2.1 Setup: Outcomes, Prospects, and Preferences

Our central question is how agents should rank and choose between uncertain alternatives, in the context of extreme tradeoffs. I will call these alternatives prospects, and understand them as probability distributions over outcomes.Footnote ⁵

An outcome is a specification of all evaluatively significant features of a possible world. For now, and in some of the following arguments, we will rely only on this characterization of outcomes. But in Section 3.2, I’ll give outcomes some structure, representing them as assignments of “local” outcomes to a set of possible “value locations.” The domain of possible outcomes is denoted $\mathbb{O}$ .

We will restrict our attention to finite outcomes – thus, every instance of “outcome” can be read as shorthand for “finite outcome.” This restriction is common in the literature on fanaticism; one motivation is to ensure that, for instance, an agent who regards some infinite outcomes (e.g., eternity in Heaven/Hell) as maximally good/bad can still count as fanatical. I won’t try to draw a precise line between “finite” and “infinite” outcomes, but roughly, a finite outcome is one in which all evaluatively significant extensive quantities, like the number of welfare subjects and the durations of their lives, are finite.

Next, I assume that outcomes can be compared in terms of value. Formally, I’ll assume a total preorder (a reflexive, transitive, and complete binary relation) ${\succsim }_{\mathbb{O}}$ on $\mathbb{O}$ , where $o_i{\succsim }_{\mathbb{O}}{o}_j$ means that $o_i$ is at least as good as $o_j$ .Footnote ⁶ If $o_i{\succsim }_{\mathbb{O}}{o}_j$ but $o_j̸{\succsim }_{\mathbb{O}}{o}_i$ , then $o_i$ is strictly better than $o_j$ , denoted $o_i{\succ }_{\mathbb{O}}{o}_j$ . If $o_i{\succsim }_{\mathbb{O}}{o}_j$ and $o_j{\succsim }_{\mathbb{O}}{o}_i$ , then $o_i$ and $o_j$ are equally good, denoted $o_i{\sim }_{\mathbb{O}}{o}_j$ . In general, I’ll understand these as relations of impersonal or moral value, but many of the arguments discussed in later sections are compatible with other interpretations (e.g., prudential).

A prospect is a probability distribution over outcomes. More specifically, we’ll focus on discrete prospects, which assign non-zero probabilities to a finite or countably infinite set of outcomes, with those probabilities summing to 1. Such a prospect can be written as a sequence of ordered pairs $⟨(o_1,p_1),(o_2,p_2),\dots ⟩$ , where $(o_i,p_i)$ indicates that outcome $o_i$ occurs with probability $p_i$ . Many of the arguments below will concern binary prospects, with two possible outcomes. Here I use the abbreviated notation $⟨o_i,p,o_j⟩$ for the prospect that yields $o_i$ with probability $p$ and $o_j$ otherwise. A prospect that yields a single outcome $o_i$ with certainty is denoted $⟨o_i⟩$ . The domain of prospects we’re interested in evaluating is denoted $\mathbb{P}$ . I’ll take for granted, as a background assumption in all the formal arguments below, that this domain includes at least all finitely supported prospects, that is, prospects that assign non-zero probability to only finitely many outcomes. At some points, I’ll explicitly assume that the domain of prospects also includes some prospects with infinitely many possible outcomes.

In giving examples, I’ll sometimes make informal reference to states of nature (or simply states). A state of nature specifies the features of the external environment that determine the outcome of an agent’s choice (e.g., what asteroids, if any, are presently on a collision course for Earth). Given a set of mutually exclusive and jointly exhaustive states of nature, with probabilities assigned to them, prospects can be represented as functions from states to outcomes, specifying which outcome occurs in which state. While this way of representing prospects is sometimes explanatorily useful, the notion of states won’t play any essential role in the formal arguments we examine.Footnote ⁷

In understanding prospects as simple probability distributions over outcomes, I’m making two significant assumptions. First, I’m assuming that an agent can and should assign precise probabilities to particular outcomes, on the supposition that she chooses one or another risky alternative (even if those probabilities are highly subjective). Second, I’m assuming that the desirability of a prospect doesn’t (or shouldn’t) depend on which state of nature any given outcome occurs in. This assumption is disputable, but is licensed by the principle of Stochastic Dominance introduced and defended in Section 4.Footnote ⁸

The final piece of our conceptual framework is the agent’s preference ranking of prospects. Formally, an agent’s preferences are represented by a relation $\succsim$ (assumed to be reflexive, but not necessarily transitive), where $P_i\succsim P_j$ means that prospect $P_i$ is preferred at least equally (or weakly preferred) to prospect $P_j$ . If $P_i\succsim P_j$ but $P_j̸\succsim P_i$ , then $P_i$ is strictly preferred to $P_j$ , denoted $P_i\succ P_j$ . If $P_i\succsim P_j$ and $P_j\succsim P_i$ , then the two prospects are equally preferred, denoted $P_i\sim P_j$ . If neither relation holds, then there is a preference gap between $P_i$ and $P_j$ , denoted $P_i\bowtie P_j$ .Footnote ⁹

Our central interest will be in the requirements of rationality with respect to an agent’s preferences between prospects.Footnote ¹⁰ More specifically, our focus will be on ideal rationality, the rational requirements that apply to an agent without cognitive limitations, and that (I take it) bounded agents like ourselves should in some sense aim to approximate. And we will focus on the rational requirements that apply to an agent who knows all the moral and other normative facts relevant to her situation, setting aside the question of how rational requirements might depend on our normative beliefs and uncertainties. Fanaticism and anti-fanaticism, as I will understand them, are conflicting theses about these idealized rational requirements. Our conceptual framework in hand, we can now give our first precise formulations of both theses.

2.2 Fanaticism

Throughout this work, I’ll use fanaticism (lowercase) to denote the informal idea that one should be willing to give arbitrarily large weight to arbitrarily small probabilities or differences in probability, if the stakes are high enough. And I will use anti-fanaticism (lowercase) to denote the informal idea that one should be willing to give only limited weight to probabilities or differences in probability below a certain size, no matter the stakes. We will introduce multiple precisifications of both theses, to which we will give capitalized names.

Let’s begin, then, with a formal statement of a version of fanaticism. Like other forms of fanaticism and anti-fanaticism we will encounter, this formal thesis has both a “positive” component (concerning, roughly, choices between certainty of a fixed gain and a small probability of an indefinitely large gain) and a “negative” component (concerning, roughly, choices between certainty of a fixed loss and a small probability of an indefinitely large loss).

Narrow Positive Fanaticism

It is rationally required that, for any outcomes $o^{+}{\succ }_{\mathbb{O}}{o}^{-}$ and probability $p>0$ , there is an outcome $o^{*+}$ such that $⟨o^{*+},p^{\prime },o^{-}⟩\succ ⟨o^{+}⟩$ for all $p^{\prime }\ge p$ .

Narrow Negative Fanaticism

It is rationally required that, for any outcomes $o^{+}{\succ }_{\mathbb{O}}{o}^{-}$ and probability $p>0$ , there is an outcome $o^{*-}$ such that $⟨o^{-}⟩\succ ⟨o^{*-},p^{\prime },o^{+}⟩$ for all $p^{\prime }\ge p$ .Footnote ¹¹

Narrow Fanaticism is the conjunction of Narrow Positive Fanaticism and Narrow Negative Fanaticism. I call it “narrow” because it concerns a restricted category of extreme tradeoffs where one prospect delivers a single outcome with certainty; in Section 6 we’ll consider a more general context where both prospects involve uncertainty.

An agent who satisfies Narrow Positive Fanaticism will forego certainty of a very good outcome, $o^{+}$ , for a risky prospect that is almost certain to yield a very bad outcome $o^{-}$ but carries some minuscule probability of an outcome $o^{*+}$ – as long as $o^{*+}$ is good enough. An agent who satisfies Narrow Negative Fanaticism will accept certainty of the very bad outcome $o^{-}$ , rather than take a risky prospect that is almost certain to deliver the very good outcome $o^{+}$ , but carries some minuscule probability of an outcome $o^{*-}$ – as long as $o^{*-}$ is bad enough. Narrow Fanaticism does not say that any particular astronomically good/bad outcomes must count as “good enough”/“bad enough” in a given situation – it merely asserts that there must be some such outcomes.

As with other versions of fanaticism and anti-fanaticism we will consider, the positive and negative components of Narrow Fanaticism can come apart, and there is at least some intuitive motivation for accepting one without the other. (In particular, many will find the negative component of fanaticism less counterintuitive than the positive component.) But the formal arguments we consider in later sections will, in every case, apply identically to corresponding positive and negative theses.

Fanaticism strikes most people as deeply counterintuitive. To illustrate the point as starkly as possible, suppose that we face a collective choice between guaranteeing a very good future for all sentient life on Earth (say, hundreds of millions of years in which large populations will enjoy prosperity, justice, and happiness), or a gamble that gives us a one-in-a-googol ( ${10}^{-100}$ ) chance of an even better collective future and a complementary ( $1-{10}^{-100}$ ) probability of the instant and permanent annihilation of all life on Earth. Fanaticism implies that for some specification of the super-utopian future we would obtain by winning the gamble, we are required to choose it over certainty of mere utopia. Most of us will find this verdict very hard to accept.Footnote ¹²

2.3 Anti-Fanaticism

Narrow Fanaticism holds that we are rationally required to give potentially unlimited weight to arbitrarily small probabilities of sufficiently extreme outcomes. A contrary thesis, Narrow Anti-Fanaticism, asserts the opposite requirement: We are rationally required to give only limited weight to sufficiently low-probability outcomes, no matter how high the stakes. It likewise consists of two theses:

Narrow Positive Anti-Fanaticism

It is rationally required that there are some outcomes $o^{+}{\succ }_{\mathbb{O}}{o}^{-}$ and probability $p>0$ such that, for any outcome $o^{*+}$ , $⟨o^{+}⟩\succ ⟨o^{*+},p^{\prime },o^{-}⟩$ for all $p^{\prime }\le p$ .

