## Abstract

Theories that use expected utility maximization to evaluate acts have difficulty handling cases with infinitely many utility contributions. In this paper I present and motivate a way of modifying such theories to deal with these cases, employing what I call “Direct Difference Taking”. This proposal has a number of desirable features: it’s natural and well-motivated, it satisfies natural dominance intuitions, and it yields plausible prescriptions in a wide range of cases. I then compare my account to the most plausible alternative, a proposal offered by Arntzenius (Philos Perspect 28(1):31–58, 2014). I argue that while Arntzenius’s proposal has many attractive features, it runs into a number of problems which Direct Difference Taking avoids.

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## Notes

- 1.
These issues will also arise for theories that take uncertainty into account and take the sum of the good-making features of outcomes to be

*one*of the factors that bears on whether an act is permissible (e.g., theories which appeals to both utility and rights to determine what acts are permissible). That said, to simplify the discussion, I’ll focus on theories which appeal only to utility in this paper. - 2.
This is the Pasadena Game from Nover and Hájek (2004).

- 3.
See Bostrom (2011) and Arntzenius (2014) for versions of this point. (Arntzenius (2014) notes that we can partially mitigate these worries by noting that most agents like us will tend to only believe their acts will bear on the utility of subjects in a finite region. But, of course, this won’t help in cases like the chocolate case described above, which only requires the agent to believe they’re changing the utility of a single subject.) For a more detailed discussion of how these cosmological theories yield these results, see Carroll (forthcoming) and the references therein.

There are also other ways to motivate the importance of this kind of infinity problem. For example, if one follows preference utilitarians and takes utility contributions to consist of satisfied/unsatisfied desires of a subject at a time, then such problems might arise immediately (e.g., if we individuate desires in a fine-grained way, subjects like us might end having infinitely many desires, and thus yield infinitely many utility contributions; since time is dense, a temporally extended subject might yield infinitely many utility contributions; and so on). Likewise, some economists have suggested that we should model questions regarding our obligations to future generations as infinite decision problems, even if this is unlikely to actually be the case ((for examples of this kind of approach, see Chichilnisky (1996) and Chichilnisky (1997)).

- 4.
For discussions of extensions of standard decision theory along these lines, see for example Hájek (2003a), Bostrom (2011), and Chen and Rubio (2018). Bartha (2007) develops a related but distinct approach, that shares many of the drawbacks of these approaches ((e.g., it doesn’t tell us how to figure out what one should do (or what preferences one should have) in cases with outcomes with infinitely many utility contributions; see Colyvan and Hájek (2016) for discussion of a related point). Note that most of these authors weren’t trying to respond to the fourth kind of infinity problem described above, so it would be unreasonable to fault them for failing to do so.

- 5.
Here are two of the reasons this approach is unsatisfactory. First, we need to choose whether to assign coarse-grained or fine-grained values to these outcomes, and both options are problematic. If we assign coarse-grained values to such outcomes (e.g., “\(\infty \)” and “\(-\infty \)”), then these assignments yield implausible verdicts (e.g., that we have no reason to increase infinitely many people’s utility from 1 to 2, since this won’t change the utility of the outcome). Whereas if we assign fine-grained values to outcomes (e.g., hyperreal numbers), then it’s not clear it’s possible to assign these values to outcomes in a way that avoids arbitrary prescriptions. (See Arntzenius (2014) for a discussion of these difficulties for accounts employing hyperreal values.)

Second, the strategy of expanding the range of values assigned to outcomes won’t help in many cases. For in many cases the problem isn’t that we lack the values we want to assign to these outcomes, it’s that it’s unclear what value

*should*be assigned. (For example, see the Egalitarian Sweetmeats case discussed in Sect. 3.3.) - 6.
The problem is first raised by Ramsey (1928). Serious discussion in the economics literature begins with Koopmans (1960) and vonWeizsacker (1965); for a up-to-date survey of this literature, see Lauwers (2014). Serious discussion in the philosophical literature begins with Segerberg (1976); later discussions include Nelson (1991), Vallentyne (1993), Garcia and Nelson (1994), Liedekerke (1995), Cain (1995), Vallentyne and Kagan (1997), Hamkins and Montero (2000), Fishkind et al. (2002), and Lauwers and Vallentyne (2004).

- 7.
I use the generic name “\({\mathbb {V}}\)” here in order to keep this characterization of a decision problem neutral with respect to what kinds of values a utility contribution can have.

- 8.
Given this way of characterizing decision problems given above, one can think of states as functions from acts to outcomes, and the assumption that there are finitely many states as the assumption that there are only finitely many distinct functions of this kind.

- 9.
The probability of

*o*conditional on*a*is usually defined as \({\frac{cr(o \wedge a)}{cr(a)}}\) (though see Hájek (2003b) for a case for taking conditional probabilities to be primitive). Letting \(w^{a}\) be the world most similar to*w*where*a*is true, the probability of*o*imaged on*a*is usually defined as \(\sum _w cr(w) \cdot {\frac{cr(o \wedge w^a)}{cr(w^a)}}\). (This assumes that there is a unique world that is most similar to*w*; for formulations of imaging that don’t assume this, see Joyce (1999), chapter 6.) - 10.
For a discussion of these alternatives, see Buchak (2013).

- 11.
Note that this includes theories which take there to be holistic utility contributions, such as various forms of egalitarianism. As long as such theories assign utilities to outcomes in finite cases by summing over these features (e.g., by adding up the utility stemming from individual happiness and the utility stemming from the value of the outcome’s (in)egalitarian features), they’ll fall within the scope of our discussion.

