In defence of fanaticism
Hayden Wilkinson

Global Priorities Institute | September 2020 (updated
January 2021)
GPI Working Paper No. 4-2020

In defence of fanaticism
Hayden Wilkinson∗
January, 2021
Comments welcome: hayden.wilkinson@anu.edu.au

Abstract: Consider a decision between: 1) a certainty of a moderately good outcome,such as one additional life saved; 2) a lottery which probably gives a worse outcome, but hasa tiny probability of some vastly better outcome (perhaps trillions of additional blissful lives
created). Which is morally better? By expected value theory (with a plausible axiology),
no matter how tiny its probability of the better outcome, (2) will be better than (1) as longthat better outcome is good enough. But this seems fanatical. So you may be tempted toabandon expected value theory.
But not so fast - denying all such fanatical verdicts brings serious problems. For one, youmust reject either that moral betterness is transitive or even a weak principle of tradeoffs.
For two, you must accept that judgements are either: inconsistent over structurally-identicalpairs of lotteries; or absurdly sensitive to small probability differences. For three, you must
accept that the practical judgements of agents like us are sensitive to our beliefs about
far-off events that are unaffected by our actions. And, for four, you may also be forced toaccept judgements which you know you would reject if you simply learned more. Better toaccept fanaticism than these implications.

∗

School of Philosophy, Australian National University

1

1

Introduction

Suppose you face the following moral decision.
Dyson’s Wager
You have $2,000 to use for charitable purposes. You can donate it to either of two
charities.
The first charity distributes bednets in low-income countries in which malaria isendemic.1 With an additional $2,000 in their budget this year, they would prevent
one additional death from malaria. You are certain of this.
The second charity does speculative research into how to do computations using‘positronium’ - a form of matter which will be ubiquitous in the far future of our universe. If our universe has the right structure (which it probably does not), then in the
distant future we may be able to use positronium to instantiate all of the operationsof human minds living blissful lives, and thereby allow morally valuable life to survive
indefinitely long into the future.23 From your perspective as a good epistemic agent,
there is some tiny, non-zero probability that, with (and only with) your donation,
this research would discover a method for stable positronium computation and would
be used to bring infinitely many blissful lives into existence.4
1

I have in mind the Against Malaria Foundation. As of 2019, the charity evaluator GiveWell estimated that

the Against Malaria Foundation prevents the death of an additional child under the age of 5 for, on average,
every US$3,710 donated (GiveWell 2020). Including other health benefits, a total benefit roughly equivalent tothat is produced for, on average, every US$1,690 donated. Of course, in reality, a donor can never be certain that
their donation will result in an additional life saved. This assumption of certainty is for the sake of simplicity.
2
Dyson (1981) was the first to suggest positronium as a medium for computation and information storage.
This follows Dyson (1979), wherein it is argued that an infinite duration of computation could be performed withfinite energy if the computation hibernates intermittently, and if the universe has a particular structure. Tipler(1986) suggests an alternative method which may work if the universe has a different structure. Sandberg (n.d.)argues that both Dyson and Tipler’s proposals are unlikely to work, as our universe appears to match neither
structure. Nonetheless, it is still epistemically possible that the universe has the right structure for Dyson’s
proposal. And possibility is sufficient for my purposes.
3
Would such artificially-instantiated lives hold the same moral value as lives led by flesh-and-blood humans?
I assume that they would, if properly implemented. See Chalmers (2010) for arguments supporting this view.
And note that, for the purposes of the example, all that’s really needed is that it is epistemically possible thatthe lives of such simulations hold similar moral value.
4
I have deliberately chosen a case involving many separate lives rather than a single person’s life containing
infinite value. Why? You might think that one individual’s life can contribute only some bounded amount of2

What ought you do, morally speaking? Which is the better option: saving a life with certainty,
or generating a tiny probability of bringing about infinitely many future lives?
A common view in normative decision theory and the ethics of risk - expected value theory- says that it’s better to donate to the speculative research. Why? Each option has someprobability of bringing about each of several outcomes, and each of those outcomes has somevalue, specified by our moral theory. Expected value theory says that one option is better thananother if and only if it has the greater probability-weighted sum of value - the greater expectedvalue.5 Here, the option with the greater expected value is donating to the speculative research(at least on certain theories of value - more on those in a moment). So perhaps that is what you
should do.
That verdict of expected value theory is counterintuitive to many. All the more counterintuitive is that it can still be better to donate to speculative research no matter how low the
probability is (short of being 0)6 , since there are so many blissful lives at stake. For instance,
the odds of your donation actually making the research succeed could be 1 in 10100 . (10100 is
greater than the number of atoms in the observable universe). The chance that the researchyields nothing at all would be 99.99... percent, with another 96 nines after that. And yet expected value theory says that it is better to take the bet, despite it being almost guaranteedthat it will actually turn out worse than the alternative; despite the fact that you will almost
certainly have let a person die for no actual benefit. Surely not, says my own intuition. On topof that, suppose that $2,000 spent on preventing malaria would save more than one life. Suppose
it would save a billion lives, or any enormous finite value. Expected value theory would saythat it’s still better to fund the speculative research - expected value theory says that it would
value to the value of the world as a whole - you might prefer for 100 people to each obtain some finite value thanfor one person to obtain infinite value. But whether this verdict is correct is orthogonal to the issue at hand, so
I’ll focus on large amounts of value spread over many people.
5
Note that expected value is distinct from the frequently-used notion of expected utility, and expected value
theory distinct from expected utility theory. Under expected utility theory, utility is given by some (indeed,any) increasing function of value - perhaps a concave function, such that additional value contributes less andless additional utility. The utility of an outcome may even be bounded, such that arbitrarily large amounts ofadditional value contribute arbitrarily little additional utility. Where expected value theory says that a lottery is
better the higher its expected value, expected utility theory says that it is better the higher its expected utility.
And, if the utility function is bounded, then the expected utilities of lotteries will be bounded as well. As a result,
expected utility theory can avoid the fanatical verdict described here. But, if it does, it faces the objections raisedin Sections 4, 5, and 6. Where relevant, I will indicate in notes how the argument applies to expected utility
theory.
6
I’ll assume throughout that probability takes on only real values from 0 to 1.

3

be better to sacrifice those billion or more lives for a minuscule chance at the infinitely many
blissful lives (and likewise if the number of blissful lives were finite but still sufficiently many).
But endorsing that verdict, regardless of how low the probability of success and how high thecost, seems fanatical. Likewise, even without infinite value at stake, it would also seem fanaticalto judge a lottery with sufficiently tiny probability of arbitrarily high finite value as better thangetting some modest value with certainty.
Fanatical verdicts depend on more than just our theory of instrumental rationality, expectedvalue theory. They also depend on our theory of (moral) value, or axiology. Various plausible
axiologies, in conjunction with expected value theory, deliver that fanatical conclusion. Foremost
among them is totalism: that the ranking of outcomes is determined by the total aggregate ofvalue of each outcome; and that this total value increases linearly, without bound, with the sumof value in all lives that ever exist. By totalism, the outcome containing infinitely many blissful
lives is indeed a much better one than that in which one life is saved. And, as we increase thenumber of blissful lives, we can increase how much better it is without bound. No matter how
low the probability of those many blissful lives, the expected total value of the speculative research is greater than that of malaria prevention. (Likewise, even if there are only finitely many
blissful lives at stake, for any tiny probability there can be sufficiently many of them to make therisky gamble better than saving a life with certainty.) But this problem isn’t unique to totalism.
When combined with expected value theory, analogous problems face most competing axiologies, including: averageism, critical-level views, prioritarianism, pure egalitarianism, maximin,
maximax, and narrow person-affecting views. Those axiologies each allow possible outcomes tobe unboundedly valuable, so it’s easy enough to construct cases like Dyson’s Wager for each.7
And some - namely, critical-level views and prioritarianism - already deliver the same result as
totalism in the original Dyson’s Wager. In this paper, I’ll focus on totalism, both to streamline
the discussion and because it seems to me far more plausible than the others.8 But suffice it
to say that just about any plausible axiology can deliver fanatical verdicts when combined withexpected value theory.
In general, we will sometimes be led to verdicts that seem fanatical if we endorse Fanaticism.9
7

