In defence of fanaticism

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 is endemic.  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 operations of 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 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. 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 with finite 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 that the 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 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 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 some probability of bringing about each of several outcomes, and each of those outcomes has some value, specified 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.  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 10 100 . (10 100 is 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 expected 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 benefit. 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 finite 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 finite value than for 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 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.  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 the cost, seems fanatical. Likewise, even without infinite value at stake, it would also seem fanatical to judge a lottery with sufficiently tiny probability of arbitrarily high finite 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 infinitely 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 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 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 axiologies, including: averageism, critical-level views, prioritarianism, pure egalitarianism, maximin, maximax, and narrow person-affecting 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.  8  But suffice 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 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, 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.  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) .  9  This 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 (finite) probability > 0, and for any finite value v, there is some finite V that is large enough that L risky is better than L safe (no matter which scale those cardinal values are represented on). L risky : value V with probability ; value 0 otherwise And our intuitions are no better with decision-making in the face of low-probability events. L safe : value v with 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 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.
 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 . 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 differences in value.  13  And as a cardinal value function, V is unique only up to affine transformations -for any V 1 or V 2 we might use here, V 2 (O) = a × V 1 (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. 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 L R ). 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 L a , L b , L c ∈ L, if L a L b and L b L c , then L a L c . 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 the same, 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(O k ) = L(O), where O k is an outcome in O such that V (O k ) = k × V (O) .  15  This can be made more precise. Let V i denote the random variable corresponding to a given lottery L i , which outputs the total value of the outcome -equivalently, the probability that V i takes on a value in the interval [a, b] is given by L i ({O|V (O) ∈ [a, b]}). Then, for any L a , L b ∈ L, define L a + L b as some lottery on O -there may be several -which corresponds to the random variable (V a + V b ). Equivalently, L a + L b is some lottery such that L a + L b ({O|V (O) ∈ [c, d]}) is equal to the probability that (V a + V b ) takes on a value in the interval [c, d]. 16 Some 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. We can express Stochastic Dominance as follows. Here, O O denotes the set of outcomes in O that are at least as good as O . Take the lottery L 0 , which might represent a stranger's life being saved. Stochastic Dominance: For any L a , L b ∈ L, if L a (O O ) ≥ L b (O O ) for all O ∈ O, then L a L b . If, as well, L a (O O ) > L b (O O ) for some O ∈ O, then L a L b . I find L 0 : value 1 with probability 1 And take another lottery L 1 by which vastly more strangers are saved, with a very slight probability of failure -perhaps 10 10 lives saved with probability 0.999999 of success. L 1 : value 10 10 with probability 0.999999; value 0 otherwise Intuitively, L 1 seems better. Accepting a slightly lower probability of success for a vastly greater payoff seems a good trade. But then consider L 2 , which has a slightly lower probability of success but, if successful, results in many more lives saved. L 2 : value 10 10 10 with probability 0.999999 2 ; value 0 otherwise  19  The reasoning behind this is as follows. We represented the value of one additional life being saved with a 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 are produced 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. This seems better than L 1 , or at least it seems that there is some number of lives high enough that it would be better. And so we could continue, with L 3 , L 4 , and so on until some L n , such that 0.999999 n 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 the sequence 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 transitivity of ; the second, Minimal Tradeoffs. Minimal Tradeoffs: There is some ratio r < 1 such that, for any real value v and any probability p, there is some real r * such that, on any cardinal representation, the lottery represented by L a is better than that represented by L b (as defined below). L a : value v with probability p; 0 otherwise L b : value r * ×v with probability r × p; 0 otherwise This principle has intuitive force. For any given lottery with the form of L a , 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 can always 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 more intuitively plausible that Fanaticism is false than that Minimal Tradeoffs is true. And, other than clashing 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 p between p and 0 (which might be unknown or vague) at which we have a discontinuity, such that no value with probability below p is better than any value with probability above p . One way to do 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, no matter how great the value. And what of it? It seems a little counterintuitive, but perhaps less so 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.
