In defence of fanaticism

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 has
atinyprobability 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 long
that better outcome is good enough. But this seems fanatical . So you may be tempted to
abandon expected value theory.
But not so fast - denying all such fanatical verdicts brings serious problems. For one, you
must reject either that moral betterness is transitive or even a weak principle of tradeos.
For two, you must accept that judgements are either: inconsistent over structurally-identical
pairs of lotteries; or absurdly sensitive to small probability dierences. For three, you must
accept that the practical judgements of agents like us are sensitive to our beliefs about
far-o events that are unaected by our actions. And, for four, you may also be forced to
accept judgements which you know you would reject if you simply learned more. Better to
accept 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 rst charity distributes bednets in low-income countries in which malaria is
endemic.1With 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 uni-
verse. 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 operations
of human minds living blissful lives, and thereby allow morally valuable life to survive
indenitely long into the future.23From 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 innitely many blissful lives into existence.4
1I 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 benets, a total benet roughly equivalent to
that 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.
2Dyson (1981) was the rst to suggest positronium as a medium for computation and information storage.
This follows Dyson (1979), wherein it is argued that an innite duration of computation could be performed with
nite 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 dierent 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 sucient for my purposes.
3Would such articially-instantiated lives hold the same moral value as lives led by
esh-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 that
the lives of such simulations hold similar moral value.
4I have deliberately chosen a case involving many separate lives rather than a single person's life containing
innite value. Why? You might think that one individual's life can contribute only some bounded amount of
2
What ought you do, morally speaking? Which is the better option: saving a life with certainty,
or generating a tiny probability of bringing about innitely 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 some
probability of bringing about each of several outcomes, and each of those outcomes has some
value, specied by our moral theory. Expected value theory says that one option is better than
another if and only if it has the greater probability-weighted sum of value - the greater expected
value .5Here, 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 counterin-
tuitive 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. (10100is
greater than the number of atoms in the observable universe). The chance that the research
yields nothing at all would be 99.99... percent, with another 96 nines after that. And yet ex-
pected value theory says that it is better to take the bet, despite it being almost guaranteed
that 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 benet. Surely not, says my own intuition. On top
of 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 nite value. Expected value theory would say
that 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 nite value than
for one person to obtain innite 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.
5Note 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 and
less additional utility. The utility of an outcome may even be bounded, such that arbitrarily large amounts of
additional 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 raised
in Sections 4, 5, and 6. Where relevant, I will indicate in notes how the argument applies to expected utility
theory.
6I'll assume throughout that probability takes on only real values from 0 to 1.
3
be better to sacrice those billion or more lives for a minuscule chance at the innitely many
blissful lives (and likewise if the number of blissful lives were nite but still suciently many).
But endorsing that verdict, regardless of how low the probability of success and how high the
cost, seems fanatical . Likewise, even without innite value at stake, it would also seem fanatical
to judge a lottery with suciently tiny probability of arbitrarily high nite value as better than
getting some modest value with certainty.
Fanatical verdicts depend on more than just our theory of instrumental rationality, expected
value 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 of
value of each outcome; and that this total value increases linearly, without bound, with the sum
of value in all lives that ever exist. By totalism, the outcome containing innitely many blissful
lives is indeed a much better one than that in which one life is saved. And, as we increase the
number 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 re-
search is greater than that of malaria prevention. (Likewise, even if there are only nitely many
blissful lives at stake, for any tiny probability there can be suciently many of them to make the
risky 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 axiolo-
gies, including: averageism, critical-level views, prioritarianism, pure egalitarianism, maximin,
maximax, and narrow person-aecting views. Those axiologies each allow possible outcomes to
be 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.8But suce it
to say that just about any plausible axiology can deliver fanatical verdicts when combined with
expected value theory.
In general, we will sometimes be led to verdicts that seem fanatical if we endorse Fanaticism .9
7For instance, take (standard, welfarist) averageism. A population containing at least one blissful life of
innite (or arbitrarily long) duration will have average value greater than any nite value we choose. And so, to
generate an averageist analogue of Dyson's Wager, we can substitute an outcome containing this population for
the outcome of arbitrarily many lives in the original wager.
8Each of the other axiologies listed falls prey to devastating objections. See Arrhenius (2000), Huemer (2008),
Greaves (2017), and chapters 17-19 of Part (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 and
our axiology must not imply Fanaticism.