Narrow Negative Anti-Fanaticism

It is rationally required that there are some outcomes $o^{+}{\succ }_{\mathbb{O}}{o}^{-}$ and probability $p>0$ such that, for any outcome $o^{*-}$ , $⟨o^{*-},p^{\prime },o^{+}⟩\succ ⟨o^{-}⟩$ for all $p^{\prime }\le p$ .

An agent who satisfies Narrow Positive Anti-Fanaticism will at least sometimes prefer certainty of a very good outcome $o^{+}$ to a tiny chance of an astronomically good outcome $o^{*+}$ (and a very bad outcome $o^{-}$ otherwise), no matter how good $o^{*+}$ may be. On the other hand, an agent who satisfies Narrow Negative Anti-Fanaticism will sometimes prefer to take a tiny risk of an astronomically bad outcome $o^{*-}$ rather than settle for $o^{-}$ , no matter how bad $o^{*-}$ may be.

2.4 Permissivism

Fanaticism and anti-fanaticism (in the preceding formulations, and others that we will encounter later) assert contrary rational requirements. These rival theses are not jointly exhaustive, since rationality might impose neither requirement on us. I will use permissivism to denote any view which denies both that we are rationally required to have fanatical preferences and that we are rationally required to have anti-fanatical preferences. Permissivism can take multiple forms. For instance, it might permit both fanatical and anti-fanatical preferences. Or it might permit (or even, its name notwithstanding, require) incomplete preferences that are neither fanatical nor anti-fanatical. But apart from noting its existence, we will say no more about this third possibility for now, returning to it only in Section 7.

2.5 Expected Value and Expected Utility

Fanaticism is a thesis worth taking seriously, despite its counterintuitive implications, because there are substantial arguments in its favor. The most compelling arguments for fanaticism, which we will spell out formally in the next section, start from the idea that there are objective quantitative facts about the value of outcomes that constrain how agents should rank prospects. For instance, here’s a simple argument for a certain kind of prudential fanaticism: “Suppose you are facing gambles where the prizes are extensions to your lifespan – more specifically, additional years of happy life, at some fixed level of happiness. The prudential value of a year of life just depends on how happy you are, not on how many years you have lived already. So two years of life at a fixed happiness level is twice as good as one year, three years is three times as good, and so on. Similarly, for any probability $p$ , a $p$ chance of $n$ years of happy life is $n$ times as good as a $p$ chance of one year of happy life – meaning that you should be willing to pay $n$ times as much for such a chance. And surely, no matter how small $p$ is, you should be willing to pay some finite price (say, a fraction of a penny) for a $p$ chance of one additional happy life-year. But these claims together imply that, for any $p$ and any finite price, we can find some outcome (namely, a very large number of years of happy life) such that you are willing to pay that price for a $p$ chance of that outcome.”

This argument illustrates the more general idea of expected value maximization (hereafter abbreviated MEV). MEV starts from the assumption that outcomes have cardinal degrees of value, independent of any agent’s ranking of prospects, which can be represented by a value function $V$ mapping each outcome to a real number.Footnote ¹³ The expected value of a prospect $P_j$ is the probability-weighted sum of the values of all its possible outcomes:

${\mathbb{E}}_V(P_j)=\sum _i{P}_j(o_i)V(o_i)$

where $P_j(o_i)$ is the probability that $P_j$ yields outcome $o_i$ .

MEV requires agents to rank prospects by expected value. More precisely, I will understand it to consist of the following claims:

Maximize Expected Value (MEV)

It is rationally required that (i) if ${\mathbb{E}}_V(P_j)>{\mathbb{E}}_V(P_k)$ , then $P_j\succ P_k$ ; and (ii) if ${\mathbb{E}}_V(P_j)$ and ${\mathbb{E}}_V(P_k)$ are both finite and ${\mathbb{E}}_V(P_j)={\mathbb{E}}_V(P_k)$ , then $P_j\sim P_k$ .Footnote ¹⁴

The informal argument for prudential fanaticism above claims that, all else being equal, the objective prudential value of an outcome increases linearly with the number of happy life-years the agent enjoys in that outcome, and that one should maximize expected value. If there is no upper limit to the number of happy life-years that it’s possible for an individual to enjoy, then a prudential value function that increases linearly with happy life-years will be unbounded above, and maximizing the expectation of that value function entails Narrow Positive Fanaticism.

It’s important to distinguish MEV from the weaker and more widely accepted idea of expected utility maximization.

Maximize Expected Utility (MEU)

It is rationally required that there is some function $U$ from outcomes to real numbers (a “utility function”) such that (i) if ${\mathbb{E}}_U(P_j)>{\mathbb{E}}_U(P_k)$ , then $P_j\succ P_k$ ; and (ii) if ${\mathbb{E}}_U(P_j)$ and ${\mathbb{E}}_U(P_k)$ are both finite and ${\mathbb{E}}_U(P_j)={\mathbb{E}}_U(P_k)$ , then $P_j\sim P_k$ .

The requirement that an agent should rank prospects according to the expectation of some utility function is the central tenet of orthodox decision theory. It is typically defended on the grounds that an agent’s preferences ought to satisfy a set of coherence constraints (e.g., the axioms of von Neumann and Morgenstern (Reference von Neumann and Morgenstern1947) or Savage (Reference Savage1972)) which guarantee that the agent can be represented as maximizing the expectation of a utility function.

Even if there are cardinal facts about the values of outcomes, MEU in itself takes no cognizance of those facts. It just requires an agent to rank prospects according to the expectation of some utility function, which need not be identical with the objective value function or bear any particular relationship to it. In a moral context, though, it’s natural to suppose that one rationally ought to prefer certainty of a better outcome to certainty of a worse outcome. (This principle will be stated officially in Section 6 under the name Minimal Dominance.) Combined with MEU, this means that one’s utility function must be increasing with respect to the value of outcomes, assigning greater utility to better outcomes. And we will generally focus on utility functions that have this property.

Even within the constraint of assigning greater utility to better outcomes, however, MEU is much more permissive than MEV. To illustrate, consider a moral decision-making context in which lives are at risk, all of which have equal value, and the only morally relevant consideration is how many lives are saved or lost. In this context, it’s natural to think that the value of an outcome (relative to a baseline in which no one is saved) is proportionate to the number of lives saved: saving two lives is twice as good as saving one, etc. If that’s right, then MEV ranks prospects in this context by the expected number of lives saved, for example, preferring a prospect that saves $9$ lives with probability $0.5$ to a prospect that saves $4$ lives with certainty. MEU does not give such precise verdicts. It allows an agent, for instance, to maximize the expected square root of lives saved, in which case she would prefer to save $4$ lives for sure (with an expected utility of $1\times \sqrt{4}=2$ ) rather than $9$ lives with probability $0.5$ (with an expected utility of $0.5\times \sqrt{9}+0.5\times 0=1.5$ ). In other words, MEV requires risk-neutrality with respect to lives saved, while MEU is compatible with risk aversion or risk seeking.

Because MEV requires risk-neutrality with respect to a privileged value function, while MEU does not, these principles differ in their penchant for fanaticism. If there are no finite upper or lower bounds on the value of outcomes, then MEV implies Narrow Fanaticism (and any other natural formalization of fanaticism). And if moral value does have a privileged cardinal structure, it seems very plausible that it should be unbounded: For instance, if adding a happy or unhappy life to the world makes a contribution to its overall value that depends only on the intrinsic characteristics of that life, and if there is no upper bound on the number of lives (of a particular intrinsic character) that the world might contain, then value is unbounded.

MEU, on the other hand, allows both fanatical and nonfanatical preferences.Footnote ¹⁵ An agent’s utility function might simply be a linear function of moral value. But it might also dampen MEV’s fanaticism, by being concave (risk-averse) with respect to sufficiently good outcomes and convex (risk-seeking) with respect to sufficiently bad outcomes. More particularly, even if value is unbounded, and an agent’s utility function is an increasing function of value, her utilities can still be bounded (see Figure 1).Footnote ¹⁶ In this case, low-probability outcomes will carry only limited weight in her decisions, no matter how astronomically good or bad they are.

Figure 1

Some increasing functions from value to utility: a linear function (1) that’s unbounded above and below, an s-shaped function (2) that’s bounded above and below, and an everywhere-concave function (3) that’s bounded above but unbounded below.

3.1 The Spectrum Argument

Let’s begin with the simplest argument for fanaticism. Focusing on positive fanaticism (here as elsewhere, the negative and positive cases are exactly formally analogous), the essence of this argument is that we can get from certainty of a good outcome ( $⟨o^{+}⟩$ ) to a tiny probability of an astronomically good outcome coupled with near-certainty of a bad outcome ( $⟨o^{*+},p,o^{-}⟩$ ) by a series of small steps each of which seems intuitively like an improvement.Footnote ¹⁸ The argument relies on two principles. The first says, roughly, that one should always be willing to accept a very slight decrease in the probability of a desired outcome in exchange for a sufficiently great increase in its desirability.

Non-Timidity

It is rationally required that there is some $q<1$ such that, for any outcomes $o^{+}{\succ }_{\mathbb{O}}{o}^{-}$ and probability $p>0$ , there is an outcome $o^{++}$ such that $⟨o^{++},q^{\prime }p,o^{-}⟩\succ ⟨o^{+},p,o^{-}⟩$ for any $q^{\prime }\ge q$ .

Suppose, for instance, that you are facing a risky medical procedure, and can choose between two options: a “safe” option that maximizes your probability of surviving the procedure, and a “risky” option that will improve your quality of life if you do survive. Non-Timidity says that if the difference in your chance of survival is sufficiently small, and the difference in quality of life sufficiently great, then you should take the risky option.