 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 to tiny changes (even more so than if you accepted Fanaticism). Scale Independence: For any lotteries L a and L b , if L a L b , then k • L a k • L b for any positive, real k. I find Scale Independence highly plausible.  21  Take any pair of lotteries, perhaps a pair that can be cardinally represented by L 1 and L 2 . 22 L 1 : value 1 with probability 1 L 2 : 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 • L 1 and 2 • L 2 -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. as above, but with the value of each outcome doubled (on the same cardinal scale). If we ranked L 1 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 L 1 is better than L 2 should plausibly also lead us to say the same about k • L 1 and k • L 2 , 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 -by definition, 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 to tangible 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 the total value of the outcome. So, in keeping with intuition, it should not matter in the slightest whether we are comparing the lotteries L 1 and L 2 above or instead some scaled-up multiples k • L 1 and k • L 2 . 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 L 1 and 2k − 2 additional lives to that outcome in L 2 ). But each additional copy should be no more or less valuable than the first; they should contribute the same to our evaluation of the lottery.  23  So, I would claim, we should rank the resulting lotteries just the same way as we ranked them without those k − 1 copies 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 of outcomes 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 to the average value of all lives no matter how many such periods of bliss have already been experienced. We can copy 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. similarly? For us to be able not to, our evaluations of lotteries must depend on more than just the probabilities 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 our theory 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 should be 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 • L 1 and k • L 2 not be compared in a similar way to L 1 and L 2 ; 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 L risky as defined below is no better than L safe , no matter how big V is. L risky : value V with probability ; 0 otherwise L safe : 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 least if we maintain Stochastic Dominance and transitivity). The first is: there is some smaller v > 0 for which L risky is better than a sure outcome with value of v , for sufficiently large V . The second is: for any such v , L risky isn't better than any sure outcome with value v , 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. L risky is better than a sure outcome with value v , for some 0 < v < v and some large V . Still, L risky won't be better than a sure outcome with value v -i.e., L safe -or any sure outcome with even greater value (by Stochastic Dominance). But somewhere below v this changes -for some positive v , the same doesn't hold. Since that v is a positive real number, we know that there will be some real k such that k × v ≥ v. And that means that a sure outcome with value k × v is no worse than L 1 , no matter how great V is. But here is the problem: a certainty of k × v is the scaled-up counterpart of v ; and the counterpart of L risky , which is k • L risky ), is just L risky with V replaced by another value which is k times higher. That still has the same form as L risky . So it cannot be better than the sure outcome, not if we reject Fanaticism. But then our verdict for v versus L risky doesn't match our verdict for k × v versus k • L risky . So we violate Scale Independence. But there was a second possibility: that there is no such v > 0; that L risky is no better than a certainty of any v > 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 , since all such v compare the same way to lotteries like L risky .  27  But 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 finely as we want: < p 1 < p 2 < . . . < p n < 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 p i , p i+1 such that no amount of value at p i is better than any amount of value with probability p i+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 p i and p i+1 can be arbitrarily close together, since the same result holds no matter how finely spaced the sequence was. No amount of value at p i is better than any amount of value at p i+1 . We could have some astronomical value at p i , and an imperceptibly small amount of value at p i+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 is some arbitrarily small number. And, again, p i and p i+1 are some tiny probabilities that are arbitrarily close together. L 0 : value 0 with probability 1 L 1 : value 10 10 10 with probability p i ; 0 otherwise L 2 : value with probability p i+1 ; 0 otherwise L 3 : value 10 10 10 with probability p i+1 ; 0 otherwise By the above, if we reject Fanaticism and maintain Scale Independence, then we are forced to rank these lotteries as follows: L 0 ≺ L 1 ≺ L 3 and L 0 ≺ L 2 ≺ L 3 , but L 1 ⊀ L 2 . And that's a peculiar ranking. L 1 and L 3 are almost indistinguishable; their probabilities may differ by an arbitrarily tiny amount. Likewise for L 0 and L 2 , 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 a sense, the better lottery is only a trivial improvement over the other, whether by a slight increase in payoff or in probability. But, despite appearances, we must accept that L 3 is much better than L 1 , as well as that L 2 is much better than L 0 . How so? Between L 1 and L 3 comes L 2 . The lottery L 2 has an astronomically smaller payoff than L 3 and so (in a sense) is vastly worse than L 3 , and yet we cannot say that it is so bad so as to be worse than L 1 . Effectively, we must accept that, starting from L 3 , it is no worse to lose almost all of the potential value via L 2 than it is to lose that sliver of probability via L 1 . And similarly for L 0 and L 2 : between them must come L 1 . L 1 has astronomically greater potential payoff than L 0 (and much higher probability of positive payoff too), yet we cannot say it's better than L 2 . Effectively, we cannot say it is any better to gain that enormous value with probability p i than 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 payoffs -they are made far worse (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 payoff. So I do not think this sensitivity -this horn of the dilemma -is any less absurd than what we faced earlier. I should briefly note that, beyond its intuitive implausibility, this sensitivity will likely lead to practical 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 down with 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 L 3 or as bad as L 1 with its imperceptibly lower probability of success. We must either require (unrealistically) that agents be arbitrarily precise in their probabilities  29  , 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 dependence, by which our judgements of lotteries can vary without any structural reasons for doing so, 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. 