- 12.
Appealing to something like difference-taking is common in the literature regarding how to compare outcomes (instead of acts), though it’s usually implemented after some kind of preliminary aggregation of subsets of utility contributions; for example, see vonWeizsacker (1965), Vallentyne (1993), Vallentyne and Kagan (1997), and Lauwers and Vallentyne (2004). Arntzenius (2014) likewise appeals to something like difference-taking when comparing acts, applied after preliminary aggregations of utility contributions (see Sect. 5). Interestingly, some have also suggested employing difference-taking to address some of the problems that arise in infinite state cases. Colyvan (2008) has suggested that we apply difference-taking to states (instead of utility contributions) in order to to derive the desired verdicts in cases where one act dominates another, and yet neither act has a well-defined finite expected value, both Lauwers and Vallentyne (2016) and Meacham (forthcoming) have defended versions of this approach.

- 13.
I’m assuming here that there is a natural way of pairing utility contributions in different outcomes that satisfies certain desirable properties: symmetry, transitivity, one-to-one, etc. We’ll look at how to proceed if we reject this assumption in Sect. 3.5.

- 14.
More precisely, the ordering and arithmetic operations are extended over these new elements as follows:

*Order:*\(\forall r \in {\mathbb {R}}, -\infty< r < \infty \).*Addition and Subtraction:*1. \(\forall r \in {\mathbb {R}}, \ r \pm \infty = \pm \infty \). 2. \(\infty + \infty = \infty \). 3. \(-\infty + -\infty = -\infty \). 4. \(\infty + - \infty =\)*undefined*.*Multiplication:*1. \(\forall r \in \mathbb {R^+}, \ r \cdot \pm \infty = \pm \infty \). 2. \(\forall r \in \mathbb {R^-}, \ r \cdot \pm \infty = \mp \infty \). 3. \(\pm \infty \cdot \pm \infty = \infty \). 4. \(\pm \infty \cdot \mp \infty = -\infty \). 5. \(0 \cdot \pm \infty =\) 0.*Division:*1. \(\forall r \in {\mathbb {R}}, \ {\frac{r}{\pm \infty }} = 0\). 2. \(\forall r \in \mathbb {R^+}, \ {\frac{\pm \infty }{r}} = \pm \infty \). 3. \(\forall r \in \mathbb {R^-}, \ {\frac{\pm \infty }{r}} = \mp \infty \). 4. \({\frac{\pm \infty }{\pm \infty }} = {\frac{\pm \infty }{\mp \infty }} =\)*undefined*. 5. \({\frac{\pm \infty }{0}} =\)*undefined*. - 15.
Arntzenius (2014) likewise assumes an extended notion of convergence to make sense of infinite-valued sums (see Arntzenius (2014), p54), though he does not commit himself to all of the details given here. In a similar vein, much of the literature on how to compare outcomes (instead of acts) assumes some way of making sense of infinite-valued sums (e.g., see Lauwers and Vallentyne (2004), section 5).

- 16.
It’s worth highlighting that this permutation invariance requirement is distinct from the (much stronger) Isomorphism requirement that if there’s a value-preserving bijection between the utility contributions in a pair of outcomes, then those outcomes should be interchangeable. I discuss Isomorphism in more detail in “Appendix A”, and consider a case in which Isomorphism and permutation invariance come apart (a case where DDT violates Isomorphism even though it satisfies permutation invariance).

That said, it’s worth noting that even the relatively weak permutation invariance requirement I defend in this section is controversial. As we’ll see in Sect. 6, Arntzenius’s (2014) proposal violates this requirement. And in the literature on how to compare outcomes (instead of acts), many of the economists and philosophers who have written on this topic reject the analog of permutation invariance. For example, vonWeizsacker (1965) and Vallentyne (1993) effectively suggest that we should order terms temporally when making our assessments, contra permutation invariance. In a similar vein, Vallentyne and Kagan (1997) and Lauwers and Vallentyne (2004) suggest that if there’s a “natural ordering”, then we should use that to order terms when making our assessments, even if this violates permutation invariance. That said, there have also been several defenders of something like permutation invariance, such as Garcia and Nelson (1994), Liedekerke (1995), Hamkins and Montero (2000) and Fishkind et al. (2002), though these authors generally argue for much stronger principles than the one advocated here (principles like the Isomorphism requirement described above).

- 17.
Another natural thought regarding how to address this problem is to try to take one of the order-independent ways of evaluating infinite sums suggested in the infinite states literature (the second source of infinity problems mentioned in Sect. 1), and adapt it to apply to cases like Egalitarian Sweetmeats. For example, one might try to adapt something like Gwiazda’s (2014) approach, by insisting on ordering utility contributions by the magnitude of their utilities. Or one might try to adapt something like the “principal value” approach described by Easwaran (2014) to handle these cases, by considering truncated sums which omit utility contributions of magnitude greater than

