position, but I will argue that it is still possible to make my case by clarifying the nature of my proposal, i.e., understanding explanations of action from the third-person point of view (section 4). Now, Fine (2007) agrees with Putnam that there is indeed a puzzle to be solved, but he argues that Putnam’s solution of it is problematic, and that his own semantic relationism is a better view. In response to this, I will recast the notion of compositionality based on a certain conception of belief individuation, namely that the semantic content of a sentence is a function of the semantic contents of its parts and of the structure of intensional discourses (sections 3 and 5). Finally the paper will end with a reconsideration of the recalcitrant Kripke’s puzzle about belief (1979/1988), since it might seem to put some pressure on my account. It turns out that my understanding of this puzzle is again different from Fine’s perspective (section 6). (shrink)

Deontologists have been slow to address decision-making under risk and uncertainty, no doubt because the standard approaches to non-moral decision theory appear superficially similar to consequentialist moral reasoning. I identify some central tenets of simple decision theory and show that they should not put deontologists off, before showing where we should go next to develop a comprehensive deontological decision theory.