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. Background Independence says that we can take any two lotteries and add a constant value to every 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, L a and L b , such that: L a is at least as good as L b ; and, if we added a certain constant b to the value 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 value that would result, including all valuable events across all of space and time. That includes the events that occurred in ancient Egypt. And we can assume that the same events occurred in ancient Egypt in every outcome of either L a or L b .  34  With those events included, L a is at least as good as L b . But what if we consider a different hypothetical pair of lotteries, identical to L a and L b 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 increased in value by b. Then the total value of every outcome would also be increased by b. And, given that L a and L b were the lotteries that violated Background Independence, we know that this changes their ranking. So our modified version of L a is no longer at least as good as the modified version of L b . Thus, if we deny Background Independence, we face a rather severe version of the Egyptology Objection: when choosing between two lotteries, which is better can vary depending on what events occurred in ancient Egypt, even if the choice doesn't change those events at all (cf Beckstead and Thomas n.d.).  35  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: L risky and L safe . If we reject Fanaticism then, for some tiny enough probability > 0 and some cardinal value v, some lottery L risky (as defined below) is no better than L risky , no matter how big V is. L risky : value V with probability ; 0 otherwise L safe : value v with probability 1 According to Background Independence, it must also hold that the lottery L risky with any constant b added to every outcome's value is also no better than L safe with that same b added to every one of its outcomes' values. If we face a decision between L risky and L safe , 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: L risky + B and L safe + B. These are obtained by modifying L risky and L safe 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 fixed value 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 these lotteries more concrete, perhaps you must decide between two lotteries in which the value of all present and future events match the lotteries L risky and L safe , 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 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 L a is at least as good as L b , then delivering that gift to everyone shouldn't change the ranking -a modified L a with additional value b should still be at least as good as L b 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 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 L risky + B and L safe + B; you cannot simply L risky and L safe . As it turns out, there are some possible lotteries B that lead us to say that L risky + B is strictly better than L safe + B, even though L risky isn't any better than L safe . And this judgement, that L risky +B is 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 L a gives at least as high a probability as L b of resulting in an outcome which is at least as good as O, for every possible outcome O, then L a is at least as good a lottery; and if L a 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, L 1 and L 2 . L 1 : value 1 with probability 1 2 ; value 0 otherwise L 2 : value 2 with probability  1  3 ; value 1 with probability 1 3 ; value 0 otherwise 1 2 0.5 1 L 2 L 1 V (O) L i (O O O ) Figure  1 : Cumulative probability graphs for L 1 and L 2 ; here and below, O O O denotes the set of 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 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 L 2 L 1 . 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. Then the probability that L risky + B turns out at least as good as value u is + B 3 (1 − ). And, with a bit of simple arithmetic  38  , we can see that this will be greater than the corresponding probability for L safe + B if the area B 2 is no greater than × B 1 . If B 2 is small enough compared to B 1 , then L risky + B has at least as high a probability of u or better as L safe + 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. u − v u ; B B 2 B 1 B 3 V (O) L i (O O O ) And there are some probability distributions that have this property, including many Cauchy distributions.  39  And some is enough. There is some such lottery L risky + B that is better than L safe + B, even though L risky is no better than L safe . 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 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 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: 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-off, 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.  42  Continuing from Footnotes 35 and 36, an analogue of the Indology Objection applies to averageism. (Analogues 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 L risky + B as strictly better than L safe + B; but, if you opened the gift yourself and discovered what was inside, you already know that L risky with the additional value b would be no better than L safe 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 from many verdicts that seem deeply fanatical. As we saw in the case above, with the right uncertainty B about the value of unaffected events you must judge L risky + B better than L safe + B. In practice, will we often have the right uncertainty about unaffected 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 L risky and L safe (where the probability is at least 1 in 10 9 , 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. 38 + B 3 (1 − ) ≥ B 2 + B 3 ⇔ B 3 + (1 − B 3 ) ≥ B 2 + B 3 ⇔ (B 1 + B 2 ) ≥ B 2 ⇔ B 2 ≤ B 1 + B 2 , for which B 2 ≤ B 1 is a sufficient condition.