Fanaticism : For any tiny (nite) probability >0, and for any nite value v, there
is some nite Vthat is large enough that Lriskyis better than Lsafe(no matter which
scale those cardinal values are represented on).
Lrisky: valueVwith probability ; value 0 otherwise
Lsafe: valuevwith probability 1
The comparison of lotteries LriskyandLsaferesembles 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,Vneed not be innite, in case you think innite 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 dened here, it requires that the ranking of
LriskyaboveLsafeholds 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 value10are unique only up to positive ane transformations -
two outcomes represented by 0 and von one scale could instead be represented by 0 a+band
va+b(for any positive aand realb). 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 LriskyandLsafeeven though every one of their
outcomes contains enormous numbers of blissful lives, or enormous amounts of suering, as long
the dierences in value between those outcomes are proportional to 0, v, andV.
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 specic to innite values and instead uses
`Recklessness' for a claim more akin to my version of Fanaticism.). My formulation is slightly stronger than each
of theirs but also, unlike theirs, applicable even if innite total value cannot exist. For discussion of whether
outcomes with innite 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 theories
of 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;
Buon 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 in
cases similar to Dyson's Wager. Notable examples include Pascal (1669) himself, Part (1984:
§3.27), and H ajek (2014). And even they most often endorse such verdicts only because they are
a consequence of expected value theory, not because they see good independent justication for
them. I suspect that even many diehard adherents of expected value theory are uncomfortable
with 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 reject
that 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 rst is
a 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 dierences in prob-
ability. And the nal nails in the con are an updated version of Part'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.
11Smith's proposal can be interpreted in two dierent ways, only one of which rules out Fanaticism. By the
other interpretation, which Smith prefers, we still ignore events with probability below some threshold but, in any
lotteries over nitely many dierent outcomes, that threshold is set below the probability of the least probable
outcome. 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
2 Intuitions against fanaticism
There are various arguments (that can be directed) against Fanaticism that appear in the liter-
ature (see D'Alembert 1761; Buon 1777; Smith 2014; Schwitzgebel 2017; Monton 2019). And
there are various counterarguments as well (see H ajek 2014; Isaacs 2016; Monton 2019; and H ajek
n.d. for some of the most compelling).
But there is one extremely simple argument against it that, I suspect, is the most persuasive
of all: Fanaticism is just frightfully counterintuitive that it must be false.12It seems so very
implausible that I should give up a certainty of a good payo 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 payo.
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 et
al.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 ngerprint
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 re
ection, 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
12In eect, 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 from
fanatical verdicts - with a little more scrutiny, those intuitions may appear foolish too. At the
very least, the intuitive case against Fanaticism is dubious enough that we should consider the
case 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. Some
outcomes are better or worse than others. So assume that there exists some binary relation on
O, denoted by L b(O0 and some cardinal value vsuch that some lottery Lriskyas dened below
is no better than Lsafe, no matter how big Vis.
Lrisky: valueVwith probability ; 0 otherwise
Lsafe: valuevwith 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 least
25An alternative to endorsing Scale Independence is to adopt a variant of totalism that assigns more than just
a cardinal value to outcomes - perhaps a version of totalism that values outcomes on a ratio or unit scale. That
version would be far less parsimonious than the version I have been using, and I cannot think of any compelling
independent justication 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 rst is: there is some smallerv0>0
for whichLrisky isbetter than a sure outcome with value of v0, for suciently large V. The
second is: for anysuchv0,Lriskyisn't better than any sure outcome with value v0, no matter how
largeVis.26As we'll see, each possibility impales us on a respective horn of the dilemma.
Assume that the rst possibility holds. Lriskyis better than a sure outcome with value v0,
for some 0 < v0< vand some large V. Still,Lriskywon't be better than a sure outcome with
valuev- i.e.,Lsafe- or any sure outcome with even greater value (by Stochastic Dominance).
But somewhere below vthis changes - for some positive v0, the same doesn't hold.