*n*, and then seeing what values these truncated sums take in the limit as*n*goes to infinity. Unfortunately, none of these adapted proposals help with the Egalitarian Sweetmeats case. The adaptation of Gwiazda’s proposal won’t help because there is no smallest magnitude utility contribution to start with. And the adaptation of Easwaran’s proposal won’t help because the values of these truncated sums will themselves be order-dependent. ((With Easwaran’s original proposal and infinite states cases, the truncated sums we consider are over expected utility contributions of the relevant states, and these truncated sums will always yield order-independent results because the sum of the magnitudes of these terms will always be finite. With this adaptation of Easwaran’s proposal to infinitely many contribution cases, the truncated sums we consider are over the relevant utility contributions, and the sum of the magnitudes of*these*terms can be infinite and order-dependent (as in the Egalitarian Sweetmeats case).) - 18.
In the context of ranking outcomes (instead of acts), Vallentyne and Kagan (1997) briefly mention the possibility of adding “empty virtual locations” in order to compare outcomes in cases where one outcome has a proper subset of the utility contributions of another outcome (see Vallentyne and Kagan (1997), footnote 18). This is similar to zero point pairing, though it won’t apply to cases like Siblings, in which neither outcome has utility contributions that are a proper subset of the other’s. While discussing an alternative to his proposal (the Weak Person Condition discussed in “Appendix B”), Arntzenius (2014) also briefly mentions the possibility of using something like zero point pairing with respect to temporally extended individuals. It’s worth noting that zero point pairing is controversial. For example, in the literature on ranking outcomes, Lauwers and Vallentyne (2004) join a number of economists in suggesting that our ranking shouldn’t depend on where we put the zero point of the scale we use to measure utility. Thus, for example, if we measure utility using a scale that assigns every contribution a value that’s 10 units higher, our ranking of outcomes shouldn’t change. The analog of zero point pairing in this context (i.e., zero point pairing applied to outcomes instead of acts) will violate this requirement. For example, if we consider an outcome with one person with 3 utility, and another outcome where both that person and another person have 1 utility, then zero point pairing will suggest that the first outcome is better than the second. But if we consider an outcome with one person with 13 utility, and another outcome where both that person and another person have 11 utility, then zero point pairing will suggest that the second outcome is better than the first.

- 19.
- 20.
Arntzenius (2014) assumes some form of counterpart theory when formulating his proposal, but does not address the issues that arise from the ways in which counterpart relations can diverge from identity relations, or issues that arise from cases of fission and fusion (I discuss this worry for Arntzenius’s proposal in Sect. 7.1). In the context of ranking outcomes (instead of acts), Vallentyne and Kagan (1997) also discuss ways of accommodating counterpart theory. But like Arntzenius, they do not address the issues that arise from the ways in which counterpart relations diverge from identity relations (they assume the counterpart relation is one-to-one), or issues that arise from cases of fission or fusion.

- 21.
This highly restricted domain minimizes the amount by which an acceptable pairing will deviate from the pairings suggested by the transworld identity facts or the ordinary counterpart relation. First, this requirement only holds with respect to whatever one takes to be providing utility contributions (e.g., mental states at a time). One may continue to pair everything else in the standard way, even though such pairings may fail to be symmetric, transitive, or one-to-one. Second, this requirement only holds with respect to the utility contributions that appear in the outcomes of the relevant decision problem. So most utility contributions will still be paired in the standard way, even though, again, such pairings may fail to be symmetric, transitive, or one-to-one.

- 22.
We can spell out these conditions more precisely as follows. (I’ll focus here on how to spell out these conditions for counterpart theorists; it should be clear how to spell out these conditions for transworld identity theorists.) Let

*O*be the outcomes of this decision problem, and let*D*be the restricted domain of utility contributions of outcomes in*O*. Following Lewis (1968), let*I*(*x*,*y*) be the relation that holds*iff**x*is in possible world*y*, and let*C*(*x*,*y*) be the relation that holds*iff**x*is a counterpart of*y*. For all \(o, o^* \in O\), and all \(w, x, y, z \in D\), we require the counterpart relation to be such that:- (a):
(symmetry)

- (b):
(transitivity)

- (c):
(one-to-one)

- 23.
- 24.
- 25.
The restriction to finite regions is important. If we could determine the utility associated with infinite spatiotemporal regions—i.e., the entire outcome—then there’d be no need for the machinery Arntzenius brings in for the second stage.

- 26.
E.g., suppose there are infinitely many planets, each labeled with a natural number. And suppose there is a narcissistic agent who desires of each planet that there be a statue of themselves on it (where each unsatisfied desire contributes 0 utility, and each satisfied desire contributes 1 utility). Finally, suppose this agent has two options: (a) making all planets have a statue of themselves on it, or (b) making only the even-numbered planets have a statue of themselves on it. Obviously, the agent should choose option (a) given their desires. And if we take each of the agent’s desires to only contribute to regions that entail that the utility contribution obtains—e.g., a region containing both the agent and the planet in question—then every finite region will only contain finitely many utility contributions, and Arntzenius’s proposal will yield the desired result that (a) is obligatory. But if we instead take the agent’s infinitely many desires to all contribute to the region containing the agent, then both acts will lead the region containing the agent to have infinite utility, and Arntzenius’s proposal will be unable to distinguish between the two acts. (Put another way, by taking these utility contributions to all lie in the region containing only the agent, we increase the prevalence of problematic cases in which there are finite regions with infinitely many utility contributions (see Sect. 7.6).)

- 27.
For the purposes of this paper, I’ll use the term “metric” loosely to refer to any spatiotemporal distance function, be it a (proper) metric function, a pseudometric function, or some other nearby relative.