 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 favourarguments 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: violations of 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 like ourselves, 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 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 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 expected value theory too.  44   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 be defined, 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 O R ⊂ 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 O R .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 the evidential probability of each outcome arising. Almost all of what follows can be read in terms of either. With minor changes, what follows could also be read in terms of objective chance or whatever notion of probability you consider relevant for decision making. Whichever notion it is, let L be the set of all lotteries on O, and L R the set of all lotteries on a given subset O R . So each 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 of an outcome O (when L(O) = 1) I'll often write L as O. To represent a lottery with the same probabilities 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 L a + L b to represent the lottery obtained fromadding together the values of the outcomes of lotteries L a and L b , run independently. 15
 Background Independence: For any lotteries L a and L b and any outcome O , if L a L b , then L a + O L b + O . Recall that the sum of two lotteries L+O is simply the lottery you get if you run each lottery independently and sum up the values of their outcomes. But, here, O is a very simple lotteryit is just the outcome O with certainty. That outcome has some cardinal value; call it b. The lottery L a + O is simply the lottery L a with value b added to the value of each outcome. So
 Figure 5 : 5 Figure 5: Probability distribution of B

 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 L infinite and L safe 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 the outcome in which no lives are saved with value 0. By Fanaticism, L risky is better than L safe , at least for large enough finite V . And Stochastic Dominance implies that L infinite is better than L risky . 19  4 A continuum argumentYou might argue for Fanaticism in the following way: expected value theory is true; expected value 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. Such verdicts 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 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. Both are 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 L safe and L risky from the definition of Fanaticism. In L safe , we can set v = 1. And define L infinite as follows. L infinite : an outcome containing infinitely many blissful lives with probability ; value 0 otherwise L risky : value V with probability ; value 0 otherwise L safe : value 1 with probability 1 In this and the next two sections, I'll give four arguments for Fanaticism but not for expected value theory. Here is the first, which originates in Beckstead (2013: 139-47) and reappears in Beckstead and Thomas (n.d.: 4-6).
 14lotteries L 1 and L 2 above; their outcomes could just as easily have been represented on another scale such that they had values k, 2k, and 0 -the same values as k • L 1 and k • L 2 had on the previous scale. On that new scale, we of course still say that L 1 is better than L 2 if and only if we said so on the original scale, else we would be inconsistent. But why not judge k • L 1 and k • L 2
 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 differ 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 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 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 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 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 -not just in the remote future but also the remote past, of which we are often clueless -then we have little guidance 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 unaffected by our actions 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 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. 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 version of the Egyptology Objection. (As we'll see below, you may also face another even more serious objection.)