Since that v0is a positive real number, we know that there will be some real ksuch that
kv0v. And that means that a sure outcome with value kv0is no worse than L1, no matter
how greatVis. But here is the problem: a certainty of kv0is the scaled-up counterpart of v0;
and the counterpart of Lrisky, which iskLrisky), is justLriskywithVreplaced by another value
which isktimes higher. That still has the same form as Lrisky. So it cannot be better than the
sure outcome, not if we reject Fanaticism. But then our verdict for v0versusLriskydoesn't match
our verdict for kv0versuskLrisky. So we violate Scale Independence.
But there was a second possibility: that there is no such v0>0; thatLriskyis no better
than a certainty of anyv0>0. If so, we can avoid the problem of scale dependence: there's no
inconsistency between the judgements for vand judgements for any `scaled-down' counterpart
v0, since allsuchv0compare the same way to lotteries like Lrisky.27But we then face another
serious problem.
Take the probabilities and 1. We can give a sequence of increasing probabilities in between
and 1, spaced evenly apart and as nely as we want:  < p 1< p 2< ::: < p n<1. By
assumption, noamount 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+1such that no amount of value at piis 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 less
likely, lotteries much like that in the previous section.
Crucially, that piandpi+1can be arbitrarily close together, since the same result holds no
matter how nely spaced the sequence was. No amount of value at piis better than anyamount
26This dierence is analogous to the dierence between weak and strong superiority in Arrhenius & Rabinowicz
(2015).
27The comparison won't be the same for values 0 and below, but that's okay. Such values cannot be scaled up
tov; 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 atpi, 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, piandpi+1are some tiny probabilities that are arbitrarily close
together.
L0: value 0 with probability 1
L1: value 101010with probability pi; 0 otherwise
L2: value0with probability pi+1; 0 otherwise
L3: value 101010with 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: L0L1L3andL0L2L3, butL1L2. And that's
a peculiar ranking. L1andL3are almost indistinguishable; their probabilities may dier by an
arbitrarily tiny amount. Likewise for L0andL2, except it's their payos that dier slightly. For
each pair, we must say that one is better than the other, but does not seem much better - in a
sense, the better lottery is only a trivial improvement over the other, whether by a slight increase
in payo or in probability. But, despite appearances, we must accept that L3ismuch better
thanL1, as well as that L2is much better than L0. How so? Between L1andL3comesL2.
The lottery L2has an astronomically smaller payo than L3and so (in a sense) is vastly worse
thanL3, and yet we cannot say that it is so bad so as to be worse than L1. Eectively, we must
accept that, starting from L3, it is no worse to lose almost all of the potential value via L2than
it is to lose that sliver of probability via L1. And similarly for L0andL2: between them must
comeL1.L1has astronomically greater potential payo than L0(and much higher probability
of positive payo too), yet we cannot say it's better than L2. Eectively, we cannot say it is any
better to gain that enormous value with probability pithan it is to gain an ever-so-slightly more
probable shot at tiny value . So those lotteries are not just made worse by those tiny changes
in probability or payos - they are made farworse (in an intuitive sense). That is the level of
sensitivity we must accept if we reject Fanaticism and maintain Scale Independence. But such
extreme 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
payo. So I do not think this sensitivity - this horn of the dilemma - is any less absurd than
17
what we faced earlier.
I should brie
y note that, beyond its intuitive implausibility, this sensitivity will likely lead to
practical diculties. 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 down
with arbitrary precision, at least not with our merely nite capacity for calculation.28But 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
L3or as bad as L1with its imperceptibly lower probability of success. We must either require
(unrealistically) that agents be arbitrarily precise in their probabilities29, or else accept that in
practice 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 depen-
dence, by which our judgements of lotteries can vary without any structural reasons for doing
so, or that those judgements are sensitive to imperceptibly small dierences 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 face
either 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
28For discussion of why we cannot require epistemic agents to settle on precise probabilities, see Schoeneld
(2012) and Joyce (2011). For discussion of how to evaluate lotteries with imprecise probabilities, see Seidenfeld
(2004), Huntley, Hable & Troaes (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 to
indierence among many of the lotteries L1toL4listed above or else indeterminacy of which is better than
which. At least in that case, with its vast dierences in value and only small dierences in probability, it \*\*\*
29In Smith's (2014:471) terms, our theory would then be `intolerant' of human imprecision, which he considers
a major problem.
30It appears earliest in MacMahan (1981: 115) but is often attributed to Part (1984: 420).