For instance, take (standard, welfarist) averageism. A population containing at least one blissful life of

infinite (or arbitrarily long) duration will have average value greater than any finite value we choose. And so, togenerate an averageist analogue of Dyson’s Wager, we can substitute an outcome containing this population forthe outcome of arbitrarily many lives in the original wager.
8
Each of the other axiologies listed falls prey to devastating objections. See Arrhenius (2000), Huemer (2008),
Greaves (2017), and chapters 17-19 of Parfit (1984).
9This use of the term ‘fanaticism’ seems to originate with Bostrom (2011) and Beckstead (2013: chapter 6)4Inversely, to succeed in avoiding fanatical verdicts, our theory of instrumental rationality andour axiology must not imply Fanaticism.
Fanaticism: For any tiny (finite) probability  > 0, and for any finite value v, there
is some finite V that is large enough that Lrisky is better than Lsafe (no matter whichscale those cardinal values are represented on).
Lrisky : value V with probability ; value 0 otherwise
Lsafe : value v with probability 1
The comparison of lotteries Lrisky and Lsafe resembles Dyson’s Wager: one option gives a slim
chance of a potentially astronomical value V ; the other a certainty of some modest value v. But,
here, V need not be infinite, in case you think infinite value impossible. And, with some minor
assumptions (see Section 3), Fanaticism in this form implies the fanatical verdict in Dyson’s
Wager. Likewise, to reject the fanatical verdict in Dyson’s Wager, we must reject Fanaticism.
Note that Fanaticism is quite a strong claim. As defined here, it requires that the ranking of
Lrisky above Lsafe holds not only when the number of lives in each outcome are proportional to V,v, and 0. It must hold whenever outcomes can be cardinally represented with those values. Recall
that cardinal representations of value10 are unique only up to positive affine transformations two outcomes represented by 0 and v on one scale could instead be represented by 0 × a + b andv × a + b (for any positive a and real b). Conversely, an outcome that contains many happy lives
might still be represented cardinally with value 0. So Fanaticism doesn’t apply only to risky
lotteries in which some possible outcome contains zero valuable lives, or zero value on net. It
also applies to lotteries that can be represented as Lrisky and Lsafe even though every one of theiroutcomes contains enormous numbers of blissful lives, or enormous amounts of suffering, as longthe differences in value between those outcomes are proportional to 0, v, and V .
Given how strong and how counterintuitive Fanaticism is, you might think it easy to reject.
And many philosophers and other thinkers have done so, rejecting either Fanaticism or similar
principles. For instance, Bostrom (2009) presents a compelling reductio ad absurdum for fanatical(however Beckstead uses the term ‘Fanaticism’ for a similar claim specific to infinite values and instead uses
‘Recklessness’ for a claim more akin to my version of Fanaticism.). My formulation is slightly stronger than eachof theirs but also, unlike theirs, applicable even if infinite total value cannot exist. For discussion of whether
outcomes with infinite moral value are possible and how we might coherently compare them, see Bostrom 2011;
Askell 2018; Wilkinson 2020; Wilkinson n.d..
10Equivalently, these are representations of value on interval scales.

5verdicts in the prudential context. Bostrom (2011), Beckstead (2013), and Askell (2019) treat
even a weak form of (moral) Fanaticism as a reductio for moral theories. Others propose theoriesof rationality with the expressed purpose of avoiding fanatical verdicts. For instance, some
propose that we simply ignore outcomes with small enough probabilities (e.g., D’Alembert 1761;
Buffon 1777; Smith 201411 ; Monton 2019). Others insist that we maximise not expected moral
value but instead the expected utility of outcomes (given by some increasing function of an
outcome’s value), and that the correct utility function is bounded above so as to keep the
expectation of utility bounded as well (e.g., Arrow 1971: 64).
Meanwhile, there are few defenders of Fanaticism or, more broadly, of fanatical verdicts incases similar to Dyson’s Wager. Notable examples include Pascal (1669) himself, Parfit (1984:§3.27), and Hájek (2014). And even they most often endorse such verdicts only because they area consequence of expected value theory, not because they see good independent justification forthem. I suspect that even many diehard adherents of expected value theory are uncomfortablewith the fanatical verdicts supplied by their theory.
This situation is unfortunate. There are compelling arguments in favour of Fanaticism that
do not rely on expected value theory, and so we have good reason to accept it even if we rejectthat particular theory. If we do not, we face disturbing implications.
The paper proceeds as follows. Section 2 addresses some common motivations for rejecting
Fanaticism. Section 3 introduces the necessary formal framework for what follows. Sections
4 through 6 present arguments in favour of Fanaticism, each premised on weaker claims than
expected value theory, and each (by my reckoning) more compelling than the last. The first isa basic continuum argument. The second is that, to deny Fanaticism, we must accept either
what I’ll call ‘scale dependence’ or an absurd sensitivity to arbitrarily small differences in probability. And the final nails in the coffin are an updated version of Parfit’s classic Egyptology
Objection and what I’ll call the Indology Objection, by which those who deny Fanaticism must
make judgements which appear deeply irrational. Section 7 is the conclusion.
11

Smith’s proposal can be interpreted in two different ways, only one of which rules out Fanaticism. By theother interpretation, which Smith prefers, we still ignore events with probability below some threshold but, in any
lotteries over finitely many different outcomes, that threshold is set below the probability of the least probableoutcome. This is compatible with Fanticism while still avoiding the problems with which Smith is more concerned:
counterintuitive verdicts in the St Petersburg and Pasadena games.

62Intuitions against fanaticism

There are various arguments (that can be directed) against Fanaticism that appear in the literature (see D’Alembert 1761; Buffon 1777; Smith 2014; Schwitzgebel 2017; Monton 2019). And
there are various counterarguments as well (see Hájek 2014; Isaacs 2016; Monton 2019; and Hájek
n.d. for some of the most compelling).
But there is one extremely simple argument against it that, I suspect, is the most persuasiveof all: Fanaticism is just frightfully counterintuitive that it must be false.12 It seems so very
implausible that I should give up a certainty of a good payoff for a tiny chance of something
better, no matter how tiny the chance. Rather, intuition says that an event with such a tiny
chance can never tip the scales against the sure payoff.
But the psychology literature tells us that our initial intuitions about matters of probability
are often misguided. We frequently commit the Conjunction Fallacy (Tversky & Kahneman
1983), the Gambler’s Fallacy (Chen et al. 2016), the Hot Hand Fallacy (Gilovich et al. 1985),
and the Base Rate Fallacy (Kahneman & Tversky 1982). Perhaps worst of all, we are prone
to simply ignoring facts of probability - many of us choose based entirely on how we anticipate
things will actually turn out, with no regard for risks of loss or chances of gain (Baron etal. 1993; Gurmankin & Baron 2005). Even when we do account for probability, we often
radically overestimate some probabilities due to availability bias (Tversky & Kahneman 1974),
and underestimate others out of indefensible optimism (Hanoch et al. 2019).
And our intuitions are no better with decision-making in the face of low-probability events.
For instance, jurors are just as likely to convict a defendant based primarily on fingerprint
evidence if that evidence has probability 1 in 100 of being a false positive as if it were 1 in 1,000,
or even 1 in 1 million (Garrett et al. 2018). In another context, when presented with a medical
operation which posed a 1% chance of permanent harm, many respondents considered it no worse
than an operation with no risk at all (Gurmankin & Baron 2005). And in yet another context,
subjects were unwilling to pay any money at all to insure against a 1% chance of catastrophic loss
(McClelland et al. 1993). So it seems that many of us, in many contexts, treat events with low
probabilities as having probability 0, even when their true probability is as high as 1% (which is
not so low). Upon reflection, this is clearly foolish. We may have the raw intuition that we may(or should) ignore all outcomes with probability 1% but it does not hold up to scrutiny, and so
12

In effect, this is a core point of Bostrom’s (2009) objection. Meanwhile, Bostrom (2011), Beckstead (2013),

and Tarsney (n.d.) all treat Fanaticism as implausible for, it seems, the same reason.