The second principle says that an agent’s ranking of prospects should be transitive.

Transitivity

It is rationally required that, if $P_i$ is weakly preferred to $P_j$ and $P_j$ is weakly preferred to $P_k$ , then $P_i$ is weakly preferred to $P_k$ .

Transitivity is a very widely accepted principle in normative decision theory. Many find it simply incoherent that you should prefer a first thing to a second, and a second to a third, but fail to prefer the first to the third. And Transitivity can also be supported by dynamic consistency arguments (see for instance] Gustafsson, Reference Gustafsson2022, §4). It is far from uncontroversial, but entering into those controversies is beyond the scope of this work, so I will mostly take Transitivity for granted.

It is fairly easy to see that Non-Timidity and Transitivity together entail Narrow Positive Fanaticism: For any $o^{+}$ and $o^{-}$ , we can start from certainty of $o^{+}$ (i.e., $⟨o^{+},1,o^{-}⟩$ ) and work our way to a prospect of the form $⟨o^{*+},p^{\prime },o^{-}⟩$ by a series of improving steps, each time improving the more desirable outcome of the prospect while slightly reducing its probability by multiplying it by a factor of $q$ or greater. And if each prospect in this sequence is strictly preferred to the one before, then the long shot at the end of the sequence must be strictly preferred to the safe bet we started off with.Footnote ¹⁹ Call this the Spectrum Argument for fanaticism.

This argument rests on an appeal to intuition: the intuition of Non-Timidity, that one should be willing to accept some extra risk in exchange for a sufficiently great increase in reward. The opponent of fanaticism can always deny this intuition (Bottomley and Williamson, Reference Bottomley and Williamson2025, fn. 18) – or, less plausibly in my opinion, deny Transitivity. But the force of the argument is to weaken any presumption against fanaticism on the basis of intuition. We saw in the last section that fanaticism can be deeply counterintuitive. We now see that avoiding fanaticism also requires an agent to exhibit strange and counterintuitive preferences, whether by violating Non-Timidity or Transitivity.

3.2 More Setup: Outcomes as Populations

My main focus, however, will be on an argument for fanaticism that’s more complicated but also, I think, more compelling. This argument is couched in a formal framework for representing outcomes which we must first introduce. In this framework, the value of an outcome is determined by “local outcomes” realized at distinct “value locations.” These concepts can be interpreted in various ways. In a prudential context, for instance, “locations” could be identified with moments in an individual’s life, and “local outcomes” with possible states (e.g., experiential states) that an individual might be in at a particular moment. But the most common interpretation of the framework of locations, and the one that will make the following argument most compelling, is ethical. On this interpretation, locations are identified with welfare subjects (beings with morally significant welfare interests) and local outcomes with welfare states (specifications of all the features of an individual’s life that influence her welfare, like how long it lasts and what experiences it includes). I will informally assume this interpretation, and so use “welfare subject” or “individual” interchangeably with “value location,” and “welfare state” or “life” interchangeably with “local outcome.” But the formal arguments we consider won’t depend on this interpretation.

Our first task, then, is to set out a formal framework for describing outcomes in terms of local outcomes realized at locations. So, let $\mathcal{L}$ be an infinite set of possible locations, and let $\mathcal{W}$ be a set of local outcomes, including an element $\omega$ representing nonexistence. Local outcomes are ranked by a total preorder ${\succcurlyeq }_{\mathcal{W}}$ , where $w_i{\succcurlyeq }_{\mathcal{W}}{w}_j$ means that $w_i$ is at least as good as $w_j$ . An outcome can now be understood as a function $o:\mathcal{L}\to \mathcal{W}$ specifying local outcomes for each location (and thereby specifying, among other things, which locations exist). Any prospect $P$ then defines a local prospect $P^l$ for each location $l$ , a probability distribution over local outcomes.

As always, we will restrict our attention to finite outcomes, meaning here outcomes in which all but finitely many locations are assigned nonexistence. And we will continue to assume, implicitly, that the domain $\mathbb{P}$ of prospects includes at least all finitely supported probability distributions over these finite outcomes. At some points below, however, we’ll explicitly make the following stronger assumption:

Full Domain

The set of prospects $\mathbb{P}$ includes all discrete probability measures on $\mathbb{O}$ , including those that assign non-zero probability to a countable infinity of outcomes.

3.3 The Argument from Anteriority

We can now develop a second, and in my opinion more compelling, argument for fanaticism.Footnote ²⁰ The argument rests on two central ideas. The first is a kind of “individualism”: Our basic objects of concern should be individual value locations, rather than the world as a whole. Prospects, therefore, should be compared “from the perspective of” each value location. (I’ll use “from the perspective of” as shorthand for “considering only the interests of”; the talk of “perspectives” shouldn’t be taken literally.) If one prospect is better than another from the perspective of some locations, but worse from the perspective of others, then we must make tradeoffs between different locations. But if one prospect is better than another from the perspective of every location, then it is better overall; and if two prospects are equally good from the perspective of every location, then they are equally good overall. These claims are, of course, particularly plausible if we are identifying value locations with welfare subjects.

One apparent implication of this individualist stance is the following principle:

Anteriority (McCarthy et al., Reference McCarthy, Mikkola and Thomas2020)

If for each possible value location $l$ , $P_i$ and $P_j$ give the same probability distribution over local outcomes (i.e., $P_i^l=P_j^l$ for all $l\in \mathcal{L}$ ), then it is rationally required that $P_i\sim P_j$ .

This follows from two arguably more basic principles:

Ex Ante Pareto Indifference

If $P_i$ and $P_j$ are equally good from the perspective of every location, then it is rationally required that $P_i\sim P_j$ .

Individual Stochasticism

If $P_i$ and $P_j$ give $l$ the same probability distribution over local outcomes, then they are equally good from $l$ ’s perspective.

The second central idea is that there is a generic, repeatable way of making outcomes better or worse, that is always capable of “bridging the gap” between a better outcome and a worse outcome. This generic method could take either of two forms: changing the welfare of an existing individual, or adding new individuals. I will focus on the latter. The idea that we can always bridge the gap between better and worse outcomes by adding individuals to either outcome is precisified by the following principles:

Positive Additions

For any outcomes $o^{+}$ and $o^{-}$ , there is a number $n$ and a welfare level $w^{+}$ such that adding $n$ or more individuals with welfare $w^{+}$ to outcome $o^{-}$ , while leaving everything else unchanged, results in an outcome strictly better than $o^{+}$ (regardless of the identities of the added individuals).

Negative Additions

For any outcomes $o^{+}$ and $o^{-}$ , there is a number $n$ and a welfare level $w^{-}$ such that adding $n$ or more individuals with welfare $w^{-}$ to outcome $o^{+}$ , while leaving everything else unchanged, results in an outcome strictly worse than $o^{-}$ (regardless of the identities of the added individuals).

These principles will be rejected by those like Bader (Reference Bader, Arrhenius, Bykvist, Campbell and Finneron-Burns2022), who hold that populations of different sizes are always incomparable in overall value. Positive Additions will also be rejected by those who hold that some forms of suffering cannot be outweighed by any amount of positive welfare. But a very wide range of axiologies endorse both principles, including not just additive views like total utilitarianism, critical-level utilitarianism, and prioritarianism but also views like “variable-value utilitarianism” (Hurka, Reference Hurka1983) which treat additions to a population as having sharply diminishing marginal value.Footnote ²¹

The argument for fanaticism will need two more apparently innocuous principles. One is Transitivity, introduced above. The other is a very weak dominance principle.

Superdominance

If every possible outcome of $P_i$ is strictly better than every possible outcome of $P_j$ , then it is rationally required that $P_i$ is strictly preferred to $P_j$ .

Superdominance is among the weakest and least controversial principles of normative decision theory. It seems to follow self-evidently from the idea that we should evaluate an uncertain prospect based on the desirability of its possible outcomes. We will shortly find a surprising reason to question this principle, but for now, let’s take it on board.

Here, then, is the promised argument for fanaticism, which in light of its central premise we will call the Argument from Anteriority:

Table 1Illustration of the Argument from Anteriority. $Pop(o^{-})$ is the set of individuals who exist in $o^{-}$ and $Dist(o^{-})$ represents their welfare distribution in $o^{-}$ .

Table 1 Long description

The table has two panels, (A) and (B), each depicting a prospect as a matrix whose rows represent groups of individuals and whose columns represent states of nature. Panel (A) shows prospect P-up-arrow. There are m plus 1 states: s0 with probability p double prime, and s1 through s m each with probability (1 minus p double prime) by m. The first row represents the population of the baseline outcome o minus, who receive their baseline welfare distribution Dist(o minus) in every state. Below them are m groups of n added individuals each, labeled p1 to p n, p n plus 1 to p 2n, and so on up to p n (m minus 1) plus 1 to p n m. Each group receives welfare level w plus in state s0 and in exactly one of states s1 through s m (with the k-th group receiving w plus in state s k), and does not exist (indicated by a dash) in all other states. Thus, in state s0 all n times m added individuals exist with welfare w plus, while in each state s k only the k-th group of n individuals exists with welfare w plus. Panel (B) shows prospect P-star, obtained from P-up-arrow by permuting local prospects. The baseline population again receives Dist(o minus) in every state. Now, however, every group of added individuals receives w plus in states s0 and s1, and does not exist in states s2 through s m. The effect is to concentrate all n times m added individuals into the outcomes associated with states s0 and s1, while the outcomes in states s2 through s m are just o minus.

3.4 An Infinitary Puzzle

But there’s a problem for this argument: in certain infinitary contexts, the principles to which we appealed in support of fanaticism turn out to be inconsistent!