18
As a brief refresher, it goes like this. Suppose you are making a moral decision here and
now in the 21st Century, and the available outcomes dier only by some small-scale changes in
the 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 in
ancient Egypt, at least not if your actions do not change them. Intuitively, it seems absurd that
those unaltered, remote events make any dierence to your evaluation.31
But, under some axiologies, what you ought to do in such cases isdependent on unaltered,
remote events. Take a (standard, welfarist) averageist view. Here and now, should you bring
an 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 aect whether it is better to now bring an additional
person into existence or not. And, further, to be condent of which is the better outcome, you
may need to do some serious research into Egyptology (along with plenty of other historical
studies). But this is implausible. By intuition, we can ignore events that are unaected by our
actions and took place millennia ago, as we do under axiologies like totalism, critical-level views,
prioritarianism, and aggregative person-aecting views (and any other axiology which admits an
additively 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 in
practice.32Some 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 version
of 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 pro-
posals do.)33
31Such dependence brings practical problems too. If what you ought to do is dependent on such events - not
just 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 ( cfMogensen 2021). This seems a major disadvantage for a moral view.
32Even on other axiologies, including averageism, we face similar (albeit slightly less compelling) objections.
See Footnote 35.
33One 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 LaandLband any outcome O0, ifLa<
Lb, thenLa+O00 and some cardinal value v, some
lotteryLrisky(as dened below) is no better than Lrisky, no matter how big Vis.
Lrisky: valueVwith probability ; 0 otherwise
Lsafe: valuevwith probability 1
According to Background Independence, it must also hold that the lottery Lriskywith any
constantbadded to every outcome's value is also no better than Lsafewith that same badded
to every one of its outcomes' values. If we face a decision between LriskyandLsafe, it would not
matter if events in ancient Egypt were vastly dierent - we'd compare the lotteries the same way
with or without that additional value b.
But consider a further pair of lotteries: Lrisky+BandLsafe+B. These are obtained by
modifyingLriskyandLsafesuch that, in all outcomes, the events of ancient Egypt are dierent
but you are uncertain of exactly how dierent they are. We are no longer dealing with a xed
valuebof 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 these
lotteries more concrete, perhaps you must decide between two lotteries in which the value of all
present and future events match the lotteries LriskyandLsafe, but you are uncertain of the value
of 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 bto the value of
every outcome in a lottery Lwould 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 Lais at least as good as Lb, then
delivering that gift to everyone shouldn't change the ranking - a modied Lawith additional value bshould still
be at least as good as Lbwith 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 as
devastating 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 the
outcome. So you must compare Lrisky+BandLsafe+B; you cannot simply LriskyandLsafe.
As it turns out, there are some possible lotteries Bthat lead us to say that Lrisky+Bis
strictly better than Lsafe+B, even though Lriskyisn't any better than Lsafe. And this judgement,
thatLrisky+Bis better, is implied by even the extremely weak principle of Stochastic Dominance
introduced above.36
As a brief refresher, recall that Stochastic Dominance simply states that: if a lottery Lagives
at least as high a probability as Lbof resulting in an outcome which is at least as good as O, for
every possible outcome O, thenLais at least as good a lottery; and if Laalso 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, L1andL2.
L1: value 1 with probability1
2; value 0 otherwise
L2: value 2 with probability1
3; value 1 with probability1
3; value 0 otherwise
1 20:51
L2L1
V(O)Li(O4OO)
Figure 1: Cumulative probability graphs for L1andL2; here and below, O4OOdenotes the set
of outcomes inOas whichOis at least as good.
36We can reach a similar result, and thus a similar reductio , with axiologies other than totalism. Following from
Footnote 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 of
the gift - your uncertainty of its value might correspond to B. But still it seems that whether you distribute that
gift 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 of
an outcome as good or worse is at least as low. So Stochastic Dominance will say that L2<L1if
and only if L2's cumulative probability is always as low or lower than that of L1, as it is on this
graph. And, here, L2often has strictly lower cumulative probability. So Stochastic Dominance
says it is strictly better than L1.37
But accepting Stochastic Dominance doesn't rule out denying Fanaticism. Lriskydoesn't
stochastically dominate Lsafe, as illustrated below.
11Lsafe
Lrisky
V(O)Li(O4OO)
Figure 2: Cumulative probability graphs for LriskyandLsafe
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 Lsafeis better than Lrisky.