 So Stochastic Dominance will say that L 2 L 1 if and only if L 2 's cumulative probability is always as low or lower than that of L 1 , as it is on this graph. And, here, L 2 often has strictly lower cumulative probability. So Stochastic Dominance says it is strictly better than L 1 . 37  But accepting Stochastic Dominance doesn't rule out denying Fanaticism. L risky doesn't stochastically dominate L safe , as illustrated below. Cumulative probability graphs for L risky and L safe 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 L safe is better than L risky .But what about L risky + B and L safe + B? Stochastic Dominance is not necessarily silent about that comparison. In particular, suppose that the lottery B looks like this. (I'll describe An example of background uncertainty B -a Cauchy distribution When B looks like that, we obtain the following graphs for L risky + B and L safe + B. Crucially, the graph for L safe + B is never higher than that for L risky + B, and sometimes it is strictly lower. So Stochastic Dominance says that L safe + B is strictly better. Cumulative probability graphs for L risky + B and L safe + BBut how does this happen?For Stochastic Dominance to hold, we need L risky + B to have at least as high a probability as L safe + B of turning out at least as good as an outcome with value u, for all possible values u.Start with L risky + B. We know that L safe just gives one value (v) with certainty. So the probability that L safe + B gives value u or better is just the probability that B gives value u − v or better. (This corresponds to the area B 2 + B 3 on the graph below.) And then, for L risky + B, we'll get value at least u either if L risky gives value V or if B gives at least value u. Denote the probability that B gives value at least u by B 3 (corresponding to that area on the graph below). 1 L i (O O O ) B V (O) ; 1 Figure 3: 1 L i (O O O ) L i (O O O ) L safe L risky L safe + B V (O) 1 L risky + B ; ; V (O) Figure 4: So take any real value u < V . What's each lottery's probability of doing at least that well? Figure 2: the required properties of B below.)
 exclude the value of events that occurred in ancient Egypt, correspond to L risky and L safe . 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 B to 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.Italy, and Egypt. So there is likely plenty left to learn in Indology.Take the same L risky + B and L safe + B from above -lotteries such that the former is strictly better, and yet L risky is no better than L safe . Such lotteries must exist, if Fanaticism is false. Butsuppose 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 L risky + B and L safe + 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 the events 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 L risky + B and L safe + B. Instead, you would choose between L risky with b added to the value of every outcome and L safe with b added.Recall Background Independence from above. If it holds, then L risky with any b added to every outcome is better than L safe with the same b added if and only if L risky is better than L safe . 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. You cannot. You must accept that L risky + B is 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 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 than 41 cf van Fraassen's (1984) principle of Reflection and its application byWhite (2010:17-8). That form of 40  6.2 The Indology ObjectionBut 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 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 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, 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 b 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 years of research? Why not just update your judgement now and say that L risky + B is no better than L safe + B?Reflection 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.
			 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.
			 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.
			 For compelling arguments in favour of Stochastic Dominance, see Easwaran (2014) and Bader (2018). 18 Schoenfield (2014) rejects a similar principle, but only for lotteries over outcomes for which we don't have a complete ordering. With our complete O relation on O, even Schoenfield would endorse Stochastic Dominance here.
			 For readers less sympathetic to totalism, much the same argument can be made in terms of any of the axiologies mentioned in the Introduction. For instance, if we assume averageism, we can start with L 0 , by which the average value of all lives will be 1 with near certainty, and 0 otherwise. L 1 can be the lottery which gives probability 0.99999 of an outcome in which the average value of all lives will be 10 10 , or 0 otherwise. We can rapidly 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 L n with probability of some astronomical average value, or 0 otherwise, but transitivity says that it must be better than L 0 .
			 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.
			 An 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 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.
			 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.
			 Egyptology and IndologyBut 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 ObjectionHailing 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 to indifference among many of the lotteries L 1 to L 4 listed above or else indeterminacy of which is better than which. 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 considers a major problem. 30  It appears earliest in MacMahan (1981: 115) but is often attributed toParfit (1984: 420).
			 If there is some pair of lotteries L a or L b that violate Background Independence, then there is also some pair that violate it and have the exact same events occur in ancient Egypt. To obtain such a pair, simply take L a and L b 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 this section, 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 outcome
			 This relationship between Stochastic Dominance and cumulative probability would break down if some outcomes were incomparable to others. There would then be a difference between O a O O b and the negation of O a ≺ O O b . But, fortunately, totalism gives us a total preorder of O, so we can sidestep this complexity.
			 The result holds for any Cauchy distribution with a 'scale factor' of at least v . See Tarsney (n.d.: §5) and Pomatto et al (2018).
			 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 the theory 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.
			 This paper has benefitted from the input of many brilliant and generous colleagues. I am indebted to Alan

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 a 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 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 tradeoffs. For two, you must accept that judgements are either: inconsistent over structurally-identical pairs 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 to accept judgements which you know you would reject if you simply learned more. Better to accept fanaticism than these implications.