7

is likely mistaken.
Given how widespread these mistakes are, we should give little weight to our intuitions about
what we should do in cases of low probability, including those which lead us to recoil fromfanatical verdicts - with a little more scrutiny, those intuitions may appear foolish too. At thevery least, the intuitive case against Fanaticism is dubious enough that we should consider thecase in favour of it.

3

Background assumptions

Before I make a case for Fanaticism, here are my basic assumptions.
First, for any decision problem, there is some set of epistemically possible outcomes O. Someoutcomes are better or worse than others. So assume that there exists some binary relation on
O, denoted by <O , which represents an ‘at least as good as’ relation between outcomes.
As mentioned earlier, for the most part I’ll be assuming a totalist view of value. So <O isalready given: for any possible outcomes O1 and O2 , we have O1 <O O2 if and only if O1 contains
at least as much total value as O2 . And total values are totally ordered, so we know that <O will
be reflexive, transitive, and complete on O. These total values of outcomes can be represented
with a cardinal value function V : O → R, at least when those outcomes have finite differencesin value.13 And as a cardinal value function, V is unique only up to affine transformations - for
any V1 or V2 we might use here, V2 (O) = a × V1 (O) + b for some positive a and real b.
We may not always be able to give a real representation of the total value for every outcome.
If an outcome O∞ contains infinitely more value than others - as when we create infinitely many
blissful lives in Dyson’s Wager - then it may fall beyond the scope of V . Then V (O∞ ) won’t bedefined, but still O∞ O O for each finitely-valued outcome O. And that’s fine - a real-valued
total value function V won’t represent <O in all cases, but it will do just fine in purely finite
cases. I’ll use OR ⊂ O to denote any given set of outcomes over which <O admits a real-valued
representation V . The arguments that follow can be run entirely in terms of OR .
What we are really interested in is lotteries. I’ll assume that the relevant aspects of a lottery
can be fully described by a (real-valued) probability measure L on a set of outcomes. I’ll also
assume that this probability represents either the agent’s subjective degree of confidence or the13

The differences in value for two pairs of outcomes are treated as finite precisely when one can be represented

as a real multiple of the other.

8evidential probability of each outcome arising. Almost all of what follows can be read in termsof either. With minor changes, what follows could also be read in terms of objective chance orwhatever notion of probability you consider relevant for decision making. Whichever notion itis, let L be the set of all lotteries on O, and LR the set of all lotteries on a given subset OR . Soeach lottery L is a function which maps sets of possible outcomes to some value in the interval[0,1], and that function must obey the standard probability axioms.
To keep things brief, I’ll use the following shorthand. When we are interested in the probability of a single outcome, I’ll shorten L({O}) to L(O). When a lottery L gives a certainty ofan outcome O (when L(O) = 1) I’ll often write L as O. To represent a lottery with the sameprobabilities as L, but outcomes which have precisely k times the value (on the same cardinal
representation), I’ll write k · L.14 And I’ll write La + Lb to represent the lottery obtained fromadding together the values of the outcomes of lotteries La and Lb , run independently.15
Fanatical verdicts, and their rejection, consist in comparisons of lotteries. So we need another‘at least as good relation’. Let < be a binary relation on L (and hence also on the subset LR ).
Strict betterness () and equality (∼) are defined as the asymmetric and symmetric components,
respectively.
I won’t assume that < is complete on L. But I will assume that it is reflexive: that L < L
for all L ∈ L. I’ll also assume that it is transitive: that, for all La , Lb , Lc ∈ L, if La < Lb andLb < Lc , then La < Lc . Both of these properties are highly plausible. If either of them do not
hold, then betterness is a peculiar thing.16
I also want to assume another highly plausible principle of instrumental rationality, which
will be useful in what follows: Stochastic Dominance. This principle says that if two lotteries
have exactly the same probabilities of exactly the same (or equally good) outcomes, then they
are equally good; and if you improve an outcome in either lottery, keeping the probabilities thesame, then you improve that lottery. And that’s hard to deny!
14

Formally, for any real k and L ∈ L, define k · L as a probability measure on O such that, for all O in O,

k · L(Ok ) = L(O), where Ok is an outcome in O such that V (Ok ) = k × V (O).
15This can be made more precise. Let Vi denote the random variable corresponding to a given lottery Li , which
outputs the total value of the outcome - equivalently, the probability that Vi takes on a value in the interval [a, b]
is given by Li ({O|V (O) ∈ [a, b]}). Then, for any La , Lb ∈ L, define La + Lb as some lottery on O - there may
be several - which corresponds to the random variable (Va + Vb ). Equivalently, La + Lb is some lottery such that
La + Lb ({O|V (O) ∈ [c, d]}) is equal to the probability that (Va + Vb ) takes on a value in the interval [c, d].
16Some argue that moral betterness (and presumably also instrumental moral betterness) is not transitive, e.g.
Temkin (2014). But the most compelling of these arguments assume pluralism with respect to moral value. But
I’m focusing on monistic theories of value here, so transitivity remains a compelling principle.

9

We can express Stochastic Dominance as follows. Here, O<O0 denotes the set of outcomes inO that are at least as good as O0 .
Stochastic Dominance: For any La , Lb ∈ L, if La (O<O0 ) ≥ Lb (O<O0 ) for all O0 ∈ O,
then La < Lb .
If, as well, La (O<O0 ) > Lb (O<O0 ) for some O0 ∈ O, then La  Lb .
I find Stochastic Dominance overwhelmingly plausible. In this setting, it is far weaker than(but necessary for) expected value theory. But, unlike the stronger theory, it does not rule out
risk aversion, nor Allais preferences. To deny it is to accept either: that you can swap an outcome
in a lottery for a better one without making the lottery better; or that evaluations of lotteries
are dependent on something other than the values of the outcomes and their probabilities. Bothare implausible.17
And Stochastic Dominance in a setting like this is fairly uncontroversial among decision
theorists, including some who reject expected value theory. As far as I know, no serious proposal
has been made in normative decision theory which violates Stochastic Dominance (at least in
this setting).18
And Stochastic Dominance ties together the fates of Fanaticism and the fanatical verdict inDyson’s Wager. Recall the lotteries Lsafe and Lrisky from the definition of Fanaticism. In Lsafe ,we can set v = 1. And define Linfinite as follows.
Linfinite : an outcome containing infinitely many blissful lives with probability ; value
0 otherwise
Lrisky : value V with probability ; value 0 otherwise
Lsafe : value 1 with probability 1
Linfinite and Lsafe should look familiar - these match the lotteries you must choose between
in Dyson’s Wager, if we represent the outcome in which one life is saved with value 1 and theoutcome in which no lives are saved with value 0. By Fanaticism, Lrisky is better than Lsafe , at
17
18

For compelling arguments in favour of Stochastic Dominance, see Easwaran (2014) and Bader (2018).
Schoenfield (2014) rejects a similar principle, but only for lotteries over outcomes for which we don’t have acomplete ordering. With our complete <O relation on O, even Schoenfield would endorse Stochastic Dominance
here.

10

least for large enough finite V . And Stochastic Dominance implies that Linfinite is better than
Lrisky .19

4

A continuum argument

You might argue for Fanaticism in the following way: expected value theory is true; expectedvalue theory implies Fanaticism; therefore, Fanaticism is true. And likewise for verdicts which
seem fanatical, like that which the expected value theorist must accept in Dyson’s Wager. Suchverdicts are rarely defended on any grounds other than ‘That’s what expected value theory says’.
But we can do better than this. Fanaticism is far weaker than expected value theory in all its
strength (and weaker even than expected utility theory), so it should be easier to justify.
In this and the next two sections, I’ll give four arguments for Fanaticism but not for expectedvalue theory. Here is the first, which originates in Beckstead (2013: 139-47) and reappears in
Beckstead and Thomas (n.d.: 4-6).
Take the lottery L0 , which might represent a stranger’s life being saved.
L0 : value 1 with probability 1
And take another lottery L1 by which vastly more strangers are saved, with a very slightprobability of failure - perhaps 1010 lives saved with probability 0.999999 of success.
L1 : value 1010 with probability 0.999999; value 0 otherwise
Intuitively, L1 seems better. Accepting a slightly lower probability of success for a vastly
greater payoff seems a good trade. But then consider L2 , which has a slightly lower probabilityof success but, if successful, results in many more lives saved.
10

L2 : value 1010
19

with probability 0.9999992 ; value 0 otherwise

The reasoning behind this is as follows. We represented the value of one additional life being saved witha 1 and additional lives saved with a 0. Any finite positive V will hence correspond to V additional lives of
equal value being saved (or produced). This implies that the outcome in which infinitely many blissful lives areproduced cannot be represented on the same scale, and indeed it is better than any outcome that can be. But,
had that outcome been representable on the same scale with some finite number, this wouldn’t hold.