Table 2An illustration of the conflict between Anteriority, Superdominance, and Positive/Negative Additions. Each individual has the same probability of existing in both prospects (with “ —” representing nonexistence), and equal probabilities of a positive and a negative welfare level conditional on existence. But every outcome of $P^{+}$ is strictly better than the baseline outcome $o_b$ , while every outcome of $P^{-}$ is strictly worse.

Table 2 Long description

The table has two panels, (A) and (B), each depicting a prospect as a matrix whose rows represent groups of individuals and whose columns represent states s1, s2, s3, s4, and so on, with probabilities 1 by 2, 1 by 4, 1 by 8, 1 by 16, and so on, respectively. Panel (A) shows prospect P plus, labeled: surely better than baseline. The first row represents the baseline population Pop(o b), who receive their baseline welfare distribution Dist(o b) in every state. Below them is a sequence of added groups. The first addition (a group of n individuals) receives welfare w1 plus in state s1 and w1 minus in all subsequent states s2, s3, s4, and so on. The second addition (a group of m individuals) does not exist in state s1, receives w2 plus in state s2, and w2 minus in all subsequent states. The third addition does not exist in states s1 or s2, receives w3 plus in state s3, and w3 minus in all subsequent states. This pattern continues indefinitely: the k-th addition does not exist in states before s k, receives w k plus in state s k, and wk minus in all subsequent states. The welfare levels are chosen so that the positive additions always make the outcome in their good state strictly better than the baseline o b, overcoming the negative welfare of all previous additions in their bad states. As a result, every possible outcome of P plus is strictly better than o b. Panel (B) shows prospect P minus, labeled: surely worse than baseline. It has exactly the same structure, except that the welfare assignments in the good and bad states are swapped for each group: the first addition receives w1 minus in state s1 and w1 plus in all subsequent states, the second addition receives w2 minus in state s2 and w2 plus in all subsequent states, and so on. These welfare levels are chosen so that every possible outcome of P minus is strictly worse than o b.

This isn’t just an isolated problem with the particular set of premises we chose when making the case for fanaticism. It reflects a basic tension, in infinitary contexts, between “localist” principles (like Anteriority) that evaluate prospects from the perspective of particular locations or sets of locations considered in isolation and “universalist” principles (like Superdominance) that evaluate prospects by looking at the overall value of their possible outcomes.Footnote ²⁴ And it is likely to afflict any argument for fanaticism that relies on principles of both types. In particular, Russell (Reference Russell2023) and Kowalczyk (forthcoming) both show that pro-fanatical arguments conceptually related to the Anteriority-based argument we’ve been discussing run into exactly the same sort of trouble: the premises that imply fanaticism in a finitary context turn out to be inconsistent in an infinitary context.

3.5 Against Anteriority

This doesn’t necessarily mean that the Anteriority-based argument for fanaticism fails. That argument only relied on strictly finite applications of the various principles involved. And so it can be rescued from inconsistency either by restricting our domain (denying that infinitary prospects like $P^{+}$ and $P^{-}$ are even possible) or, more plausibly, restricting the scope of one or more of its premises. In particular, we might restrict either Anteriority or Superdominance so that they only apply to prospects with finitely many possible outcomes.Footnote ²⁵ This would require some sacrifice of intuition – after all, both these principles seem very plausible in their unrestricted forms, and the need to deny either of them in any context is counterintuitive. But infinities are counterintuitive in lots of ways, and many intuitive generalizations turn out to be true in finite contexts but false in infinite contexts. If we take this line, then the case for fanaticism is still going strong.

On the other hand, we might think that the need to give up either Anteriority or Superdominance in infinite contexts is telling us that one of those principles, despite its appeal, rests on a mistake. If the basic thoughts that motivated those principles do not distinguish between finite and infinite contexts, then we cannot restrict either principle without giving up its basic motivation; and once we have done that, we should no longer feel particularly attached to its finite rump. On this view, infinitary inconsistency results like the one above undercut the case for fanaticism.

I favor the latter response, though for somewhat idiosyncratic reasons. I will conclude this section by setting out those reasons, though I don’t expect them to convince everyone.

To my mind, the conflict between “individualist”/“localist” and “universalist” principles in infinite contexts indicates that we must make a fundamental choice between these two ways of evaluating prospects: focusing either on the perspective of each individual location, or on the overall value of each possible outcome. And it is the individualist perspective that rests on a mistake. The mistake is not in thinking that individuals are the fundamental bearers of value, or that our fundamental concern should be with their welfare. Rather, the mistake is in thinking that there is a morally privileged way of “matching up” individuals across different possible outcomes (a transworld identity or counterpart relation) that allows us to compare different outcomes or prospects from the perspective of one and the same individual. I’m skeptical that any such relation is metaphysically substantial enough to bear moral weight: I don’t know what it means, for instance, to ask whether some individual in a different possible world or a counterfactual scenario would be “identical with” some actual individual, and I don’t see how any “counterpart” relation between individuals in different possible worlds could be anything other than an arbitrary human creation (even if it is a great conceptual convenience in many contexts). But individualistic principles like Anteriority fundamentally depend on such identifications, since their application requires us to compare the welfare of the same individual, or counterpart individuals, across different outcomes.Footnote ²⁶

Without such identifications, the truth that individuals are the fundamental loci of value leads in a different direction: The absolute value of an outcome is grounded in the welfare of each particular individual who exists in that outcome. But the comparative value of two different outcomes is grounded not in an individual-by-individual comparison (which would require a privileged way of matching up individuals), but in the absolute values of each outcome. Likewise, the comparative rational desirability of two prospects is grounded not in an individual-by-individual comparison of local prospects, but in the absolute desirability of each prospect, which is grounded in the absolute values and probabilities of its various possible outcomes. Thus, the fact that the interests of individuals are what fundamentally matter (and ultimately ground the comparative value of outcomes and prospects) doesn’t imply that we must compare outcomes and prospects “from the perspective of” each possible individual. The appeal of principles like Anteriority rests on the natural but mistaken belief in the meaningfulness of cross-world identifications.

This picture supports an unrestricted version of Superdominance and, I will later argue, of stronger dominance principles as well. And while it does not rule out that Anteriority or similar principles might hold true at least in finite contexts, it undercuts our justification for believing them.

But there is also positive reason to doubt Anteriority in finite contexts. For instance, consider the following example:

Great Escape

Humanity has just learned, much to our collective chagrin, that all life on Earth will be inevitably wiped out in just fifty years: An intermediate-mass black hole is headed our way on a trajectory that will take it through the inner Solar System, either swallowing Earth or ejecting it into interstellar space. Our only hope for the long-term survival of humanity is to rapidly build an interstellar spacecraft that will let a few individuals escape to another star system and start over. There are two candidate designs for the drive system of this spacecraft, either of which would require such an investment of resources that it’s not possible to build both. Unfortunately, which design will allow us to successfully reach and settle another system depends on still-unresolved questions in fundamental physics: If Theory 1 is correct, then only Design 1 will be successful, while if Theory 2 is correct, only Design 2 will be successful. Scientists regard these two theories as equally likely. As it turns out, the theories also have distinctive cosmological implications. Theory 1 implies that the universe as a whole is very large (many, many orders of magnitude larger than the observable universe), so large that it almost certainly contains millions of other intelligent civilizations – though so far away that we will almost certainly never come in contact with them. Theory 2 implies that the universe is relatively small (not much larger than the observable universe), in which case – we now believe – humanity’s existence was a lucky fluke, and it is extremely unlikely that there is or will be intelligent life anywhere else in the universe.

The choice of which drive system to build, we can stipulate, makes no difference to any individual’s probability of ending up with a given individual outcome. The same individuals will be chosen to crew the spacecraft either way, and while the choice might affect which future individuals are born, we have no ex ante knowledge of those effects, and cannot say that any particular individual is more likely to be born or more likely to achieve a given welfare level if we choose one design rather than the other. Nevertheless, in light of the cosmological implications of the two theories, it strikes me as entirely reasonable to strictly prefer Design 2: In choosing Design 1, we run an especially great risk, for if Theory 2 turns out to be true, then we will be left with a universe forever more devoid of life and intelligence, and in which those things only ever existed as a tiny, ephemeral flicker. In choosing Design 2, we greatly reduce that risk, for it is very likely that either humanity will survive and flourish (if Theory 2 is correct) or other civilizations, elsewhere in the universe will do so (if Theory 1 is correct). That seems like a sufficient justification for preferring Design 2. It would even be rationally permissible, I think, to prefer and to choose Design 2 if it came at some significant extra cost (e.g., requiring a small additional tax on every member of the present population).

This case provides not just a motivation for rejecting Anteriority but one that is particularly germane to the question of fanaticism. What the case illustrates is that it is prima facie reasonable to have non-neutral risk attitudes toward the overall value of outcomes, like wanting to maximize the probability that the world achieves some threshold of value (e.g., the threshold corresponding to a long-lived, flourishing civilization). These risk attitudes are exactly what supply our intuitions against fanaticism: in both prudential and moral decision-making contexts, we are often driven by goals like maximizing the probability of a “good enough” outcome or minimizing the probability of a “catastrophic” outcome.Footnote ²⁷ These concerns with particular thresholds make us sensitive to correlations between individual prospects – correlations which Anteriority tells us to ignore (as, for instance, when it tells us that $P^{\uparrow }$ and $P^{*}$ are equally good, in the argument for fanaticism from Section 3.3). If Anteriority is well-motivated (in particular, by the idea that we must compare prospects by comparing them from the point of view of each individual), then it shows us that these concerns are irrational. But if those motivations are undercut, then we should stick with our judgment that threshold concerns of the sort illustrated by The Great Escape are reasonable and rational.Footnote ²⁸

4.1 Generalized St. Petersburg Prospects

The St. Petersburg lottery is a game in which a fair coin is flipped repeatedly until it lands tails. The player then receives a payoff of $ $2^n$ , where $n$ is the total number of flips. What is interesting about this game is that, because the player wins $2 with probability $\frac{1}{2}$ , $4 with probability $\frac{1}{4}$ , $8 with probability $\frac{1}{8}$ , and so on, the expected monetary reward from playing the game (the probability-weighted sum of its possible monetary payoffs) is infinite: $\mathrm{\$}1+\mathrm{\$}1+\mathrm{\$}1+\dots$ . A gambler who aims to maximize their expected wealth, therefore, should be willing to pay any finite price for a St. Petersburg lottery ticket. What makes this conclusion especially puzzling is that every possible payoff of the game is finite. Thus, expectational reasoning appears to tell us that the St. Petersburg game is worth more than any of its possible payoffs.