- 28.
That is, there should be fixed distances \(d_i\) such that each successive region in a sequence contains all and only the points

*x*such that for some point*y*in the previous region, \(m_i(x,y) \le d_i\) for all*i*. - 29.
This description is a bit of a simplification; it describes Arntzenius’s suggestion for how to pick out regions in cases involving classical spacetimes, like Newtonian spacetime. Arntzenius suggests a different way of picking out regions in Lorentzian spacetimes (because the region containing all of the points within (say) 1 unit of spacetime interval of some point will be infinite). But these complications are orthogonal to the issues we’re concerned with, so I gloss over these complications in the text.

- 30.
The reason we require the difference to always be at least

*x*, instead just requiring the difference to be positive, is that if the partial sums of these differences converges to zero from above (e.g., the partial sums are \(1, 1/2, 1/4, 1/8, \ldots \)) we don’t want to say that the first act is better than the second, since the difference between the two vanishes in the limit. - 31.
It’s worth nothing that there are structurally similar cases in which Arntzenius’s proposal also fails to give fine-grained prescriptions, such as the Intensity or Duration case discussed in “Appendix B.3”.

- 32.
What if one

*does*adopt the implausible stance that utility contributions are directly assigned to spatiotemporal regions? Then many of these objections (e.g., the objections raised in Sects. 7.4, 7.5, 7.6, and 7.7) will fail to differentiate between DDT and Arntzenius’s proposal, since they’ll apply to both proposals or neither proposal. That said, simply endorsing a picture of utility contributions which directly assigns values to spatiotemporal regions won’t suffice to shield Arntzenius’s proposal from these criticisms. For part of the project Arntzenius and I are engaged in is to provide a way to yield plausible infinite generalizations of moral theories. And if Arntzenius’s proposal has difficulty doing so in a plausible way for preference utilitarians, hedonic utilitarians, and the like, then this is a mark against his proposal for how to generalize moral theories. - 33.
Arntzenius (2014) briefly considers two other approaches to handling cases with infinitely many utility contributions, which he calls the

*Weak Person Condition*and the*Weak Location Condition*. Although these proposals are interesting, it would distract too much from the comparison of Arntzenius’s proposal and DDT to discuss them in the text. So I delegate my discussion of these views to “Appendix B”, in which I describe these proposals, and assess the pros and cons of these proposals relative to Arntzenius’s proposal and DDT - 34.
Here’s a simple case in which ordinary counterpart relations over spatiotemporal regions fail to be symmetric. Consider a pair of spatially finite outcomes,

*o*and \(o^\prime \), where \(o^\prime \) is twice the size of*o*. The first half of \(o^\prime \), \(r_1\), is a perfect duplicate of the region at*o*,*r*. The second half of the \(o^\prime \), \(r_2\), is an almost perfect duplicate of*r*. In this case, \(r_2\)’s counterpart at*o*is*r*. But \(r_2\) is not*r*’s counterpart at \(o^\prime \), \(r_1\) is. (Similar cases yield counterexamples to transitivity.) - 35.
Note that one can’t dismiss this problem by suggesting that such possibilities are unlikely. For if there are possible agents with non-zero credences in such possibilities, then one’s proposal needs to able to accommodate such possibilities (regardless of how implausible one might personally take them to be). Moreover, it’s not entirely clear that we shouldn’t take such possibilities seriously; see Bricker (1996) and Bricker (2001) for discussions of such possibilities.

- 36.
Of course, this modification won’t help with worlds with infinitely many islands. Since the initial region is required to be finite, worlds with infinitely many islands will require the parts of each island in the initial region to get arbitrarily small (e.g., a cubic meter of the first island, half a cubic meter of the second, a quarter of a cubic meter of the third, and so on). But once we perform our first uniform expansion of such a region (e.g., adding all points within a centimeter of the previous region), we’ll get a region that’s infinite. And since the first stage of Arntzenius’s proposal only assigns values to finite regions, such infinite regions won’t be assigned values. As a result, the second stage of Arntzenius’s proposal (which employs the first stage’s assignment of values to regions to evaluate acts) won’t get off the ground. Now, one could try to modify the first stage of Arntzenius’s proposal so that it also assigns values to some infinite regions as well. But this raises other problems for Arntzenius’s proposal, since such regions will generally have infinitely many utility contributions, and Arntzenius’s proposal is ill-equipped to handle such regions (cf. Sect. 7.6).

A different requirement one might consider is requiring the initial region of acceptable expanding sequences to be located in the island in which the agent is located, leading the proposal to effectively ignore other islands. After all, the agent’s acts will generally be probabilistically and causally independent of the utility contributions at other islands. Given this, it seems unproblematic to simply ignore other islands, since they won’t bear on what the agent should do. But this modification won’t be able to accommodate more exotic possibilities in which utility contributions at other islands aren’t probabilistically or causally independent of the agent’s acts. (Here are some examples of such cases. EX 1. Suppose one deduces from the laws that there exists an island that’s a duplicate of this one. Then one’s acts won’t be probabilistically independent of what happens at that island. EX 2. Suppose a counterfactual analysis of causation is correct, and that certain counterfactuals hold between your acts and outcomes in other islands. Then one’s acts won’t be causally independent of what happens at other islands. Of course, one might question whether a

*temporally*disconnected region could have any causal relations to this one. But those who are attracted to counterfactual analyses of causation may feel some pressure to admit the possibility of non-temporally mediated causal relations, since the appropriate counterfactual relations can hold despite the lack of temporal relations. See Baron and Miller (2014) and Baron and Miller (2015) for defenses of such a view.) - 37.
One might be tempted to escape this problem by adopting the alternate account of when a utility contribution is in a region discussed in Sect. 5, on which utility contributions due to satisfied desires lie in the region containing only the subject of the desire. This alternate account would allow Arntzenius’s proposal to handle Remembrance, since it would now be easy to find finite regions containing the additional subject’s utility contribution. But this temptation should be resisted. First, as mentioned earlier, this alternate account is unattractive for other reasons (cf. footnote 26). Second, there are other cases where this alternate account won’t help. For example, suppose subjective hedonic utilitarianism is true—you should act to maximize expected happiness—and that some functionalist account of the mind is correct. And consider the following case:

**Big Brain.**:The world is infinitely extended, spatially and temporally, and contains infinitely many subjects with various degrees of happiness. You can choose to bring about one of two outcomes.