But what about Lrisky+BandLsafe+B? Stochastic Dominance is not necessarily silent
about that comparison. In particular, suppose that the lottery Blooks like this. (I'll describe
the required properties of Bbelow.)
37This relationship between Stochastic Dominance and cumulative probability would break down if some out-
comes were incomparable to others. There would then be a dierence between Oa<OOband the negation of
OaOOb. But, fortunately, totalism gives us a total preorder of O, so we can sidestep this complexity.
23
1
;B
V(O)Li(O4OO)
Figure 3: An example of background uncertainty B- a Cauchy distribution
WhenBlooks like that, we obtain the following graphs for Lrisky+BandLsafe+B. Crucially,
the graph for Lsafe+Bis never higher than that for Lrisky+B, and sometimes it is strictly lower.
So Stochastic Dominance says that Lsafe+Bis strictly better.
1
;;Lsafe+B
Lrisky+B
V(O)Li(O4OO)
Figure 4: Cumulative probability graphs for Lrisky+BandLsafe+B
But how does this happen?
For Stochastic Dominance to hold, we need Lrisky+Bto have at least as high a probability
asLsafe+Bof turning out at least as good as an outcome with value u, for allpossible 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 Lsafejust gives one value ( v) with certainty. So the
probability that Lsafe+Bgives value uor better is just the probability that Bgives value uv
or better. (This corresponds to the area B2+B3on the graph below.) And then, for Lrisky+B,
24
we'll get value at least ueither ifLriskygives value Vor ifBgives at least value u. Denote the
probability that Bgives value at least ubyB3(corresponding to that area on the graph below).
Then the probability that Lrisky+Bturns out at least as good as value uis+B3(1). And,
with a bit of simple arithmetic38, we can see that this will be greater than the corresponding
probability for Lsafe+Bif the areaB2is no greater than B1.
uvu
;B
B2B1B3V(O)Li(O'OO)
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;andB3corresponding to the areas mentioned in the last paragraph.
IfB2is small enough compared to B1, thenLrisky+Bhas at least as high a probability of uor
better asLsafe+Bdoes. And for it to be at least as high for allrealu, we just need the interval
betweenuanduvto have a tiny enough area under it. And, for that to happen, we just need
Bto go down slowly enough as we approach 1, 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.39And some is enough. There is some such lottery Lrisky+Bthat is better than
Lsafe+B, even though Lriskyis 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+(1B3)B2+B3
,(B1+B2)B2
,B2B1+B2, for which B2B1is a sucient condition.
39The result holds for any Cauchy distribution with a `scale factor' of at leastv
. See Tarsney (n.d.: §5) and
Pomatto et al (2018).
25
exclude the value of events that occurred in ancient Egypt, correspond to LriskyandLsafe. 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 events
of 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 is
sensitive just to your uncertainty about what happened there. That is perhaps less devastating
a problem, as an agent may know enough about events in ancient Egypt to constrain Bto a less
troublesome distribution. Still, for judgements to be sensitive to the agent's beliefs about events
in ancient Egypt at all is still, I think, rather absurd - absurd enough that we should be willing
to 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, classical
Indology includes the study of the Indus Valley Civilisation, a Bronze Age civilisation which
lasted from 3300 BCE to 1300 BCE. Althought less well-known and understood than its contem-
poraries - 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 know
even less about what happened in the ancient Indus Valley than in ancient Egypt - archaeological
research 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+BandLsafe+Bfrom above - lotteries such that the former is strictly
better, and yet Lriskyis no better than Lsafe. Such lotteries must exist, if Fanaticism is false. But
40This 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 the
theory when deciding on a strategy for all of those decisions at once. Essentially, the combination of two lotteries
can be evaluated quite dierently 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 BinLrisky+BandLsafe+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 Bdown. 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 the
events of the ancient Indus Valley - which value bthe lottery Bactually results in. If you did
rst do that research, you would no longer need to choose between Lrisky+BandLsafe+B.
Instead, you would choose between Lriskywithbadded to the value of every outcome and Lsafe
withbadded.