The classic response to this puzzle, in its original formulation, is to point out that money has diminishing marginal utility: the richer you are, the less you benefit from one more dollar. And the rational way to evaluate monetary gambles is by expected utility, not expected monetary reward. (Thus, for instance, anyone who is not already very rich should prefer certainty of $1 million to a 1% chance of $101 million.) Given a realistically concave function from money to utility, the St. Petersburg game has only finite expected utility.Footnote ²⁹

But this response by itself doesn’t resolve the basic decision-theoretic paradox. For as long as an expected utility maximizer’s function from wealth to utility has no upper bound, we can find a St. Petersburg-like gamble with infinite expected utility, to which she must apparently assign infinite value (Menger, Reference Menger1934). We can again imagine a fair coin flipped until it lands tails, and construct a prospect where if the coin lands tails on the first flip, the agent receives a monetary payoff with utility $x$ (or greater), if it lands tails on the second flip, she receives a monetary payoff with utility $2x$ (or greater), $4x$ on the third flip, $8x$ on the fourth flip, and so on. Thus, in the context of expected utility theory, the St. Petersburg game has often been taken to show that utility functions must be bounded.Footnote ³⁰

But the St. Petersburg paradox isn’t confined to expected utility theory. It turns out that Narrow Fanaticism, with modest auxiliary assumptions, implies that we can construct “Generalized St. Petersburg prospects” with the same paradoxical features as the original – specifically, prospects that are “improper” in that they must be strictly preferred, or strictly dispreferred, to any of their possible outcomes.Footnote ³¹ Here’s how we do it (focusing, as usual, on the positive case): Take two arbitrary outcomes $o^{+}$ and $o^{-}$ , with $o^{+}$ strictly better than $o^{-}$ . Narrow Fanaticism implies that there is an outcome $o_1$ such that $⟨o_1,\frac{1}{2},o^{-}⟩$ is strictly preferred to $⟨o^{+}⟩$ ; an outcome $o_2$ such that $⟨o_2,\frac{1}{4},o^{-}⟩$ is strictly preferred to $⟨o_1⟩$ ; an outcome $o_3$ such that $⟨o_3,\frac{1}{8},o^{-}⟩$ is strictly preferred to $⟨o_2⟩$ ; and so on. Now, consider the following prospect:

Generalized St. Petersburg

A fair coin is flipped until it lands tails, yielding the outcome $o_n$ , where $n$ is the total number of flips. This generates the prospect $⟨(o_1,\frac{1}{2}),(o_2,\frac{1}{4}),(o_3,\frac{1}{8}),\dots ⟩$ .

To show that Generalized St. Petersburg is strictly preferable to each of its possible outcomes (or, strictly speaking, to the prospects that yield those outcomes with certainty), we need one new principle.

Stochastic Dominance

If for any outcome $o$ , $P_i$ is at least as likely as $P_j$ to yield an outcome at least as good as $o$ , then it is rationally required that $P_i\succsim P_j$ . If in addition, for some outcome $o$ , $P_i$ is strictly more likely than $P_j$ to yield an outcome at least as good as $o$ , then it is rationally required that $P_i\succ P_j$ .

When the first of the above conditions is met, we say that $P_i$ weakly stochastically dominates $P_j$ . When the second condition is also met, we say that $P_i$ strictly stochastically dominates $P_j$ .

Though it is substantially stronger than the principle of Superdominance we relied on in the last section, Stochastic Dominance is still one of the least controversial principles in normative decision theory.Footnote ³² Various arguments can be made in its favor. But the most compelling argument, to my mind, appeals to the idea that what ultimately matters in evaluating prospects, and gives us reason to form particular preferences between them, are the values and probabilities of their potential outcomes. When $P_i$ and $P_j$ offer the same probabilities of achieving any given level of value, then there is no evaluatively significant consideration favoring either prospect, and the only rationally appropriate attitude is indifference. When $P_i$ strictly stochastically dominates $P_j$ , there is a contrastive consideration in favor of $P_i$ (there is some threshold or “aspiration level” of value that it gives you a better chance of achieving), but no contrastive consideration in favor of $P_j$ : for any outcome one might point to as an inducement to choose $P_j$ , $P_i$ is at least as likely to yield an outcome at least that good. And these seem like decisive grounds for preferring $P_i$ .Footnote ³³

Stochastic Dominance and Transitivity together imply that Generalized St. Petersburg is strictly preferable to $⟨o_n⟩$ , for each of its possible outcomes $o_n$ . Narrow (Positive) Fanaticism allowed us to stipulate that, for every $n$ , the prospect $⟨o_{n+1},\frac{1}{2^{n+1}},o^{-}⟩$ is strictly preferred to $⟨o_n⟩$ . By Stochastic Dominance, Generalized St. Petersburg is strictly preferred to $⟨o_{n+1},\frac{1}{2^{n+1}},o^{-}⟩$ (since it has the same probability of yielding outcome $o_{n+1}$ , and all its other possible outcomes are strictly better than $o^{-}$ ). So, by Transitivity, Generalized St. Petersburg is strictly preferred to $⟨o_n⟩$ , for every $n$ .

4.2 Money-Pumping Fanatics

The existence of prospects like Generalized St. Petersburg is not just odd, but has a number of seriously undesirable consequences. In particular, an agent who strictly prefers (or disprefers) some prospect to all its possible outcomes will, given plausible principles of sequential choice, sometimes make sequences of choices that are certainly worse than available alternatives.

A sequential choice situation that leads an agent, choosing on the basis of her preferences, to make a sequence of choices that is unambiguously worse than an available alternative is called a money pump.Footnote ³⁴ Here is an example of the sort of money pump to which fanatics are vulnerable:Footnote ³⁵

Don’t Let the Green Grass Fool You

A fair blue coin will be flipped until it lands tails. You are offered, initially, a choice between two prospects based on these coin flips. The possible outcomes of these prospects come from a sequence $o_1$ , $o_2$ , etc where, as above, $⟨o_{n+1},2^{-(n+1)},o^{-}⟩$ is always strictly preferred to $⟨o_n⟩$ , for some baseline outcome $o^{-}$ strictly worse than $o_1$ . The prospect ${\mathrm{B}\mathrm{l}\mathrm{u}\mathrm{e}}^{+}$ will give you outcome $o_2$ if the blue coin lands tails on the first flip, $o_3$ if it lands tails on the second flip, etc. The prospect ${\mathrm{B}\mathrm{l}\mathrm{u}\mathrm{e}}^{++}$ will give you $o_3$ if the blue coin lands tails on the first flip, $o_4$ if it lands tails on the second flip, etc. Thus, for each possible outcome of the blue coin tosses, ${\mathrm{B}\mathrm{l}\mathrm{u}\mathrm{e}}^{++}$ will give you a strictly better outcome than ${\mathrm{B}\mathrm{l}\mathrm{u}\mathrm{e}}^{+}$ .

But there’s a catch: If you choose ${\mathrm{B}\mathrm{l}\mathrm{u}\mathrm{e}}^{++}$ , then after seeing the flips of the blue coin, you will be offered the chance to switch to a different prospect. This alternative prospect is based on the flips of a green coin, also fair and uncorrelated with the blue coin. The prospect Green will give you $o_1$ if the green coin lands tails on the first flip, $o_2$ if it lands tails on the second flip, etc.

This sequence of choices is illustrated in Figure 2. (In this diagram, squares represent choices, circles represent chance events, and numbers on the lines connecting two nodes represent probabilities.)

Figure 2

A money pump for fanatics

Figure 2 Long description

The tree starts with a square node on the left labeled 0. One branch leads from this node to the Blue superscript plus. Another leads to a circular node labeled blue coin. Three branches from this node are shown, with ellipses indicating infinitely more unshown. The three shown branches are labeled with probabilities one-half, one-quarter, and one-eighth, leading to square nodes labeled 1, 2, and 3, respectively. Each of these nodes has two branches, with the top branches leading to the prospect Green and the bottom branches leading to the outcomes: O subscript 3, O subscript 4, and O subscript 5, respectively.

How should you act in this situation, given your fanatical preferences? Intuitively, you should reason as follows: You know that you prefer Green to each of its possible outcomes, and therefore to each of the possible outcomes of ${\mathrm{B}\mathrm{l}\mathrm{u}\mathrm{e}}^{+}$ and ${\mathrm{B}\mathrm{l}\mathrm{u}\mathrm{e}}^{++}$ . If you choose ${\mathrm{B}\mathrm{l}\mathrm{u}\mathrm{e}}^{++}$ , therefore, after learning its outcome – whatever that outcome turns out to be – you will prefer to switch to Green. Thus your initial choice (at node 0) is really not between ${\mathrm{B}\mathrm{l}\mathrm{u}\mathrm{e}}^{+}$ and ${\mathrm{B}\mathrm{l}\mathrm{u}\mathrm{e}}^{++}$ , but between ${\mathrm{B}\mathrm{l}\mathrm{u}\mathrm{e}}^{+}$ and Green. And you should prefer ${\mathrm{B}\mathrm{l}\mathrm{u}\mathrm{e}}^{+}$ to Green, since it gives you the prospect $⟨(o_2,\frac{1}{2}),(o_3,\frac{1}{4}),(o_4,\frac{1}{8}),\dots ⟩$ instead of $⟨(o_1,\frac{1}{2}),(o_2,\frac{1}{4}),(o_3,\frac{1}{8}),\dots ⟩$ (and $o^{n+1}$ is in every case strictly better than $o_n$ ). So you should choose ${\mathrm{B}\mathrm{l}\mathrm{u}\mathrm{e}}^{+}$ . But this means that, with certainty, you will end up with a worse outcome than you could have had by choosing ${\mathrm{B}\mathrm{l}\mathrm{u}\mathrm{e}}^{++}$ and declining the option to switch to Green. And that seems bad.