These outcomes are the same in almost every respect relevant to happiness, with one exception: at the second outcome, the distribution of subjects in the world is such that there’s a further emergent consciousness—an infinitely extended “big brain” (along the lines of Block’s (1978) China Brain example)—which is happy.

- 38.
One might be tempted to reply that cases with infinitely many utility contributions are themselves recherché cases. Thus if we’re bracketing such cases, we should bracket the kinds of cases that motivate adopting Arntzenius’s proposal in the first place. But this is too quick; as we saw in Sect. 1, cases with infinitely many utility contributions

*are*cases that agents like us encounter. Moreover, many of the debates in decision theory (e.g., the debate between causal and evidential decision theorists) already concern recherché cases (e.g., Newcomb’s Case) that agents like us never encounter (e.g., see Horwich (1985)). But these debates are nonetheless taken to be important because they bear on what the correct theory of decision-making is. - 39.
“The new picture turns everything elegantly inside out. What had seemed merely abstract and symbolic in the old picture (that is, the wave functions, and the high-dimensional space in which they undulate) becomes real and physical and concrete in the new one, and what had seemed exact and fundamental in the old picture (that is, the talk of particles in a three-dimensional space) becomes vague and approximate and emergent in the new one” (Albert (2015), p 146).

- 40.
For example, if one rejects metaphysical indeterminacy, then Albert’s argument that three-dimensional space is “vague and approximate” (Albert (2015), p 146) suggests such a stance.

- 41.
See Allori (2016) and the references therein. As before, one might take these results different ways; one might take them to suggest that the ordinary four dimensional spacetime of our acquaintance still exists, if only as an emergent non-fundamental entity (Huggett and Wüthrich 2013). Or one might take them to suggest that ordinary spacetime doesn’t exist.

- 42.
Assuming here a notion of convergence to extended reals along the lines sketched in Sect. 3.2.

- 43.
Does DDT do better in this case? It does, because DDT tells us how to obtain verdicts in this case. First we pair the utility contributions (in this case, desires) in the two different outcomes. Since there are no desires in the first outcome, this will be a case of zero-point pairing. We then consider the infinite sum over the differences between these utility contributions, which is identical to the sum of the utility contributions in the second outcome. Since the value of this sum is order dependent, we’ll get the result that both options are permissible. (I owe an anonymous referee for encouraging me to address this question.)

- 44.
In the comparing outcomes literature, Garcia and Nelson (1994) and Cain (1995) raise the worry that views which appeal to a privileged temporal order are effectively concerned with how quickly utility is produced, rather than how much utility there is. One can see this worry as being in a similar vein.

One might wonder whether Population Density is essentially a Repugnant Conclusion-style worry (see “Appendix A” for discussion), and thus something that can be recreated as an objection to DDT as well. (I owe an anonymous referee for encouraging me to address this worry.) Despite some superficial similarities, Population Density is quite different from Repugnant Conclusion-style worries. Population Density is a same-numbers case, in which the same individuals (and the same number of individuals) exist regardless of what you do. This contrasts with different-numbers cases like those that yield the Repugnant Conclusion. Since both Arntzenius’s proposal and DDT are subject to Repugnant Conclusion-style worries, we can bracket those worries for the purposes of comparison. And after bracketing such worries, same-numbers case like Population Density raise further problems for Arntzenius’s proposal that do not arise for DDT.

- 45.
I owe an anonymous referee for encouraging me to address these objections.

- 46.
There are, of course, a number of alternatives to utilitarianism that one might consider to try to avoid these worries, such as averaging views, critical levels views, and the like. For an in-depth discussion of these options, see Arrhenius et al. (2017).

- 47.
This is a variant of case discussed by Vallentyne and Kagan (1997).

- 48.
Both Arntzenius’s proposal and DDT respect dominance considerations, but the dominance considerations they respect are not identical. Arntzenius’s proposal respects dominance with respect to spatiotemporal regions (e.g., if there are two outcomes with the same spatiotemporal regions, and in one outcome every region is at least as good, and sometimes better, than the corresponding region in the other outcome, than the first outcome is better than the second). DDT respects dominance with respect to utility contributions, which in this case we’re assuming to be individuals.

- 49.
I own an anonymous referee for encouraging me to discuss these proposals.

- 50.
Though as we’ll see below, there are at least some cases of this kind in which WPC arguably yields more plausible verdicts than DDT.