11

This seems better than L1 , or at least it seems that there is some number of lives high enoughthat it would be better. And so we could continue, with L3 , L4 , and so on until some Ln , suchthat 0.999999n is less than , for any arbitrarily small  you want. Intuition suggests that vastly
increasing the payoff can compensate for a slightly lower probability; that each lottery in thesequence is better than the one that precedes it. So the final lottery in the sequence must be
better than the first. But the final lottery has a probability less than  of any positive payoff at
all. So we have Fanaticism.20
As I see it, this argument rests on two intuitively plausible principles: the first, the transitivityof <; the second, Minimal Tradeoffs.
Minimal Tradeoffs: There is some ratio r < 1 such that, for any real value v andany probability p, there is some real r∗ such that, on any cardinal representation, thelottery represented by La is better than that represented by Lb (as defined below).
La : value v with probability p; 0 otherwise
Lb : value r ∗ ×v with probability r × p; 0 otherwise
This principle has intuitive force. For any given lottery with the form of La , surely there
is any lottery with only that ever-so-slightly lower probability of success is better. Intuition
suggests that we can always make at least some tradeoff between probability and value: we canalways compensate for a, perhaps, 0.000001% lower probability of success with a vastly greater
payoff.
But you might be unconvinced by the continuum argument. You may find it far moreintuitively plausible that Fanaticism is false than that Minimal Tradeoffs is true. And, other thanclashing with intuition, I see nothing obviously problematic about denying Minimal Tradeoffs.
Perhaps we should just reject it; perhaps there is at least one threshold of probability p0 between
p and 0 (which might be unknown or vague) at which we have a discontinuity, such that no value
with probability below p0 is better than any value with probability above p0 . One way to do20

For readers less sympathetic to totalism, much the same argument can be made in terms of any of theaxiologies mentioned in the Introduction. For instance, if we assume averageism, we can start with L0 , by which
the average value of all lives will be 1 with near certainty, and 0 otherwise. L1 can be the lottery which gives
probability 0.99999 of an outcome in which the average value of all lives will be 1010 , or 0 otherwise. We canrapidly scale up the average value of lives just as we scaled up the total values above, while gradually scaling
down the probability of non-zero value. Eventually, we reach a lottery Ln with probability  of some astronomical
average value, or 0 otherwise, but transitivity says that it must be better than L0 .

12this is by adopting expected utility theory with a bounded utility function, which would imply
that the former lottery is better than the latter. But, whichever approach we use to reach that
conclusion, there is some point at which we can no longer trade off probability against value, nomatter how great the value. And what of it? It seems a little counterintuitive, but perhaps lessso than Fanaticism.
If you accept Minimal Tradeoffs then you must accept Fanaticism. And, inversely, to reject
Fanaticism you must also reject Minimal Tradeoffs and the continuum argument it generates.
But you might think that a small price to pay.

5

A dilemma for the unfanatical

The next argument for Fanaticism comes in the form of a nasty dilemma that you face if you
deny it. In brief, if you deny Fanaticism then you must accept at least one of: abandoning Scale
Independence, as defined below; and allowing comparisons of lotteries to be absurdly sensitive totiny changes (even more so than if you accepted Fanaticism).
Scale Independence: For any lotteries La and Lb , if La < Lb , then k · La < k · Lb for
any positive, real k.
I find Scale Independence highly plausible.21 Take any pair of lotteries, perhaps a pair thatcan be cardinally represented by L1 and L2 .22
L1 : value 1 with probability 1
L2 : value 2 with probability 0.9; value 0 otherwise
I’ll remain agnostic on which is better; < might recommend either (or neither). But suppose
we scaled up each lottery by 2, to obtain 2 · L1 and 2 · L2 - lotteries with the same probabilities
21

One way to violate it is to adopt expected utility theory with a bounded utility function (see Rabin 2000). I

find this good reason to reject such a form of expected utility theory. On the other hand, there are also theories
that avoid Fanaticism without a bounded utility function and hence scale dependence, e.g., Buchak’s (2013)
risk-weighted expected utility theory and the Buffon-Smith method of ignoring outcomes with sufficiently low
probabilities.
22
For readers unsympathetic to totalism, these payoffs could just as easily represent: the average values of lives
in each outcome; the priority-weighted total value in each outcome; the total value in the lives of all persons who
would exist regardless of the agent’s choice; and so on.

13as above, but with the value of each outcome doubled (on the same cardinal scale). If we ranked
L1 as better then, intuitively, that ranking shouldn’t change as we double the values. After all,the scaled-up lotteries have the same structure as the originals. Whatever general principles lead
us to say whether L1 is better than L2 should plausibly also lead us to say the same about k · L1
and k · L2 , for any positive k you like.
And remember that here we are dealing with moral value, not dollars or some other commodity that is merely instrumentally valuable. Unlike such commodities, when we are dealing
with value itself, it is incoherent to say that additional units of value are worth less and less - bydefinition, adding one unit of value is always an improvement of one unit of value. And under
totalism, some version of which I’m assuming here to be true, units of value can correspond totangible objects such as (identical) human lives. By the lights of totalism and by at least my own
intuition, an additional (identical) human life is always worth the same amount no matter how
many lives already exist - and of course it always contributes precisely the same amount to thetotal value of the outcome. So, in keeping with intuition, it should not matter in the slightest
whether we are comparing the lotteries L1 and L2 above or instead some scaled-up multiples
k · L1 and k · L2 . In the latter pair, we have done the equivalent of adding k − 1 additional copies
to the contents of each outcome (perhaps k − 1 additional lives to L1 and 2k − 2 additional lives
to that outcome in L2 ). But each additional copy should be no more or less valuable than thefirst; they should contribute the same to our evaluation of the lottery.23 So, I would claim, weshould rank the resulting lotteries just the same way as we ranked them without those k − 1copies added in.
It is also worth noting that totalism provides only that value may be represented cardinally,
not that it can be represented on a ratio or unit scale (as do various other axiologies).24 It does
not provide any absolute zero of value, nor any unique scale on which to represent it. A pair ofoutcomes represented with values 1 and 2 on one scale can just as easily be represented with any
values k and 2k (for positive k) on other scales. When evaluating those outcomes with totalism,
we cannot say that any such representation is more valid than any other. Suppose we took
23

A similar story can be told for averageism and other axiologies, although a little more awkwardly. On

plausible versions of averageism, an additional period of bliss given to everyone contributes the same amount tothe average value of all lives no matter how many such periods of bliss have already been experienced. We cancopy the experiences each person has in much the same way we can copy the number of additional lives in the
totalist scenario. It seems plausible then that we should rank lotteries with k − 1 copies of each experience added
into outcome just the same way as we rank them without those copies.
24
Indeed, all of the axiologies listed in the introduction, with the exception of pure egalitarianism, typically do
as well.