To turn this into a more formal argument against fanaticism, we need three principles of rationality for sequential choice:

Preferential Choice

When an agent faces a terminal choice (one not followed by any further choice nodes), she is rationally required to choose an act whose prospect is maximally preferred. That is, she is rationally prohibited from choosing an act if there is another act whose prospect she strictly prefers.

Sophisticated Choice

When an agent faces a choice where, conditional on any act she might choose, she can predict with certainty what choice she will make at any subsequent choice nodes, then she is rationally required to treat those predictable future choices as fixed and to choose an act that leads to a maximally preferred prospect given those choices.

Non-Domination

If an agent has rational preferences, chooses rationally based on those preferences, knows the structure of the sequential choice situation she faces, and is certain that at future nodes she’ll choose rationally, update her beliefs rationally, and not change her preferences, then she will not take any action that is dominated in the sense that any pure strategy that takes that action is stochastically dominated by a strategy that does not.

In the preceding example, Stochastic Dominance and Transitivity imply that you strictly prefer the prospect Green to every possible outcome of ${\mathrm{B}\mathrm{l}\mathrm{u}\mathrm{e}}^{++}$ (as we saw in Section 4.1). Preferential Choice therefore implies that you will be rationally required to choose Green at each of nodes 1, 2, 3, etc. Sophisticated Choice then implies that, if you know the structure of the choice situation and are certain of your own future rationality, you should choose ${\mathrm{B}\mathrm{l}\mathrm{u}\mathrm{e}}^{+}$ at node 0 if and only if you prefer ${\mathrm{B}\mathrm{l}\mathrm{u}\mathrm{e}}^{+}$ to Green. Stochastic Dominance implies that you do prefer ${\mathrm{B}\mathrm{l}\mathrm{u}\mathrm{e}}^{+}$ to Green. Therefore, if you choose rationally based on your preferences, you’ll go up at node 0. Any pure strategy that goes up at node 0 yields the prospect ${\mathrm{B}\mathrm{l}\mathrm{u}\mathrm{e}}^{+}$ . But there’s an available strategy that yields the stochastically dominant prospect ${\mathrm{B}\mathrm{l}\mathrm{u}\mathrm{e}}^{++}$ – namely, going down at every choice node, that is, choosing ${\mathrm{B}\mathrm{l}\mathrm{u}\mathrm{e}}^{++}$ and sticking with it. (Indeed, ${\mathrm{B}\mathrm{l}\mathrm{u}\mathrm{e}}^{++}$ statewise dominates ${\mathrm{B}\mathrm{l}\mathrm{u}\mathrm{e}}^{+}$ , since it yields a better outcome regardless of the outcome of the blue coin flips.) Thus, by Non-Domination, your preferences must be irrational.

Although this example relied on Narrow Positive Fanaticism, we could give an exactly parallel argument for Narrow Negative Fanaticism, using a “negative” St. Petersburg prospect with an infinite sequence of increasingly bad outcomes. We can therefore conclude the following:

I see this as quite a strong argument against fanaticism. Stochastic Dominance and Transitivity are very plausible and widely accepted, and the uses to which those principles were put in the preceding argument are more clear-cut than some of the edge cases in which they have been challenged. Sophisticated Choice strikes me as clearly correct, and in any case, it’s not obvious that giving it up would help the fanatic avoid money pumps.Footnote ³⁷ One option for the fanatic is to give up Preferential Choice in favor of “resolute choice,” which holds roughly that at the start of a sequential choice situation, you should adopt a strategy that yields a maximally preferred prospect from your present perspective, then choose according to that strategy even if it contradicts your later preferences. This is a very general strategy for avoiding money pumps. But it relies on attributing a dubious normative authority to past decisions. It’s easy to see why you should follow through on your plans when the temptation to deviate from them stems from weakness of will or other forms of irrationality, but much harder to see why the preferences of your past self should override the rationally formed preferences of your present self in determining what it’s rational for you to do right now.Footnote ³⁸

The best option for the fanatic, I think, is to deny Non-Domination. It certainly seems strange that rationality should require an agent to make dominated choices. One thing we would like norms of practical rationality to do is to make us effective at pursuing desirable outcomes; forcing us to settle for prospects that are unambiguously worse than available alternatives seems antithetical to that. But perhaps in infinitary contexts, absolute invulnerability to this sort of failure is simply too much to ask of rationality (Arntzenius et al., Reference Arntzenius, Elga and Hawthorne2004). If a combination of basic moral and decision-theoretic principles (like Anteriority, Positive/Negative Additions, Superdominance, and Transitivity) require us to have certain preferences, then we must follow those preferences where they lead. And the fact that we cannot reliably coordinate our choices at different times and different branches of the decision tree to secure the prospect we might ideally like (e.g., cannot commit ourselves to going down at nodes 1, 2, 3, etc, thereby securing the prospect ${\mathrm{B}\mathrm{l}\mathrm{u}\mathrm{e}}^{++}$ ) should be regarded as an unfortunate feature of our situation, not proof of irrationality. If I were convinced of principles that entailed fanaticism, then I would at least be open to biting the bullet in this way. Nevertheless, vulnerability to money pumps is clearly an unwelcome conclusion, and puts significant weight on the scales against fanaticism.

6.1 Small Probabilities and Small Differences in Probability

Narrow Fanaticism and Narrow Anti-Fanaticism are attractively simple theses. But they don’t fully capture the question of whether we should give potentially unlimited weight to arbitrarily small probabilities. That’s because they only consider a special case, where an agent is choosing between certainty on the one hand and a binary prospect involving a tiny probability of an astronomically good/bad outcome on the other. The more general (and more realistic) case is a choice between two prospects that differ slightly in how they distribute probability between much better and much worse outcomes – in other words, a case where the agent must decide how much she is willing to pay in order to shift a small amount of probability from a much worse outcome to a much better outcome.

To see the difference, consider the two choice situations described in Tables 3a and 3b, each of which involves a choice between two prospects, with uncertainty between two possible states of nature. The first is a simple choice between certainty of a small gain ( $1$ ) and a small probability ( $2\epsilon$ ) of a large gain ( $\frac{1}{\epsilon }$ ). The second case is slightly more complicated: Here the astronomically large gain $\frac{1}{\epsilon }$ has a “baseline” probability of $0.5-\epsilon$ , and the choice is between taking a sure gain (improving each outcome by 1) or slightly increasing that baseline probability (by $2\epsilon$ ) – in this case, by moving the astronomically good outcome from a less probable to a more probable state. In the second case, small probabilities are nowhere to be seen – every state, and every outcome, has a quite substantial probability. What is small is the difference in probability between $s_1$ and $s_2$ , and hence between the probabilities of a very good outcome associated with $P_3$ and $P_4$ respectively. It seems to me, though, that anyone who finds expected value maximization counterintuitively fanatical in the first case will have the same intuition about the second case.

Table 3Left: A simple choice between certainty of a small gain and a small probability of a large gain. Right: A small probability difference without a small probability. The choice is between a sure gain of $1$ and a small $(2\epsilon )$ increase in the probability of a large gain $\left(\frac{1}{\epsilon }\right)$ .

Table 3 Long description

The table has two panels, (A) and (B), each showing a two-by-two decision table with rows corresponding to prospects and columns corresponding to states of nature. Outcomes of each prospect in each state are represented by numerical values. Panel (A), labeled Small Probability, has states s1 with probability 1 to 2 epsilon and s2 with probability 2 epsilon. Prospect P1 yields 1 in both states. Prospect P2 yields 0 in state s1 and 1 by epsilon in state s2. Panel (B), labeled Small Probability Difference, has states s1 with probability 0.5 to epsilon and s2 with probability 0.5 plus epsilon. Prospect P3 yields (1 by epsilon) plus 1 in state s1 and 1 in state s2. Prospect P4 yields 0 in state s1 and 1 by epsilon in state s2.

Choices like 3b are also much more common and realistic than choices like 3a. Consider, for instance, two important real-world cases: voting and existential risk mitigation. If you are deciding whether it is worth your while to vote in an election, in terms of the difference you might make to the outcome, it’s never the case that your preferred candidate is certain to lose if you don’t vote, but has a tiny probability of winning if you do; nor that they are certain to win if you vote, but have a tiny probability of losing if you don’t. Rather, they have some intermediate probability of winning, which your vote would slightly increase. Similarly, if you are deciding whether to devote some unit of resources to mitigating existential risks (from engineered pandemics, nuclear war, AI or the like) or to providing some more certain good (like direct cash transfers to the poor), the situation is not that near-term human extinction is certain if you do not act to prevent it, and your intervention represents humanity’s only slim hope of survival; nor that humanity is certain to survive if you act to assure its survival, and the only chance of doom comes from your inaction. Rather, there is some nontrivial probability that humanity will succumb to a near-term existential catastrophe, and some nontrivial probability that it will not, and your intervention (if well-chosen) might very slightly reduce the former probability and increase the latter.