- 51.
Let me go through these three worries in reverse order. First, worries similar to those discussed in Sect. 7.6 arise for WPC in cases in which infinitely many utility contributions are associated with a temporally extended individuals. Second, as with Arntzenius’s proposal, worries will arise in cases in which some outcomes are spatiotemporal and some are not (cf. Sect. 7.5). WPC works by (1) summing up utility contributions to assigns values to the complete temporal lives of people, and then (2) using those values to assess acts. But in cases in which the outcomes don’t have temporal structure there won’t be any temporally extended individuals, so WPC will fall silent. Third, as with Arntzenius’s proposal, worries will arise in cases in which different outcomes have different temporal structure (cf. Sect. 7.4). In particular, worries will arise here that intersect worries WPC faces regarding how to work out details regarding personal identity. If we adopt a psychological continuity-over-time or physical continuity-over-time account of personal identity, then questions will arise regarding how to pick out temporally extended individuals at temporally discontinuous spacetimes, or spacetimes without a clear spatial/temporal distinction, or spacetimes with multiple temporal dimensions. And even with an answer to these questions, further questions will arise regarding which temporally extended individuals should be identified with each other in cases in which different outcomes have different temporal structures.

- 52.
For example, if we adopt a psychological continuity-over-time or physical continuity-over-time account of personal identity, then tricky questions arise regarding how to pick out temporally extended individuals at temporally discontinuous spacetimes, or spacetimes without a clear spatial/temporal distinction, or spacetimes with multiple temporal dimensions.

- 53.
I thank an anonymous referee for suggesting a version of this case.

- 54.
One might be tempted to defend DDT by arguing that those who think DDT yields the wrong verdicts in Intensity or Duration should find fault with the notion of utility contributions we’re employing, not DDT. After all, if we took utility contributions to be assigned to something like the complete lives of subjects, DDT would not yield those verdicts. I think we should resist this temptation. The goal of proposals like Arntzenius’s proposal and DDT is to provide generalizations of theories like hedonic utilitarianism that yield plausible results in cases with infinitely many utility contributions. And trying to avoid implausible consequences of such proposals by changing the theory we’re trying to generalize doesn’t help the proposal achieve this goal, it just changes the topic (cf. footnote 32).

- 55.
Since WLC joins Arntzenius’s proposal in taking spatiotemporal regions to be the bearers of utility in the second stage of assessment, it might be surprising that WLC doesn’t do any better than DDT or WPC in cases like Favoring the Positive. But unlike Arntzenius’s proposal, WLC doesn’t work by looking at increasing uniform expansions, and seeing what happens in the limit. Instead, WLC just partitions spacetime and sums over the utility contributions in those regions (in some arbitrary order), depriving it of much of the spatiotemporal information that Arntzenius’s proposal makes use of.

- 56.
The immediate worries described in Sect. 7.4 and the “no spacetime structure” part of Sect. 7.5 won’t apply to the version of WLC described in “Appendix B.2”, because it effectively builds in the first amendment suggested in response to each of these worries. But it is still subject to the worries raised for these amendments.

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## Acknowledgements

I’d like to thank Maya Eddon, Itai Sher, two anonymous referees, participants of the Spring 2018 UMass Brown Bag group, participants of the 2018 Recent Work in Decision Theory and Epistemology conference at Columbia, and the audience of my 2019 colloquium talk at the University of Maryland, College Park, for helpful comments and discussion. I’d also like to thank Sophie Horowitz for suggesting both the title of this paper and the title of Sect. 5. Finally, I’d like to thank Frank Arntzenius for inspiring my interest in these issues and many others

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## Appendices

### Objections to both proposals

I’ve discussed objections that apply to DDT but not Arntzenius’s proposal in Sect. 6, and objections that apply to Arntzenius’s proposal but not DDT in Sect. 7. In this appendix, I’ll present and respond to two objections that apply to both proposals.^{Footnote 45}

Here is the first objection. As Parfit (1984) notes, aggregative theories like utilitarianism are generally subject to Repugnant Conclusion-style worries. For such theories will generally entail that an outcome with a large number of very high utility people is worse than an outcome with a much larger number of low utility people, as long as the latter outcome has more overall utility. For example, they’ll generally hold that given a choice between a million people with 100 utility and a trillion people with 0.01 utility, we’re obligated to choose the latter. And most find this conclusion repugnant.

Both Arntzenius’s proposal and DDT will yield this verdict. But this is not a demerit of these proposals. For both Arntzenius’s proposal and DDT are, in part, attempts to describe how one should generalize moral theories like utilitarianism so that they yield plausible verdicts in infinite cases. To qualify as infinite generalizations of utilitarianism, these proposals must yield the same verdicts as utilitarianism in finite cases. Since utilitarianism yields Repugnant Conclusion-style worries in finite cases (like the case described above), any infinite generalization of utilitarianism must yield these verdicts as well. Thus Arntzenius’s proposal and DDT must yield these verdicts, given what they’re trying to do.^{Footnote 46}

Here is the second objection. Hamkins and Montero (2000) suggest that a plausible condition on a satisfactory theory is that if there’s a value-preserving bijection between the utility contributions of a pair of outcomes, then these outcomes should be treated interchangeably. Call this condition *Isomorphism*. Now suppose utility contributions are assigned to complete lives, and consider the following case:^{Footnote 47}

In this case DDT says that only Act 1 is permissible, even though there’s a value-preserving bijection between them. Thus DDT will violate Isomorphism. And if we assume all of the people in these outcomes are equally spaced along some spatial axis, and occupy the same locations regardless of which act one performs, then Arntzenius’s proposal will yield the same prescriptions as DDT in this case. Thus Arntzenius’s proposal violates Isomorphism as well.