Recall Background Independence from above. If it holds, then Lriskywith anybadded to
every outcome is better than Lsafewith the same badded if and only ifLriskyis better than
Lsafe. 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 of
byou 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 years
of research? Why not just update your judgement now and say that Lrisky+Bis no better than
Lsafe+B? You cannot. You must accept that Lrisky+Bis strictly better, even though you can
predict 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 howBturns 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 than
41cfvan Fraassen's (1984) principle of Re
ection and its application by White (2010:17-8). That form of
Re
ection implies that, if you know that you would set the probability of some hypothesis to pwhether you
learned either proposition Por 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 then
we are forced to accept a version of the Egyptology Objection wherein the judgements of agents
like us will be sensitive to their uncertainty about what occurred in ancient Egypt, even though
their choices do not aect ancient Egypt at all. If we reject both Fanaticism and Background
Independence, then we face an even more severe version of the Egyptology Objection: how
such 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 (as
I 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-o,
unrelated events in the ancient Indus Valley. Any one of these implications - let alone any two
of them - is absurd. In light of these, it seems a lot less appealing to deny Fanaticism.
42Continuing 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 Bthat mean that you will judge Lrisky+Bas strictly
better than Lsafe+B; but, if you opened the gift yourself and discovered what was inside, you already know
thatLriskywith the additional value bwould be no better than Lsafewith the additional value b. And this seems
absurd, just as its analogue was for the totalist.
43A closely related objection to rejecting Fanaticism is that, even if we do reject it, we will not be spared from
many verdicts that seem deeply fanatical. As we saw in the case above, with the right uncertainty Babout the
value of unaected events you must judge Lrisky+Bbetter than Lsafe+B. In practice, will we often have the
right uncertainty about unaected events to lead us to judge otherwise extremely risky lotteries as better than
otherwise safe ones? Tarnsey (n.d.: §6) argues that we often will, for an epistemically rational agent and for
a wide range of cases, including many resembling Lrisky andLsafe(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 and
the Drake equation together motivate a probability distribution over that number of inhabited planets which is
relevantly 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 take
the probability-weighted average of many such distributions to get our overall distribution, it will then have the
right 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
7 Conclusion
These are just some of the compelling arguments for accepting Fanaticism and, with it, fanatical
verdicts in cases like Dyson's Wager. There are other compelling arguments too - for instance, all
of 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 of
these, 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 to
escape 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 Tradeos or transitivity,
and accept the counterintuitive verdicts which follow. Likewise, we must accept either: violations
of Scale Independence; or absurd sensitivity to the tiniest dierences in probabilities and value.
So too, to deny Fanaticism, we must accept a version of the Egyptology Objection: for agents like
ourselves, which judgement is correct can depend on our beliefs about what occurred in far-o,
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 we
might learn.
Given all of this, it no longer seems so attractive to reject Fanaticism. As it turns out, the
cure 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 to
produce some tiny probability of innite moral gain (or arbitrarily high gain), no matter how
tiny the probability, than it is to produce some modest nite 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 expected
value theory too.44
44This paper has benetted from the input of many brilliant and generous colleagues. I am indebted to Alan
29
References
[1] Arrhenius, Gustaf 2000. Future Generations: A Challenge for Moral Theory (Uppsala: Uni-
versity Press).
[2] Arrhenius, Gustaf, & Wlodek Rabinowicz 2015. `Value superiority', The Oxford Handbook of
Value Theory : 225-48.
[3] Arrow, Kenneth 1971. Essays in the Theory of Risk-Bearing (Chicago: Markham).
[4] Askell, Amanda 2019. Pareto Principles in Innite Ethics , PhD dissertation, New York Uni-
versity.
[5] Bader, Ralf 2018, `Stochastic dominance and opaque sweetening', Australasian Journal of
Philosophy 96, no. 3:498-507.
[6] Baron, Jonathan, Laura Granato, Mark Spranca, & Eva Teubal 1993. `Decision making biases
in children and early adolescents: Exploratory studies', Merrill Palmer Quarterly 39: 23{47.
[7] Beckstead, Nicholas 2013. On the Overwhelming Importance of Shaping the Far Future , PhD
dissertation, Rutgers University.
[8] Beckstead, Nicholas and Teruji Thomas. `A paradox for tiny probabilities and enormous
values', unpublished manuscript.
[9] Bostrom, Nick 2009. `Pascal's mugging', Analysis 69, no. 3: 443-5.