14lotteries L1 and L2 above; their outcomes could just as easily have been represented on anotherscale such that they had values k, 2k, and 0 - the same values as k · L1 and k · L2 had on theprevious scale. On that new scale, we of course still say that L1 is better than L2 if and only if
we said so on the original scale, else we would be inconsistent. But why not judge k · L1 and k · L2
similarly? For us to be able not to, our evaluations of lotteries must depend on more than just theprobabilities of outcomes and cardinal values that totalism assigns them; they must also depend
on which cardinal representation is being used when those values are assigned. In effect, they
depend on richer details of the value of outcomes than is provided by the cardinal representation
given by totalism. And this makes for a troubling shift in how we evaluate lotteries - doing so
is no longer a matter of using our axiology to assign values to outcomes and then using ourtheory of instrumental rationality to turn those values and each outcome’s probabilities into an
evaluation. Instead, our evaluation is sensitive to facts about the values of outcomes that even
our theory of value is not sensitive to. That is awfully strange and, I think, absurd. We shouldbe able to apply our theory of rationality to the verdicts of our axiology without ‘double-dipping’
into facts about the value of outcomes. And so we should reject the suggestion that k · L1 andk · L2 not be compared in a similar way to L1 and L2 ; we should accept Scale Independence.25
For those who reject Fanaticism, the dilemma they face is between rejecting Scale Independence as well and accepting an absurd level of sensitivity to tiny changes in lotteries. I’ll
illustrate below what I mean by ‘sensitivity to tiny changes’ but, for now, be assured that it’s
just as implausible as violating Scale Independence.
The argument for the dilemma goes like this. First, for Fanaticism to be false, there must be
some probability  > 0 and some cardinal value v such that some lottery Lrisky as defined below
is no better than Lsafe , no matter how big V is.
Lrisky : value V with probability ; 0 otherwise
Lsafe : value v with probability 1
But what about values below v? How would getting some lower value with certainty compare
to an -probability of arbitrarily enormous V ? There are only two distinct possibilities (at least25

An alternative to endorsing Scale Independence is to adopt a variant of totalism that assigns more than justa cardinal value to outcomes - perhaps a version of totalism that values outcomes on a ratio or unit scale. Thatversion would be far less parsimonious than the version I have been using, and I cannot think of any compelling
independent justification for it. It seems at least a considerable cost to adopt such a version of totalism to avoid
endorsing Scale Independence and the implication of Fanaticism.

15

if we maintain Stochastic Dominance and transitivity). The first is: there is some smaller v 0 > 0
for which Lrisky is better than a sure outcome with value of v 0 , for sufficiently large V . The
second is: for any such v 0 , Lrisky isn’t better than any sure outcome with value v 0 , no matter how
large V is.26 As we’ll see, each possibility impales us on a respective horn of the dilemma.
Assume that the first possibility holds. Lrisky is better than a sure outcome with value v 0 ,for some 0 < v 0 < v and some large V . Still, Lrisky won’t be better than a sure outcome withvalue v - i.e., Lsafe - or any sure outcome with even greater value (by Stochastic Dominance).
But somewhere below v this changes - for some positive v 0 , the same doesn’t hold.
Since that v 0 is a positive real number, we know that there will be some real k such thatk × v 0 ≥ v. And that means that a sure outcome with value k × v 0 is no worse than L1 , no matter
how great V is. But here is the problem: a certainty of k × v 0 is the scaled-up counterpart of v 0 ;and the counterpart of Lrisky , which is k · Lrisky ), is just Lrisky with V replaced by another value
which is k times higher. That still has the same form as Lrisky . So it cannot be better than thesure outcome, not if we reject Fanaticism. But then our verdict for v 0 versus Lrisky doesn’t matchour verdict for k × v 0 versus k · Lrisky . So we violate Scale Independence.
But there was a second possibility: that there is no such v 0 > 0; that Lrisky is no better
than a certainty of any v 0 > 0. If so, we can avoid the problem of scale dependence: there’s no
inconsistency between the judgements for v and judgements for any ‘scaled-down’ counterpart
v 0 , since all such v 0 compare the same way to lotteries like Lrisky .27 But we then face anotherserious problem.
Take the probabilities  and 1. We can give a sequence of increasing probabilities in between
 and 1, spaced evenly apart and as finely as we want:  < p1 < p2 < . . . < pn < 1. By
assumption, no amount of value with probability  (and value 0 otherwise) is better than any(positive) amount of value with probability 1. But then there must be some pair of successive
pi , pi+1 such that no amount of value at pi is better than any amount of value with probability
pi+1 . If there were no such pair, we could give a sequence of better and better, and less and lesslikely, lotteries much like that in the previous section.
Crucially, that pi and pi+1 can be arbitrarily close together, since the same result holds nomatter how finely spaced the sequence was. No amount of value at pi is better than any amount
26

This difference is analogous to the difference between weak and strong superiority in Arrhenius & Rabinowicz(2015).
27
The comparison won’t be the same for values 0 and below, but that’s okay. Such values cannot be scaled up
to v; multiply 0 or any negative number by any real positive value you want and you still won’t reach v.

16

of value at pi+1 . We could have some astronomical value at pi , and an imperceptibly small
amount of value at pi+1 , and the latter would still be better. So we have a radical discontinuity
in how we value lotteries.
This will make our judgements of betterness absurdly sensitive to tiny changes in probability.
To illustrate that sensitivity, consider the following four lotteries. Here, 0 > 0 is some arbitrarily
small number. And, again, pi and pi+1 are some tiny probabilities that are arbitrarily close
together.
L0 : value 0 with probability 1
10

L1 : value 1010

with probability pi ; 0 otherwise

L2 : value 0 with probability pi+1 ; 0 otherwise10

L3 : value 1010

with probability pi+1 ; 0 otherwise

By the above, if we reject Fanaticism and maintain Scale Independence, then we are forced
to rank these lotteries as follows: L0 ≺ L1 ≺ L3 and L0 ≺ L2 ≺ L3 , but L1 ⊀ L2 . And that’s
a peculiar ranking. L1 and L3 are almost indistinguishable; their probabilities may differ by anarbitrarily tiny amount. Likewise for L0 and L2 , except it’s their payoffs that differ slightly. For
each pair, we must say that one is better than the other, but does not seem much better - in asense, the better lottery is only a trivial improvement over the other, whether by a slight increasein payoff or in probability. But, despite appearances, we must accept that L3 is much better
than L1 , as well as that L2 is much better than L0 . How so? Between L1 and L3 comes L2 .
The lottery L2 has an astronomically smaller payoff than L3 and so (in a sense) is vastly worse
than L3 , and yet we cannot say that it is so bad so as to be worse than L1 . Effectively, we must
accept that, starting from L3 , it is no worse to lose almost all of the potential value via L2 than
it is to lose that sliver of probability via L1 . And similarly for L0 and L2 : between them mustcome L1 . L1 has astronomically greater potential payoff than L0 (and much higher probabilityof positive payoff too), yet we cannot say it’s better than L2 . Effectively, we cannot say it is any
better to gain that enormous value with probability pi than it is to gain an ever-so-slightly moreprobable shot at tiny value . So those lotteries are not just made worse by those tiny changes
in probability or payoffs - they are made far worse (in an intuitive sense). That is the level ofsensitivity we must accept if we reject Fanaticism and maintain Scale Independence. But suchextreme sensitivity in our evaluations of lotteries seems absurd. Evaluations of lotteries should
not change quite so wildly - so discontinuously - with arbitrarily tiny changes in probability or
payoff. So I do not think this sensitivity - this horn of the dilemma - is any less absurd than17

what we faced earlier.
I should briefly note that, beyond its intuitive implausibility, this sensitivity will likely lead topractical difficulties. The probabilities that human agents have access to are subjective degrees
of belief and evidential probabilities. In practice, neither sort of probability can be pinned downwith arbitrary precision, at least not with our merely finite capacity for calculation.28 But we’ll
sometimes need arbitrary precision to compare lotteries, at least by this horn of the dilemma. If
not, in cases like that immediately above, we would have no idea whether a lottery is as good as
L3 or as bad as L1 with its imperceptibly lower probability of success. We must either require(unrealistically) that agents be arbitrarily precise in their probabilities29 , or else accept that inpractice our decision theory will be unable to tell agents that (at least some of) these lotteries
are better than others.
This is a dilemma we must face if we reject Fanaticism: we must accept either scale dependence, by which our judgements of lotteries can vary without any structural reasons for doingso, or that those judgements are sensitive to imperceptibly small differences in either probability
and value. Both options seem absurd to me, indeed more absurd than Fanaticism itself.

6

Egyptology and Indology

But it gets worse. There are even greater costs you must pay to reject Fanaticism: you faceeither one or both of the following objections, each of which is intuitively absurd.