I concede that there is some force to this objection. But as Hamkins and Montero (2000) and Lauwers and Vallentyne (2004) note, what this shows is that in these kinds of infinite cases we’re forced to choose between two plausible but conflicting principles: Isomorphism and dominance. Act 1 and Act 2 are isomorphic, so Isomorphism requires both acts to be permissible. But Act 1 dominates Act 2, so dominance requires Act 2 to be impermissible. So we have to reject either Isomorphism or dominance. Arntzenius and I follow Lauwers and Vallentyne (2004) in taking dominance considerations to be more compelling than Isomorphism.^{Footnote 48} Thus while both Arntzenius’s proposal and DDT reject Isomorphism, I take this rejection to be defensible, as it’s required in order to hold on to dominance.

### The weak person condition and the weak location condition

In section 10 of his paper, Arntzenius (2014) briefly mentions two other proposals for how to handle cases with infinitely many utility contributions. In what follows I’ll describe these proposals, and compare them to Arntzenius’s proposal and DDT.^{Footnote 49}

### The weak person condition

The first proposal is what Arntzenius calls the *Weak Person Condition* (WPC). Like Arntzenius’s proposal, WPC has two stages: first it agglomerates utility contributions to assign values to temporally extended individuals (i.e., people’s lives), and second it lays out a procedure for using those values to assess the available acts. As with Arntzenius’s proposal, the first stage is largely left implicit, though it is presumably determined by something like summing over utility contributions associated with the subject’s life. The second stage of assessment is effectively the conjunction of difference-taking (as described in Sect. 3.1.1, but applied to people’s lives instead of utility contributions), infinite convergence (Sect. 3.2.1), permutation invariance (Sect. 3.3.1), zero-point pairing (Sect. 3.4.1), and expectation taking (Sect. 3.6.1).

WPC differs from Arntzenius’s proposal in that its first stage agglomerates utility contributions associated with people’s lives instead of spatiotemporal regions, it employs zero-point pairing, and it does not employ the expanding sequence structure Arntzenius’s proposal employs.

WPC differs from DDT in that it has two stages (agglomerating over utility contributions to determine the utility of people’s lives, and then using these values to assess acts) instead of one (using the utility contributions to directly assess acts), and does not employ acceptable pairings to overcome the problems that arise due to the failure of counterpart relations to be symmetric, transitive, or one-to-one (as in cases of fission or fusion). It also leaves the order of some of the relevant quantifiers ambiguous in a way that might yield a further deviation from DDT. (For example, does WPC tell us that an act *B* is permissible *iff* there doesn’t exist an act *A* such that, for every acceptable counterpart pairing, \(EU(A)>EU(B)\)? Or does WPC tell us that an act *B* is permissible iff for every acceptable counterpart pairing, there doesn’t exist an act *A* such that \(EU(A)>EU(B)\)? These two disambiguations can yield different verdicts, and only the latter will line up with DDT.)

### The weak location condition

The second proposal is what Arntzenius calls the *Weak Location Condition* (WLC). Roughly, WLC can be seen as a compromise between WPC and Arntzenius’s proposal. Like both of these views, WLC has two stages: first it agglomerates utility contributions to assign values to spatiotemporal regions, and second it lays out a procedure for using those values to assess the available acts. And the second stage of assessment is effectively the conjunction of difference-taking (as described in Sect. 3.1.1, but applied to spatiotemporal regions instead of utility contributions), infinite convergence (Sect. 3.2.1), permutation invariance (Sect. 3.3.1), and expectations taking (Sect. 3.6.1).

Since Arntzenius only briefly describes WLC, we’re not told what the locations we’re summing over and comparing are. To make things concrete, let’s fill in these details by assuming that the regions WLC sums over must be mutually exclusive and exhaustive, that these regions must be finite, and that *if* the spacetime structure is the same in a pair of outcomes, then the counterpart relations between such outcomes need to pair regions of the same size.

WPC differs from WLC in assigning values to spatiotemporal regions instead of people’s lives, and not appealing to zero-point pairing. WPC differs from Arntzenius’s proposal in that it doesn’t appeal to Arntzenius’s machinery for finding initial regions in the outcomes of the same size, uniformly expanding them, and seeing what the difference between the utility of these regions goes to in the limit.

WLC differs from DDT in that it has two stages (agglomerating over utility contributions to determine the utility of spatiotemporal regions, and then using these values to assess acts) instead of one (using the utility contributions to directly assess acts), and does not employ zero-point pairing or acceptable pairings. And like WPC, WLC leaves the order of some of the relevant quantifiers ambiguous in a way that might yield further deviations from DDT.

### Assessing the weak person condition

In Sects. 6 and 7 we looked at objections to DDT and Arntzenius’s proposal. Now let’s look at how WPC compares to these views.

WPC is generally subject to the same worries as DDT regarding coarse-grained prescriptions (cf. Sect. 6).^{Footnote 50} Like DDT, it doesn’t employ the the spatiotemporal structure of outcomes to effectively provide a privileged ordering or grouping of terms. So it will yield coarse-grained prescriptions in cases like Favoring the Positive in much the same way as DDT.