[10] Bostrom, Nick 2011, `Innite ethics', Analysis and Metaphysics 10:9.
[11] Buchak, Lara 2013, Risk and Rationality (Oxford: Oxford University Press).
[12] Bradley, Seamus 2015. `How to choose among choice functions', Proceedings of the Ninth
International Symposium on Imprecise Probability: Theories and Applications : 57{66.
H ajek, Christian Tarsney, Teru Thomas, Jasper Hedges, Timothy L. Williamson, Andreas Mogensen, and two
anonymous reviewers for detailed comments. Likewise to Philip Trammell, Anders Sandberg, Shang Long Yeo and
audiences at both the Australian National University and the Forethought Foundation for valuable discussions.
For their generous nancial support, I thank the Australian Government Research Training Program and the
Forethought Foundation for Global Priorities Research. As well, I am grateful to all of the sta at the Global
Priorities Institute at the University of Oxford for inviting me to visit for several months in late 2019 and early
2020, and to convene a reading group on innitary fanaticism while I was there. Were it not for their hospitality,
support, and philosophical insights, this paper would not have been written.
30
[13] Bradley, Seamus and Katie Steele 2015. `Can free evidence be bad? Value of information
for the imprecise probabilist', Philosophy of Science 83: 1-28.
[14] Briggs, Rachael 2015. `Costs of abandoning the sure-thing principle', Canadian Journal of
Philosophy 45, no.5: 827-40.
[15] Buon, Georges-Louis Leclerc de 1777. `Essai d'Arithm etique Morale', in Supplement a
l'Histoire Naturelle , vol. 4 (Paris: L'Imprimerie Royale).
[16] Chalmers, David 2010. `The singularity: A philosophical analysis', Journal of Consciousness
Studies , 17, no. 9: 7-65.
[17] Chen, Daniel, Tobias Moskowitz, & Kelly Shue 2016. `Decision making under the gambler's
fallacy: Evidence from asylum judges, loan ocers, and baseball umpires', The Quarterly
Journal of Economics 131, no. 3: 1181-242.
[18] d'Alembert, Jean le Rond 1761. Opuscules Math ematiques ou M emoires sur di erens sujets
de G eom etrie, de M echanique, d'Optique, d'Astronomie, Tome Second (Paris: David).
[19] Dyson, Freeman 1981. `Life in the universe', unpublished Darwin Lecture, Darwin College,
Camrbridge, Nov 10, 1981.
[20] Dyson, Freeman 1979. `Time without end: Physics and biology in an open universe', Reviews
of Modern Physics , 51, no. 3.
[21] Easwaran, Kenny 2014, `Decision theory without representation theorems', Philosopher's
Imprint 14, no. 27:1-30.
[22] Garrett, Brandon, Gregory Mitechell & Nicholas Scurich 2018. `Comparing categorical and
probabilistic ngerprint evidence', Journal of Forensic Sciences , 63, no. 6: 1712-7.
[23] GiveWell 2020. `GiveWell's cost-eectiveness analyses', GiveWell , available at:
https://www.givewell.org/how-we-work/our-criteria/cost-eectiveness/cost-eectiveness-
models. (Accessed April 2020)
[24] Greaves, Hilary 2017. `Population axiology', Philosophy Compass , 12, no. 11: 12442.
[25] Gurmankin, Levy, & Jonathan Baron 2005, `How bad is a 10% chance of losing a toe?:
Judgments of probabilistic conditions by doctors and laypeople', Memory and Cognition 33:
1399{406.
31
[26] H ajek, Alan 2014. `Unexpected expectations', Mind , 123, no. 490: 533-67.
[27] H ajek, Alan n.d. `Most counterfactuals are false', unpublished manuscript.
[28] H ajek, Alan & Wlodek Rabinowicz n.d. `Degrees of commensurability and the repugnant
conclusion', unpublished manuscript.
[29] Hanoch, Yaniv et al. 2019. `Reaping the benets and avoiding the risks: Unrealistic optimism
in the health domain', Risk Analysis 39, no. 4: 792-804.
[30] Huemer, Michael 2008. 'In defence of repugnance', Mind 117.468: 899-933.
[31] Huntley, Nathan, Robert Hable, and Matthias Troaes 2014. `Decision making', in T. Au-
gustin et al. (eds) Introduction to Imprecise Probabilities (New York: John Wiley Sons):
190{206.