6.1

The Egyptology Objection

Hailing from population ethics, the Egyptology Objection is a classic argument against various
axiologies, including averageism, egalitarianism, and maximin.30
28

For discussion of why we cannot require epistemic agents to settle on precise probabilities, see Schoenfield(2012) and Joyce (2011). For discussion of how to evaluate lotteries with imprecise probabilities, see Seidenfeld(2004), Huntley, Hable & Troffaes (2014), Bradley & Steele (2015), and Bradley (2015). On any of the methods
suggested, a combination of imprecise probabilities and sensitivity to the degree described above will lead toindifference among many of the lotteries L1 to L4 listed above or else indeterminacy of which is better thanwhich. At least in that case, with its vast differences in value and only small differences in probability, it ***29
In Smith’s (2014:471) terms, our theory would then be ‘intolerant’ of human imprecision, which he considersa major problem.
30
It appears earliest in MacMahan (1981: 115) but is often attributed to Parfit (1984: 420).

18

As a brief refresher, it goes like this. Suppose you are making a moral decision here andnow in the 21st Century, and the available outcomes differ only by some small-scale changes inthe very near future. Your actions will not change what happens in distant galaxies or what
happened in, say, ancient Egypt. Then, intuitively, what you ought to do should depend only
on the events altered by your choice. It should not depend on events in distant galaxies or inancient Egypt, at least not if your actions do not change them. Intuitively, it seems absurd thatthose unaltered, remote events make any difference to your evaluation.31
But, under some axiologies, what you ought to do in such cases is dependent on unaltered,remote events. Take a (standard, welfarist) averageist view. Here and now, should you bringan additional person into existence? It depends - will that additional life raise or lower the
average value of lives that ever exist? Sometimes yes, sometimes no, depending on what actually
happened in ancient Egypt, in distant galaxies, and everywhere else. Averageism will thus imply
that what happened in ancient Egypt can affect whether it is better to now bring an additional
person into existence or not. And, further, to be confident of which is the better outcome, you
may need to do some serious research into Egyptology (along with plenty of other historicalstudies). But this is implausible. By intuition, we can ignore events that are unaffected by ouractions and took place millennia ago, as we do under axiologies like totalism, critical-level views,
prioritarianism, and aggregative person-affecting views (and any other axiology which admits anadditively separable representation). To me, this is one of the main appeals of those views.
But even if we accept an axiology like totalism, we may still face much the same objection inpractice.32 Some methods of comparing lotteries give rise to an updated Egyptology Objection,
including all of those that deny Fanaticism. That’s right: to deny Fanaticism, you must not
only endure the costs detailed in the previous two sections, you must also fall prey to a versionof the Egyptology Objection. (As we’ll see below, you may also face another even more serious
objection.)
You will most easily fall prey to it if you reject Background Independence. (And some proposals do.)33
31

Such dependence brings practical problems too. If what you ought to do is dependent on such events - notjust in the remote future but also the remote past, of which we are often clueless - then we have little guidance
for what we ought to do (cf Mogensen 2021). This seems a major disadvantage for a moral view.
32
Even on other axiologies, including averageism, we face similar (albeit slightly less compelling) objections.
See Footnote 35.
33
One such proposal is expected utility theory with a utility function that is concave and/or bounded (e.g.,
Arrow 1971). As Beckstead and Thomas (n.d.: 15-16) point out, this results in comparisons of lotteries being
strangely dependent on events that are unaltered in every outcome and indeed some irrelevant to the comparison.

19

Background Independence: For any lotteries La and Lb and any outcome O0 , if La <
Lb , then La + O0 < Lb + O0 .
Recall that the sum of two lotteries L+O0 is simply the lottery you get if you run each lottery
independently and sum up the values of their outcomes. But, here, O0 is a very simple lottery it is just the outcome O0 with certainty. That outcome has some cardinal value; call it b. The
lottery La + O0 is simply the lottery La with value b added to the value of each outcome. So
Background Independence says that we can take any two lotteries and add a constant value toevery one of their possible outcomes, and this won’t change their ranking.
Suppose for now that Background Independence is false. Then there is some pair of lotteries,
La and Lb , such that: La is at least as good as Lb ; and, if we added a certain constant b to thevalue of every one of their outcomes, that would change their ranking. Since we’re assuming
totalism, the value of each outcome in each lottery must represent the total aggregate of valuethat would result, including all valuable events across all of space and time. That includes theevents that occurred in ancient Egypt. And we can assume that the same events occurred inancient Egypt in every outcome of either La or Lb .34 With those events included, La is at least
as good as Lb .
But what if we consider a different hypothetical pair of lotteries, identical to La and Lb
except that the events of ancient Egypt were different? Would the ranking be any different, had
events gone differently back then? Yes, it would, if Background Independence is false and if
those events differed drastically enough. They might differ in such a way that they increasedin value by b. Then the total value of every outcome would also be increased by b. And, giventhat La and Lb were the lotteries that violated Background Independence, we know that thischanges their ranking. So our modified version of La is no longer at least as good as the modified
version of Lb . Thus, if we deny Background Independence, we face a rather severe version of theEgyptology Objection: when choosing between two lotteries, which is better can vary dependingon what events occurred in ancient Egypt, even if the choice doesn’t change those events at all(cf Beckstead and Thomas n.d.).35
34

If there is some pair of lotteries La or Lb that violate Background Independence, then there is also some pairthat violate it and have the exact same events occur in ancient Egypt. To obtain such a pair, simply take La andLb and replace each of their outcomes with an outcome of equal total value but an identical history of ancient
Egypt.
35
This new version of the Egyptology Objection, along with the other objections described in the rest of thissection, can also be applied under axiologies other than totalism. It will just look a little different.
Suppose that (standard, welfarist) averageism is the correct axiology. Then to add b to value of an outcome20

To avoid this implication, we must at minimum accept Background Independence. I think
we ought to accept it regardless - I find it highly plausible for much the same reasons as Scale
Independence was. In any case, I’ll assume for the remainder of this section that Background
Independence holds.
But even if we accept Background Independence, if we still reject Fanaticism then it turns
out that a version of the Egyptology Objection still arises, albeit a less severe one.
To see why, first recall the lotteries from the definition of Fanaticism: Lrisky and Lsafe . If we
reject Fanaticism then, for some tiny enough probability  > 0 and some cardinal value v, somelottery Lrisky (as defined below) is no better than Lrisky , no matter how big V is.
Lrisky : value V with probability ; 0 otherwise
Lsafe : value v with probability 1
According to Background Independence, it must also hold that the lottery Lrisky with any
constant b added to every outcome’s value is also no better than Lsafe with that same b added
to every one of its outcomes’ values. If we face a decision between Lrisky and Lsafe , it would not
matter if events in ancient Egypt were vastly different - we’d compare the lotteries the same way
with or without that additional value b.
But consider a further pair of lotteries: Lrisky + B and Lsafe + B. These are obtained bymodifying Lrisky and Lsafe such that, in all outcomes, the events of ancient Egypt are different
but you are uncertain of exactly how different they are. We are no longer dealing with a fixedvalue b of which you are certain. We are still adding the same value to all of them, but you are
uncertain of what value it is. And that uncertainty is described by the lottery B. To make theselotteries more concrete, perhaps you must decide between two lotteries in which the value of allpresent and future events match the lotteries Lrisky and Lsafe , but you are uncertain of the valueof past events such as those in ancient Egypt. As a devout totalist, you must evaluate outcomes
would be to increase the average value obtained by all persons in that outcome. And to add b to the value of
every outcome in a lottery L would be to increase the average value in every outcome by b. We might imagine
doing this by delivering a gift to every inhabitant of the outcomes of L, with that gift producing the same boost
in value, b, for each recipient. Even by the lights of averageism it seems that, if La is at least as good as Lb , thendelivering that gift to everyone shouldn’t change the ranking - a modified La with additional value b should still
be at least as good as Lb with additional value b. And so we will have an averageist analogue of the Egyptology
Objection presented above. And this objection will still be worrying for averageists, even if it is not quite asdevastating as its analogue is for totalists.