Insofar as we find fanaticism counterintuitive, its counterintuitiveness is not confined to simple choices between risk and certainty, but extends to the more general case of small changes to an uncertain baseline prospect. And insofar as anti-fanaticism is meant to resist fanatical applications of expected value reasoning in real-world contexts, it must apply to this more general case as well. The thesis must be that it is irrational to trade certainty of a substantial gain for arbitrarily small shifts in probability from one outcome to another.

6.2 Anti-Fanaticism Generalized

Our next task, then, is to make that informal thesis precise. To do this, we need to introduce two new concepts. In the simple context of small probabilities, the “safe” option could be characterized as yielding a single outcome with certainty. In the more general context, we assume that astronomically better and astronomically worse outcomes will have some non-zero probability whatever option one chooses, and so the safe option can no longer be characterized in that way. We can characterize it instead, however, as offering a sure improvement to the outcome of the baseline prospect. To illustrate: In the case of voting, if you choose not to vote, you can instead spend an hour watching television. This doesn’t deliver a single outcome with certainty since (among other things) it leaves you uncertain who will win the election – but it does mean that both possible election outcomes will be improved, from your perspective, by the addition of one hour of television. Similarly, if instead of spending a fixed sum of money to mitigate existential risks, you spend it on direct cash transfers to the very poor, you are not left with certainty of any particular outcome, but instead with certainty that whatever otherwise would have happened will be improved by certain very poor people being made slightly less poor.Footnote ⁴⁰

We therefore introduce the following two concepts:

• An improvement is a function $I:\mathbb{O}\to \mathbb{O}$ that maps every outcome to a strictly better outcome if one exists, or else to an equally good outcome.

• A worsening is a function $W:\mathbb{O}\to \mathbb{O}$ that maps every outcome to a strictly worse outcome if one exists, or else to an equally good outcome.

Paradigmatically, an improvement might be a fixed change that can be applied to any outcome, like giving one individual an additional year of life at a fixed level of positive well-being. But the concept is more flexible than that, and does not presuppose that there is any concrete change that can be applied to every possible outcome and that always makes an outcome (even weakly) better. An improvement (or worsening) could correspond to intuitively very different concrete changes to different outcomes.

In rough terms, the generalized form of anti-fanaticism will say (in the positive case) that a large enough improvement to every outcome in a baseline prospect should always be preferred to a small enough shift in probability from one outcome to another, no matter how disparate those outcomes might be. But this gloss requires an important caveat: One extreme way of being an anti-fanatic is to hold that some outcomes and prospects are “good enough,” in the sense that a rational agent may be completely indifferent to any further improvements. Formally, let’s say that a prospect $P$ is maximal if no other prospect is strictly preferred to it, and that an outcome $o$ is maximal if the prospect $⟨o⟩$ is maximal. (Likewise, $P$ is minimal if no other prospect is strictly dispreferred to it, and $o$ is minimal if $⟨o⟩$ is minimal.) In a choice between a sure improvement and a small probability shift, the prospect resulting from the small shift might be maximal, either because the baseline prospect itself consisted entirely of maximal outcomes or because it involved only a small probability of a nonmaximal outcome, which the small probability shift eliminates. In this case, the anti-fanatic need not require that the sure improvement be strictly preferred.

These points in hand, we can now state a more general version of anti-fanaticism:

General Positive Anti-Fanaticism

It is rationally required that there is some improvement $I$ and probability $p>0$ such that, for any outcomes $o^{*+}{\succ }_{\mathbb{O}}{o}^{-}$ and any probability $q<1$ , the prospect $⟨I(o^{*+}),q,I(o^{-})⟩$ is weakly preferred to $⟨o^{*+},q+p^{\prime },o^{-}⟩$ for any $p^{\prime }\le p$ (and $\le 1-q$ ). Moreover, this preference is strict unless $o^{*+}$ and $o^{-}$ are both maximal, or $o^{*+}$ is maximal and $q+p^{\prime }=1$ .

General Negative Anti-Fanaticism

It is rationally required that there is some worsening $W$ and probability $p>0$ such that, for any outcomes $o^{+}{\succ }_{\mathbb{O}}{o}^{*-}$ and any probability $q<1$ , the prospect $⟨o^{*-},q+p^{\prime },o^{+}⟩$ is weakly preferred to $⟨W(o^{*-}),q,W(o^{+})⟩$ for any $p^{\prime }\le p$ (and $\le 1-q$ ). Moreover, this preference is strict unless $o^{+}$ and $o^{*-}$ are both minimal, or $o^{*-}$ is minimal and $q+p^{\prime }=1$ .

Intuitively, General Positive Anti-Fanaticism says that there is some improvement $I$ large enough, and (non-zero) probability $p$ small enough, that given a choice between (i) improving every outcome in your “baseline” prospect by $I$ or (ii) increasing the probability of an astronomically good outcome by $p$ , you always at least weakly prefer the former. For example, perhaps you should always prefer adding twenty happy years to someone’s life over increasing the probability of any outcome (no matter how good) by ${10}^{-30}$ or less. And General Negative Anti-Fanaticism says that there is some worsening $W$ large enough, and (non-zero) probability $p$ small enough, that given a choice between (i) worsening every outcome in your baseline prospect by $W$ or (ii) increasing the probability of an astronomically bad outcome by $p$ , you always at least weakly prefer the latter. For instance, perhaps you should always be willing to increase the risk of any outcome (no matter how bad) by ${10}^{-30}$ or less rather than shorten a happy life by twenty years. General Anti-Fanaticism is the conjunction of these two theses.

General Anti-Fanaticism is stronger than Narrow Anti-Fanaticism, since it universally quantifies over the baseline probability $q$ , where Narrow Anti-Fanaticism only covers the special case of $q=0$ .Footnote ⁴¹

6.3 The Argument from Acyclicity

Anyone who wants to do justice to our anti-fanatical intuitions and to resist fanatical real-world applications of expected value reasoning must, I have argued, endorse not just Narrow but General Anti-Fanaticism. But unfortunately, this thesis is subject to a very powerful objection. To state the objection, we need to introduce a few new principles.

No Best Outcome

For every outcome, there is a strictly better outcome.

No Worst Outcome

For every outcome, there is a strictly worse outcome.

Minimal Dominance

If $o_i{\succ }_{\mathbb{O}}{o}_j$ , then it is rationally required that $⟨o_i⟩\succ ⟨o_j⟩$ .

Acyclicity

It is rationally required that, if $P_1\succ P_2$ , $P_2\succ P_3$ , $\dots$ , $P_{n-1}\succ P_n$ , then it’s not the case that $P_n\succ P_1$ .

These principles are very weak, and hard to deny. No Best Outcome and No Worst Outcome do not imply anything like unboundedness of cardinal value or utility. They only assert that there is always some way of making an outcome at least a little better/worse – for instance, by extending a happy/unhappy life, or adding a positive/negative experience to a life without changing its duration. Minimal Dominance is perhaps the weakest possible expression of the idea that the desirability of a prospect depends on the value of its possible outcomes. It’s much weaker than Stochastic Dominance, and even weaker than Superdominance.Footnote ⁴² Finally, Acyclicity is the least controversial of the standard coherence constraints on rational preference associated with expected utility theory. It is implied by, but significantly weaker than, Transitivity. And it is supported by a particularly compelling and straightforward money-pump argument (Gustafsson, Reference Gustafsson2022, Ch. 2).

But surprisingly, General Anti-Fanaticism is incompatible with these very weak principles. Specifically:

Table 4An illustration of the cyclicity objection to General Anti-Fanaticism

Table 4An illustration of the cyclicity objection to General Anti-Fanaticism

	${\mathit{s}}_{\mathbf{\text{1}}}$ $\left(\frac{\mathbf{\text{1}}}{\mathbf{\text{n}}}\right)$	${\mathit{s}}_{\mathbf{\text{2}}}$ $\left(\frac{\mathbf{\text{1}}}{\mathbf{\text{n}}}\right)$	${\mathit{s}}_{\mathbf{\text{3}}}$ $\left(\frac{\mathbf{\text{1}}}{\mathbf{\text{n}}}\right)$	$\mathbf{\cdots }$	${\mathit{s}}_{\mathbf{\text{n}}}$ $\left(\frac{\mathbf{\text{1}}}{\mathbf{\text{n}}}\right)$
$P_0$	$I^n(o)$	$I^n(o)$	$I^n(o)$	$\cdots$	$I^n(o)$
$P_1$	$I(o)$	$I^{n+1}(o)$	$I^{n+1}(o)$	$\cdots$	$I^{n+1}(o)$
$P_2$	$I^2(o)$	$I^2(o)$	$I^{n+2}(o)$	$\cdots$	$I^{n+2}(o)$
$⋮$	$⋮$	$⋮$	$⋮$	$\ddots$	$⋮$
$P_n$	$I^n(o)$	$I^n(o)$	$I^n(o)$	$\cdots$	$I^n(o)$

General Positive Anti-Fanaticism implies that each step from $P_{i-1}$ to $P_i$ is a strict improvement: In a choice between improving every outcome by $I$ or shifting probability $\frac{1}{n}\le p$ to an astronomically better outcome, we always prefer the former. Thus, $P_1\succ P_0$ , $P_2\succ P_1$ , and so on. But at the end of this sequence of strict improvements, we end up exactly where we started – $P_n$ is identical to $P_0$ ! So we have a cycle: $P_n\succ P_{n-1}\succ \dots P_1\succ P_n$ .