[32] Isaacs, Yoaav 2016. `Probabilities cannot be rationally neglected', Mind 125, no. 499: 759-
62.
[33] Joyce, James 2011. `A defence of imprecise credences in inference and decision making',
Philosophical Perspectives 24: 281-323.
[34] Kahneman, Daniel & Amos Tversky 1982. `Evidential impact of base rates', in D. Kah-
neman, P. Slovic & A. Tversky (ed.). Judgment under uncertainty: Heuristics and biases
(Cambridge: Cambridge University Press): 153{60.
[35] McMahan, Jeerson 1981. `Problems of population theory', Ethics 92, no. 1: 96-127.
[36] McClelland, Gary, William Schulze & Don Coursey 1993. `Insurance for low-probability
hazards: A bimodal response to unlikely events', in Making Decisions About Liability and
Insurance (Dordrecht: Springer): 95-116.
[37] Mogensen, Andreas 2021. `Maximal cluelessness', Philosophical Quarterly 71.1:141-62.
[38] Monton, Bradley 2019. `How to avoid maximising expected utility', Philosophers' Imprint ,
19.
[39] Ng, Yew-Kwang 2012. `A case for happiness, cardinalism, and interpersonal comparability',
The Economic Journal , 107: 1848-58.
[40] Part, Derek 1984. Reasons and Persons (Oxford: Oxford University Press).
32
[41] Pascal, Blaise 1852 (1669). Pens ees (Paris: Dezobry et E. Magdeleine).
[42] Pomatto, Luciano, Philipp Strack, & Omer Tamuz 2018. `Stochastic dominance under in-
dependent noise', arXiv e-prints , arXiv:1807.06927.
[43] Rabin, Matthew 2000. `Risk aversion and expected-utility theory: A calibration theorem',
Econometrica , 68, no. 5: 1281-92.
[44] Sandberg, Anders, Grand Futures , unpublished manuscript.
[45] Schoeneld, Miriam 2012. `Chilling out on epistemic rationality', Philosophical Studies 158:
197-219.
[46] Schoeneld, Miriam 2014. `Decision making in the face of parity', Philosophical Perspectives
28.
[47] Schwitzgebel, Eric 2017. `1% skepticism', No^ us 51.2: 271-90.
[48] Seidenfeld, Teddy 2004. `A contrast between two decision rules for use with (convex) sets of
probabilities: Gamma-maximin versus E-admissibility', Synthese 140: 69{88.
[49] Smith, Nicholas J.J. 2014.`Is evaluative compositionality a requirement of rationality?', Mind
123.490: 457-502.
[50] Tarsney, Christian. `Exceeding expectations: Stochastic dominance as a general decision
theory', unpublished manuscript.
[51] Temkin, Larry 2014. Rethinking the Good (Oxford: Oxford University Press).
[52] Thoma, Johanna 2018. `Risk aversion and the long run', Ethics , 129, no. 2: 230-53.
[53] Tipler, Frank 1986. `Cosmological limits on computation', Internation Journal of Theoretical
Physics , 25, no. 6.
[54] Gilovich, Thomas, Amos Tversky & Robert Vallone 1985. `The hot hand in basketball: On
the misperception of random sequences', Cognitive Psychology , 17, no. 3: 295-314.
[55] Tversky, Amos & Daniel Kahneman 1974. `Judgment under uncertainty: Heuristics and
biases', Science , 185: 1124{31.
[56] Tversky, Amos & Daniel Kahneman 1983. `Extensional versus intuitive reasoning: The
conjunction fallacy in probability judgment', Psychological Review , 90: 293{315.
33
[57] van Fraassen, Bas 1984. `Belief and the will,' Journal of Philosophy 81: 235-56.
[58] White, Roger 2010. `Evidential symmetry and mushy credence,' Oxford Studies in Episte-
mology 3: 161-86.
[59] Wilkinson, Hayden 2020. `Innite aggregation: Expanded addition', Philosophical Studies .
[60] Wilkinson, Hayden. `Chaos, add innitum', unpublished manuscript.
[61] Wright, Rita 2009. The Ancient Indus: Urbanism, Economy, and Society , (Cambridge:
Cambridge University Press).
34