21

and lotteries based on the total aggregate of value across all of space and time. And even if you
know that those past events will turn out the same way no matter what you do, your uncertainty
over exactly how they would turn out is a part of your uncertainty over the total value of theoutcome. So you must compare Lrisky + B and Lsafe + B; you cannot simply Lrisky and Lsafe .
As it turns out, there are some possible lotteries B that lead us to say that Lrisky + B isstrictly better than Lsafe + B, even though Lrisky isn’t any better than Lsafe . And this judgement,that Lrisky +B is better, is implied by even the extremely weak principle of Stochastic Dominanceintroduced above.36
As a brief refresher, recall that Stochastic Dominance simply states that: if a lottery La gives
at least as high a probability as Lb of resulting in an outcome which is at least as good as O, forevery possible outcome O, then La is at least as good a lottery; and if La also gives a strictly
greater probability of turning out at least as good as some O, then it is strictly better.
Stochastic dominance is easy to spot graphically. To illustrate, consider the cumulative
probability graphs of the following two lotteries, L1 and L2 .
L1 : value 1 with probability 21 ; value 0 otherwise
L2 : value 2 with probability 13 ; value 1 with probability 13 ; value 0 otherwise1

Li (O4O O )

L1
L2

0.5
V (O)
1

2

Figure 1: Cumulative probability graphs for L1 and L2 ; here and below, O4O O denotes the setof outcomes in O as which O is at least as good.
36

We can reach a similar result, and thus a similar reductio, with axiologies other than totalism. Following fromFootnote 35, we can adopt averageism and, instead of improving/worsening events in ancient Egypt, distributing
an identical and morally valuable gift to each person in the world. You might be uncertain of the exact value ofthe gift - your uncertainty of its value might correspond to B. But still it seems that whether you distribute thatgift is irrelevant to whether the risky lottery is better or worse than the safe one.

22

On a graph like this, one can easily see when Stochastic Dominance says that L2 < L1 .
Cumulative probability, on the vertical axis, is just the probability that the lottery produces
an outcome no better than an outcome with some particular value. Meanwhile, Stochastic
Dominance says that one lottery is at least as good as another if its probability of producing an
outcome as good or better is just as high for all outcomes - or, equivalently, if the probability ofan outcome as good or worse is at least as low. So Stochastic Dominance will say that L2 < L1 if
and only if L2 ’s cumulative probability is always as low or lower than that of L1 , as it is on thisgraph. And, here, L2 often has strictly lower cumulative probability. So Stochastic Dominancesays it is strictly better than L1 .37
But accepting Stochastic Dominance doesn’t rule out denying Fanaticism. Lrisky doesn’tstochastically dominate Lsafe , as illustrated below.

1

Li (O4O O )Lsafe
LriskyV (O)1Figure 2: Cumulative probability graphs for Lrisky and Lsafe

Sometimes one is higher; sometimes the other. So Stochastic Dominance remains silent. And
it’s a good thing it does - to deny Fanaticism, we must not say that Lsafe is better than Lrisky .
But what about Lrisky + B and Lsafe + B? Stochastic Dominance is not necessarily silent
about that comparison. In particular, suppose that the lottery B looks like this. (I’ll describe
the required properties of B below.)37

This relationship between Stochastic Dominance and cumulative probability would break down if some out-

comes were incomparable to others. There would then be a difference between Oa <O Ob and the negation of
Oa ≺O Ob . But, fortunately, totalism gives us a total preorder of O, so we can sidestep this complexity.

23

1

Li (O4O O )
B

V (O)
;
Figure 3: An example of background uncertainty B - a Cauchy distribution

When B looks like that, we obtain the following graphs for Lrisky + B and Lsafe + B. Crucially,
the graph for Lsafe + B is never higher than that for Lrisky + B, and sometimes it is strictly lower.
So Stochastic Dominance says that Lsafe + B is strictly better.

1

Li (O4O O )Lsafe + B
Lrisky + B
V (O);Figure 4: Cumulative probability graphs for Lrisky + B and Lsafe + B

But how does this happen?
For Stochastic Dominance to hold, we need Lrisky + B to have at least as high a probability
as Lsafe + B of turning out at least as good as an outcome with value u, for all possible values u.
So take any real value u < V . What’s each lottery’s probability of doing at least that well?
Start with Lrisky + B. We know that Lsafe just gives one value (v) with certainty. So theprobability that Lsafe + B gives value u or better is just the probability that B gives value u − v
or better. (This corresponds to the area B2 + B3 on the graph below.) And then, for Lrisky + B,24

B
we’ll get value at least u either if Lrisky gives value V or if B gives at least value u. Denote theprobability that B gives value at least u by B3 (corresponding to that area on the graph below).
Then the probability that Lrisky + B turns out at least as good as value u is  + B3 (1 − ). And,
with a bit of simple arithmetic38 , we can see that this will be greater than the corresponding
probability for Lsafe + B if the area B2 is no greater than  × B1 .

Li (O'O O )

B1

B2

B3

V (O)

u−v u
;
Figure 5: Probability distribution of B

Here we have a probability graph of one possible B (importantly, not a cumulative probability
graph), with areas B1 , B2 , and B3 corresponding to the areas mentioned in the last paragraph.
If B2 is small enough compared to B1 , then Lrisky + B has at least as high a probability of u or
better as Lsafe + B does. And for it to be at least as high for all real u, we just need the interval
between u and u − v to have a tiny enough area under it. And, for that to happen, we just need
B to go down slowly enough as we approach −∞, and rise and fall quickly enough as we pass
the peak of the curve.
And there are some probability distributions that have this property, including many Cauchy
distributions.39 And some is enough. There is some such lottery Lrisky + B that is better thanLsafe + B, even though Lrisky is no better than Lsafe .
And so we have a new version of the Egyptology Objection, which I think is slightly milder
than the previous version. You may be faced with a decision between two lotteries that, if you
38

 + B3 (1 − ) ≥ B2 + B3 ⇔ B3 + (1 − B3 ) ≥ B2 + B3
⇔ (B1 + B2 ) ≥ B2
⇔ B2 ≤ B1 + B2 ,39

for which B2 ≤ B1

is a sufficient condition.

The result holds for any Cauchy distribution with a ‘scale factor’ of at least v . See Tarsney (n.d.: §5) andPomatto et al (2018).

25

exclude the value of events that occurred in ancient Egypt, correspond to Lrisky and Lsafe . And
which lottery is better overall? That depends on what happened in ancient Egypt, even though
you know the same events will have happened there no matter which you choose. If the eventsof ancient Egypt have value 0 (or b), then the risky lottery is no better than the safe one. But
if ancient Egypt might have contained greater or lesser value, and you aren’t certain how much,then it may be that the risky lottery is better after all. Much like under the classic Egyptology
Objection, your evaluation is sensitive to facts that seem irrelevant. But, unlike the original
version, the evaluation is not merely sensitive to what actually occurred there; instead, it issensitive just to your uncertainty about what happened there. That is perhaps less devastatinga problem, as an agent may know enough about events in ancient Egypt to constrain B to a lesstroublesome distribution. Still, for judgements to be sensitive to the agent’s beliefs about eventsin ancient Egypt at all is still, I think, rather absurd - absurd enough that we should be willingto accept Fanaticism to avoid it.40

6.2

The Indology Objection

But it gets worse. If you reject Fanaticism, you may also face what I’ll call the Indology Objection.
For context, Indology is the study of the history and culture of India. In particular, classicalIndology includes the study of the Indus Valley Civilisation, a Bronze Age civilisation whichlasted from 3300 BCE to 1300 BCE. Althought less well-known and understood than its contemporaries - ancient Egypt and Mesopotamia - it spanned a greater area than either and rivalled
them in population, technology, and urban infrastructure (see Wright 2009). We happen to knoweven less about what happened in the ancient Indus Valley than in ancient Egypt - archaeologicalresearch and excavations of key sites in India began centuries later than similar work in Britain,
Italy, and Egypt. So there is likely plenty left to learn in Indology.
Take the same Lrisky + B and Lsafe + B from above - lotteries such that the former is strictly
better, and yet Lrisky is no better than Lsafe . Such lotteries must exist, if Fanaticism is false. But
40

This version of the Egyptology Objection loosely resembles one which has been levelled against decision

theories that violate the Sure-Thing Principle (by, e.g., Briggs 2015). It turns out that following such a decision
theory in each of several consecutive, independent decisions will sometimes be inconsistent with following thetheory when deciding on a strategy for all of those decisions at once. Essentially, the combination of two lotteries
can be evaluated quite differently to how we evaluate those lotteries separately. And that is troubling. It is
unclear whether we should evaluate such decisions together or apart, globally or individually. But the correct
verdict depends on which way we do it.