This result constitutes an argument against General Anti-Fanaticism, which I will call the Argument from Acyclicity. This argument strikes me as quite compelling. In particular, it relies on much weaker premises than extant arguments for fanaticism, like the Spectrum Argument, the Argument from Anteriority, and other arguments discussed in the recent literature (e.g., by Wilkinson, Reference Wilkinson2022; Russell, Reference Russell2023; Beckstead and Thomas, Reference Beckstead and Thomas2024), which rely on substantial ethical assumptions like Prospect Separability (fn. 28), or substantive constraints on agents’ risk attitudes. No Best/Worst Outcome and Minimal Dominance strike me as almost undeniable. The easiest principle for the anti-fanatic to give up, I think, is Acyclicity. The cost of this move is somewhat mitigated by the fact that the cycles into which General Anti-Fanaticism leads us may be extremely long. For instance, an agent who ignores probability differences smaller than ${10}^{-15}$ need not have any preference cycles shorter than ${10}^{15}$ steps. (This will, among other things, make her fairly difficult to money pump – it will require a setup in which she faces, or believes herself to face, a potential sequence of choices at least ${10}^{15}$ nodes long.) Still, we should not let the juxtaposition with other, even-less-deniable principles fool us: Acyclicity is an extremely plausible principle, and denying it is a very serious cost. I conclude, therefore, that we should reject General Anti-Fanaticism.Footnote ⁴⁴

6.4 Bounded MEU

Let’s now consider what the preceding argument tells us about two views in normative decision theory that have been treated as paradigmatic forms of anti-fanaticism: bounded expected utility maximization (Bounded MEU) and small-probability discounting.

Beginning with the former: Bounded MEU combines MEU (as stated in Section 2.5) with the claim that utilities should be bounded: There should be some real numbers $\bar{u}$ and $\underset{\_}{u}$ such that no outcome receives a utility greater than $\bar{u}$ or less than $\underset{\_}{u}$ . This need not imply that there is a highest-utility or a lowest-utility outcome, since the utilities of outcomes may approach the bounds without ever reaching them. It does, however, satisfy Narrow Anti-Fanaticism: For instance, given a choice between certainty of an outcome with near-maximal utility and a risky prospect that is almost certain to yield a significantly worse outcome, any long-shot reward will be at most slightly better than the sure thing, and so insufficient to justify the risk.

Bounded MEU is perhaps the most orthodox form of orthodox decision theory. The requirement of boundedness is commonly motivated by the need to avoid St. Petersburg-like prospects (see fn. 30), whose paradoxical features are not just counterintuitive but also in tension with central features of expected utility theory.Footnote ⁴⁵ But in addition to these more technical arguments, some expected utility theorists have seen the avoidance of counterintuitive fanaticism as a sufficient justification for boundedness.Footnote ⁴⁶

On the other hand, Bounded MEU itself has some fairly counterintuitive features. It requires extreme risk-aversion with respect to outcomes near the upper bound of the utility function, and extreme risk-seeking for outcomes near the lower bound. (For discussion of this point, see § $3.3$ of Beckstead and Thomas (Reference Beckstead and Thomas2024).) And, in the moral context, the combination of Bounded MEU with many plausible outcome axiologies will result in violations of individualist principles like Anteriority.Footnote ⁴⁷

But my main purpose here isn’t to argue against Bounded MEU. (In particular, as discussed in Section 3.5, I don’t think that conflict with Anteriority is a decisive objection.) Rather, the upshot of the preceding discussion is that despite appearances (and despite satisfying Narrow Anti-Fanaticism), Bounded MEU in its standard form is not generally anti-fanatical: Since Bounded MEU is acyclic, Proposition 4 implies that, given No Best Outcome, No Worst Outcome, and Minimal Dominance, it will not satisfy General Anti-Fanaticism.

The intuitive explanation for this fact (focusing, as usual, on the positive case) is that as we approach the upper bound of the utility function, the marginal utility of any given improvement must decrease very rapidly. Consider, for instance, a choice between shifting a small amount of probability from a mediocre outcome (near the middle of the utility function) to an astronomically good outcome (near the upper bound), or else applying some fixed improvement to every outcome. If the baseline probability of the astronomically good outcome is already high, then the increment to expected utility from the latter option will be very small, since the improvement is most likely to be applied to an outcome that already has nearly maximal utility.

As a concrete illustration, consider a choice between slightly reducing the risk of premature human extinction and creating some fixed benefit, for example, saving a life. Suppose you think that the baseline risk of premature extinction is fairly low, and that if we avoid it, we are very likely to achieve a very good future. Then, as a bounded expected utility maximizer, you will be likely to prefer even a tiny reduction in the already-small risk of human extinction to a sure improvement that is very likely to improve an already-very-good world, and so counts for very little.

It’s possible for a bounded expected utility maximizer to satisfy General Anti-Fanaticism, by assigning some outcome(s) maximal utility and some outcome(s) minimal utility. (We might call this Compact MEU, since it requires the range of the utility function to be not just bounded but compact.) General Anti-Fanaticism is then satisfied because (focusing on the positive case) we can find an improvement that maps every outcome to an outcome with maximal utility. Applying this improvement to every outcome in a baseline prospect yields a prospect with maximal expected utility, which is weakly preferred to any long-shot alternative, and strictly preferred unless that alternative consists entirely of maximal outcomes. Unlike the standard form of Bounded MEU, however, this view must give up either No Best/Worst Outcome or Minimal Dominance. I won’t say more about Compact MEU, for the sake of space and because it strikes me as quite implausible. (I say a bit more in Tarsney (2025a, §6).) But it shows that it’s possible, in principle, to satisfy General Anti-Fanaticism within the confines of expected utility theory, without preference cycles.

6.5 Small-Probability Discounting

The other commonly proposed strategy for avoiding both fanaticism and the paradoxical implications of St. Petersburg-like prospects is to simply ignore small probabilities. This idea has a long history, going back at least to a 1714 letter from Nicolaus Bernoulli to Pierre Rémond de Montmort. And it has recently experienced something of a revival, being advocated by Smith (Reference Smith2014) and Monton (Reference Monton2019), and seriously entertained by Buchak (Reference Buchak2013, 73–74) and Hong (Reference Hong2024).Footnote ⁴⁸

Small-probability discounting can take many forms. The simplest forms are state discounting, which ignores very improbable states, and outcome discounting, which ignores very improbable outcomes. The very simplest versions of these approaches, which tell us to ignore all states or outcomes with probabilities below some fixed threshold $t$ , seem unworkable, since there will be situations where all states and outcomes have probabilities below that threshold (Arrow, Reference Arrow1951, 414–415). But more sophisticated versions of state/outcome discounting avoid this problem, for example, by ignoring all states/outcomes that are at least $n$ times less probable than the most probable state/outcome, or ignoring the least probable states/outcomes up to a total probability of $t$ . Even with these amendments, though, state and outcome discounting are subject to powerful objections. They are implausibly sensitive to small differences that can turn a single high-probability state/outcome into many low-probability states/outcomes (Beckstead and Thomas, Reference Beckstead and Thomas2024, §2.3). And they can easily run afoul of dominance principles (Isaacs, Reference Isaacs2016; Kosonen, Reference Kosonen2022, 151–164; Beckstead and Thomas, Reference Beckstead and Thomas2024, §2.3).

A more promising way of ignoring small probabilities is tail discounting.Footnote ⁴⁹ Roughly, tail discounting tells us to ignore the very-worst-case and very-best-case outcomes of every prospect, up to a certain probability (say, 0.01%) – either simply removing those worst-case and best-case outcomes from each prospect altogether (and renormalizing the remaining probabilities), or rounding those outcomes up/down (so that, for instance, any possible outcomes better than the 99.99th percentile of possible outcomes are “rounded down” to the 99.99th percentile outcome, and any possible outcomes worse than the 0.01st percentile are “rounded up” to the 0.01st percentile outcome). We can then apply expected value maximization, or another decision rule, to these “truncated” prospects.

Tail discounting has significant advantages over state and outcome discounting. In particular, its verdicts don’t depend on how we individuate states or outcomes, and it therefore avoids extreme sensitivity to small differences between outcomes. Moreover, it can be straightforwardly reconciled with Stochastic Dominance and other dominance principles, by stipulating that the truncated portions of a prospect are used as tiebreakers (Beckstead and Thomas,Reference Beckstead and Thomas2024, §2.3).

On the other hand, tail discounting is still subject to significant objections. Since it violates the Independence axiom of expected utility theory, it’s susceptible to money pumps (Kosonen, Reference Kosonen2022, 178–189; Kosonen, Reference Kosonen2024). And in the moral context, as with Bounded MEU, it’s prone to conflict with principles like Anteriority. (For instance, since tail-discounted MEV satisfies Transitivity and Superdominance but not Narrow Fanaticism, Proposition 1 implies that it will violate Anteriority given any outcome axiology that satisfies either Positive or Negative Additions.)

But as with Bounded MEU, my main purpose here isn’t to adjudicate the arguments for and against tail discounting. Rather, once again, the upshot of the arguments from Sections 6.1–6.3 is that whatever its other merits and despite initial appearances, tail discounting does not fully vindicate our anti-fanatical intuitions. This is easier to see for tail discounting than for Bounded MEU. If, for instance, we are considering how much we should be willing to sacrifice to reduce the probability of near-term human extinction from $0.5$ to $0.5-\epsilon$ , tail-discounted expected value maximization will be just as fanatical as ordinary expected value maximization.Footnote ⁵⁰

As with Bounded MEU, it’s possible to devise a version of small-probability discounting that satisfies General Anti-Fanaticism. I will describe one such version, which is based on representing prospects by their quantile functions. For a prospect $P_i$ on a set of totally ordered outcomes, the quantile function $Q_i$ of $P_i$ is a function from probabilities to outcomes, mapping any probability $p\in (0,1)$ to the worst outcome $o$ such that the probability of an outcome no better than $o$ is greater than or equal to $p$ . Thus, for instance, $Q_i(0.5)$ gives the median outcome of $P_i$ , $Q_i(0.75)$ gives the 75th percentile outcome, and so on.