WPC is also subject to many of the worries raised for Arntzenius’s proposal in Sect. 7, such as the worries raised in Sect. 7.1, and worries similar to those raised in Sects. 7.4, 7.5, and 7.6.^{Footnote 51} Finally, WPC will face worries arising from the need to resolve issues like a subject’s identity over time in order to make sense of the first stage value assignments the view employs.^{Footnote 52}

That said, WPC does have an arguable advantage over DDT in certain cases in which agglomerating utility by subject seems natural. Let’s suppose hedonic utilitarianism is true, so that the utility contributions are provided by experiences of pleasure or pain. Furthermore, let’s assume that given the laws that obtain at the outcomes we’ll be concerned with, the time required to experience an episode of pleasure or pain is one minute. Now consider the following case:^{Footnote 53}

**Intensity or Duration.**:There are countably infinitely many babies, each who will live for three minutes. You have two options. The first (Candy Dump) is to give each baby three pieces of candy as soon as they come into existence, in which case each will have a utility of 2 during their first minute, and a utility of 0 in the next two minutes. The second (Candy Rationing) is to give each baby one piece of candy each minute, in which case each will have a utility of 1 for each minute.

According to WPC, Candy Dump is impermissible and Candy Rationing is obligatory. For the utility assigned to each subject’s life will be 2 given Candy Dump and 3 given Candy Rationing, and the sum of the difference between Candy Dump and Candy Rationing will be \(-\,\infty \).

But according to DDT both Candy Dump and Candy Rationing will be permissible. For the sum over differences in utility contributions will yield infinitely many positive contributions and infinitely many negative contributions. And since different permutations of these terms will yield different verdicts—summing over two positive terms per negative term will yield \(\infty \), while summing over two negative terms per positive term will yield \(-\,\infty \)— DDT will take both options to be permissible.

(What verdict will Arntzenius’s proposal yield? It depends on how the babies are spatiotemporally arranged (and this is true even if we assume the babies occupy the same spatiotemporal locations in both outcomes). For example, suppose that all outcomes are Newtonian, that the babies exist at the same time, and that they are equally spaced along some spatial axis. Then Arntzenius’s proposal will yield the verdict that Candy Dump is impermissible and Candy Rationing is obligatory. Suppose instead that the babies aren’t equally spaced along the spatial axis, but become exponentially more densely packed (spatially) as we move away from Baby 1, and suppose they don’t exist at the same time, but are born later and later as we progress down the axis (at a rate linear with distance). Then there will be some acceptable expanding sequences (e.g., intuitively, ones which expand slowly in the temporal direction and quickly in the spatial direction) which favor Candy Dump (since each expansion will let in more positive utility contributions than negative ones), and some acceptable expanding sequences (e.g., intuitively, ones which expand quickly in the temporal direction and slowly in the spatial direction) which favor Candy Rationing (since each expansion will let in more negative utility contributions than positive ones). So Arntzenius’s proposal will yield the verdict that both options are permissible.)

As an objection to DDT, this worry is structurally identical to the Favoring the Positive case discussed in Sect. 6. In both cases, we’re presented with cases where differences in utility contributions give us a sum over infinitely many plusses and minuses of non-trivial magnitudes, and since such sums aren’t permutation invariant, DDT will take both options to be permissible. And as with Favoring the Positive, how compelling one takes this case to be will depend on how one feels about these kinds of aggregative intuitions. One might argue that the intuition that Candy Rationing is better than Candy Dump seems to hang on the thought that the size of the set of positive \(\tau \)s is greater (in the relevant sense) than the size of the set of negative \(\tau \)s. And one might question whether we should rely on such intuitions regarding the sizes of these sets in such cases.^{Footnote 54}

All said and done, I take WPC to be roughly on a par with Arntzenius’s proposal. For while it can’t provide fine-grained prescriptions in cases like Favoring the Positive, it always (instead of merely sometimes) takes Candy Dump to be impermissible in Intensity or Duration, and it avoids a couple of the worries facing Arntzenius’s account.

Like Arntzenius’s proposal, I take WPC to be less attractive than DDT. While it can arguably yield more plausible verdicts in cases like Candy Rationing, it yields the same verdicts as DDT in cases like Favoring the Positive. And it encounters many of the same problems Arntzenius’s proposal runs into, in addition to worries regarding a subject’s identity over time.

### Assessing the weak location condition

Now let’s look at how WLC compares to these views. Like WPC, WLC is generally subject to the same worries as DDT regarding coarse-grained prescriptions that (cf. Sect. 6 and “Appendix B.3”). Like DDT, it doesn’t employ the the spatiotemporal structure of outcomes (in the case of Favoring the Positive) or the temporal structure of subjects (in the case of Intensity or Duration) to effectively provide a privileged ordering or grouping of terms.^{Footnote 55} So it will yield coarse-grained prescriptions in cases like Favoring the Positive and Intensity or Duration in much the same way as DDT. WLC is also subject to almost all of the worries raised for Arntzenius’s proposal in Sect. 7, such as the worries raised in Sects. 7.1, 7.3, 7.4, 7.5, 7.6, and 7.7.^{Footnote 56}

All said and done, I take WLC to be strictly less attractive than the other views we’ve considered. If we take the inability to provide fine-grained prescriptions in cases like Favoring the Positive to be a demerit of a view, then WLC is essentially dominated by Arntzenius’s proposal. For WLC faces virtually all of the same problems as Arntzenius’s proposal, but without the benefit or being able to provide fine-grained prescriptions in cases like Favoring the Positive. On the other hand, if we think that a view shouldn’t be providing fine-grained prescriptions in cases like Favoring the Positive, then WLC is dominated by DDT. For it’s not clear that WLC has any virtues relative to DDT, and it has a lot of vices: it faces virtually all of the same problems as Arntzenius’s proposal.

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Meacham, C.J.G. Too much of a good thing: decision-making in cases with infinitely many utility contributions.
*Synthese* (2020). https://doi.org/10.1007/s11229-019-02522-0

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