26

suppose that you are no longer uncertain about the events of ancient Egypt; instead, you are
uncertain of what occurred in the ancient Indus Valley. The lottery B in Lrisky + B and Lsafe + B
represents only those events.
But, in Indology, there is a great deal more work that can be done! Had you the option,
you could spend many years researching the ancient Indus Valley, peeling back that uncertainty
and narrowing B down. Let’s suppose that with enough years of intensive research you could
eventually remove that uncertainty entirely and determine exactly which value to assign to theevents of the ancient Indus Valley - which value b the lottery B actually results in. If you did
first do that research, you would no longer need to choose between Lrisky + B and Lsafe + B.
Instead, you would choose between Lrisky with b added to the value of every outcome and Lsafe
with b added.
Recall Background Independence from above. If it holds, then Lrisky with any b added toevery outcome is better than Lsafe with the same b added if and only if Lrisky is better thanLsafe . And if it doesn’t, as we saw above, we face an even more severe version of the Egyptology
Objection. So we can safely assume that Background Independence holds.
From Background Independence we know that, whatever you might uncover in your research,
you would conclude that the risky lottery is no better than the safe lottery. For any value ofb you might pin down, you’ll establish that it’s no better to take the risky lottery. To judge
otherwise would be to accept Fanaticism. So you know what judgement you would make if you
simply learned more, no matter what it is you would actually learn. So why do the many yearsof research? Why not just update your judgement now and say that Lrisky + B is no better thanLsafe + B? You cannot. You must accept that Lrisky + B is strictly better, even though you canpredict with certainty that you would change your mind if you knew more. To be able to change
your mind you must go through those years of gruelling research and digging, even though you
already know which judgement you would conclude to be correct.41
This ambivalence seems deeply irrational. Surely we can sidestep those years of research
into how B turns out, and make the judgement required by every possible value of b. Surely
rationality requires that we do so, rather than require that we do not. But, if we deny Fanaticism,we must accept this inconsistency - an inconsistency which, to me, seems far more absurd than41

cf van Fraassen’s (1984) principle of Reflection and its application by White (2010:17-8). That form ofReflection implies that, if you know that you would set the probability of some hypothesis to p whether you
learned either proposition P or not-P , then its current probability should already be p. The version I am
assuming is somewhat similar but need not be quite as strong.

27simply accepting Fanaticism and even more absurd than the Egyptology Objection.42 43
Taking stock of this section, we have further reasons to accept Fanaticism. If we deny it thenwe are forced to accept a version of the Egyptology Objection wherein the judgements of agentslike us will be sensitive to their uncertainty about what occurred in ancient Egypt, even though
their choices do not affect ancient Egypt at all. If we reject both Fanaticism and Background
Independence, then we face an even more severe version of the Egyptology Objection: howsuch agents should compare their options will be sensitive to not just their uncertainty but also
what actually occurred in ancient Egypt. And even if we accept Background Independence (asI think we should), if we try to reject Fanaticism then a bizarre form of inconsistency arises
in how agents should compare their options - agents are sometimes forced to make judgements
which are inconsistent with anything they might learn to resolve their uncertainty about far-off,unrelated events in the ancient Indus Valley. Any one of these implications - let alone any twoof them - is absurd. In light of these, it seems a lot less appealing to deny Fanaticism.
42

Continuing from Footnotes 35 and 36, an analogue of the Indology Objection applies to averageism. (Ana-logues can also be constructed for other axiologies.) You might be facing two lotteries - one risky and one safe
- in which you know that, whatever the outcome, an identical gift will be given to everyone in the world. And
you’re uncertain of just how much value that gift will bring, with your uncertainty corresponding to B. Then, as
we saw above, there are some probability distributions for B that mean that you will judge Lrisky + B as strictly
better than Lsafe + B; but, if you opened the gift yourself and discovered what was inside, you already knowthat Lrisky with the additional value b would be no better than Lsafe with the additional value b. And this seems
absurd, just as its analogue was for the totalist.
43
A closely related objection to rejecting Fanaticism is that, even if we do reject it, we will not be spared frommany verdicts that seem deeply fanatical. As we saw in the case above, with the right uncertainty B about thevalue of unaffected events you must judge Lrisky + B better than Lsafe + B. In practice, will we often have theright uncertainty about unaffected events to lead us to judge otherwise extremely risky lotteries as better thanotherwise safe ones? Tarnsey (n.d.: §6) argues that we often will, for an epistemically rational agent and fora wide range of cases, including many resembling Lrisky and Lsafe (where the probability  is at least 1 in 109 ,and likely also a lot lower). His reasoning for this is that the moral value of distant events in our universe will
be roughly correlated with the number of inhabited planets in the universe; and our cosmological evidence andthe Drake equation together motivate a probability distribution over that number of inhabited planets which isrelevantly similar to the above B. But even apart from that reasoning, an epistemically rational agent would
still have some uncertainty over which probability distribution best characterises our cosmological evidence. So
they would still put some non-zero credence in distributions of the right sort. And this is enough - when we takethe probability-weighted average of many such distributions to get our overall distribution, it will then have theright properties (see ibid.: 37-9 for details). Given this, Stochastic Dominance by itself will lead us to making
judgements which seem fanatical. So we would gain little from rejecting Fanaticism.

28

7Conclusion

These are just some of the compelling arguments for accepting Fanaticism and, with it, fanaticalverdicts in cases like Dyson’s Wager. There are other compelling arguments too - for instance, allof the arguments for the stronger claims of expected value theory, or for expected utility theory
with an unbounded utility function. A fortiori, these also justify Fanaticism. For a sampling ofthese, I refer interested readers to Feller (1968), Ng (2012), Briggs (2015), and Thoma (2018).
Some philosophers still reject expected value theory (e.g., Buchak 2013, Smith 2014, Monton
2019, Tarsney n.d.), and presumably the arguments that imply it. But this is not enough toescape Fanaticism. As I have demonstrated here, there are compelling arguments in its favour arguments stronger than many of those for expected value theory.
To recap, if we deny Fanaticism then we must deny either Minimal Tradeoffs or transitivity,
and accept the counterintuitive verdicts which follow. Likewise, we must accept either: violationsof Scale Independence; or absurd sensitivity to the tiniest differences in probabilities and value.
So too, to deny Fanaticism, we must accept a version of the Egyptology Objection: for agents likeourselves, which judgement is correct can depend on our beliefs about what occurred in far-off,unrelated events, such as those in ancient Egypt. And, unless we wish to accept an even more
severe version of the Egyptology Objection, we also face the Indology Objection: we sometimes
ought to make judgements that we know we would reject if we learnt more, no matter what wemight learn.
Given all of this, it no longer seems so attractive to reject Fanaticism. As it turns out, thecure is worse than the disease. I would suggest that it is better to simply accept Fanaticism and,with it, fanatical verdicts such as that in Dyson’s Wager. We should accept that it is better toproduce some tiny probability of infinite moral gain (or arbitrarily high gain), no matter how
tiny the probability, than it is to produce some modest finite gain with certainty.
All of this has implications for normative decision theory more broadly, at least dialectically.
Philosophers often reject expected value theory because it implies Fanaticism, or because it
implies fanatical verdicts in cases like Dyson’s Wager (e.g., Monton 2019, Tarsney 2020). The
arguments I have given here suggest that this rejection is a little hasty. Doing so invites greater
problems than it solves. So I, for one, am thankful that expected value theory does imply
Fanaticism. We have little choice but to accept Fanaticism, so we might as well accept expectedvalue theory too.44
44This paper has benefitted from the input of many brilliant and generous colleagues. I am indebted to Alan

29

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