What calibrating variable-value population ethics suggests

Variable-Value axiologies avoid Parﬁt’s Repugnant Conclusion while satisfying some weak instances of the Mere Addition principle. We apply calibration methods to two leading members of the family of Variable-Value views conditional upon: ﬁrst, a very weak instance of Mere Addition and, second, some plausible empirical assumptions about the size and welfare of the intertemporal world population. We ﬁnd that such facts calibrate these two Variable-Value views to be nearly totalist, and therefore imply conclusions that should seem repugnant to anyone who opposes Total Utilitarianism only due to the Repugnant Conclusion.


Introduction
Much research in population ethics is motivated by the quest to avoid what Parfit (1984) called the Repugnant Conclusion, one version of which states that: 1 The Repugnant Conclusion.For any perfectly equal population of very well-off people, there is a better population consisting entirely of lives that are barely worth living.
Total Utilitarianism (TU), according to which a population is better the greater sum of welfare it contains, is widely recognized to entail the Repugnant Conclusion.No matter how well-off people are in some population A, and independently of A's size, there is some (potentially much bigger) imaginable population Z that contains a greater sum of welfare than A does -even though people in Z have lives that are each barely worth living (understood as having barely positive welfare).

Most paths to avoiding the Repugnant Conclusion begin by abandoning what Parfit
called the Mere Addition principle, which can be stated thus: Mere Addition.By adding any life worth living to any population, without making anyone else worse off, we do not make the population worse.
Total Utilitarianism implies the Mere Addition principle.But this principle is violated by Average Utilitarianism (AU), according to which a population is better the greater average welfare it contains.And Average Utilitarianism avoids the Repugnant Conclusion: Population Z, whose members all have lives that are barely worth living, contains lower average welfare than A. So A is better than Z, according to Average Utilitarianism.
Somebody who abandons the Mere Addition principle thinks that adding a life worth living, without making anyone worse off, can make a population worse.But what about adding a life well worth living?Consider merely adding a person who lives a very good life by modern standards: say, a happy professor living in a developed country in 2023.Surely by adding a person like that to any population, without thereby making anyone else worse off, we have not made the population worse?Not according Average Utilitarianism.To see that are barely worth living."(Parfit 1984: 388).Our formulation is closer to Arrhenius's (ming).Spears and Budolfson (2021) have argued that formalizations of the Repugnant Conclusion should be broaderincluding, for example, additions to unaffected, intersecting populations -but for this paper we ignore that proposal and focus on what they call a "restricted" formalization.this in an absurd example: adding our professor to a single-person "population" whose member is only a tiny bit better-off than the professor makes the resulting population worse, according to Average Utilitarianism.In fact, if the future of humanity is as long and wonderful as some hope (see, e.g., Ord, 2020), then adding a person likes this to the actual intertemporal world population2 might make the resulting population worse, according to Average Utilitarianism.This anti-natalist implication of Average Utilitarianism violates even a very weakened version of the Mere Addition principle.
In light of the above counterintuitive implications of on, the one hand, Total Utilitarianism (implying the Repugnant Conclusion) and, on the other hand, of Average Utilitarianism (violating even a highly weakened Mere Addition principle), some theorists have been attracted to a family of views that are often called Variable-Value views.3Some views within this family avoid the Repugnant Conclusion altogether while capturing the intuition that adding a well-off person to a small population always makes the resulting population better.More specifically, these views hold that the quantity that added persons (with fixed levels of welfare) contribute towards the overall value of a population decreases as the size of the population increases, cumulatively contributing only a bounded amount, which is how such views escape the Repugnant Conclusion.
Various versions of Variable-Value views have been rigorously formalized.These formalizations and the ensuing analysis have focused on qualitative properties of Variable-Value views: with which axioms do they comply?However, there has not been a similar focus on the quantitative implications of Variable-Value views.In particular, one might wonder how fast the quantity that an added person contributes towards the overall value of a population diminishes as the size and average welfare of the population increases, and what implications that will have for various trade-offs between increasing the size and the average welfare of a population.Similarly, one might wonder precisely which weakenings of the Mere Addition principle these views can accommodate without implying seemingly repugnant instances of the Repugnant Conclusion.
Our aim in this paper is to offer some suggestive examples that fit in this quantitative gap in the population ethics literature. 4So, our paper has more in common with the population ethics sub-literature that argues using striking, carefully chosen examples than with the sub-literature that provides qualitative, axiomatic proofs.In particular, we shall, first, conditionally assume that a very weak and intuitively compelling version of the Mere Addition principle holds even under what we take to be plausible (optimistic) empirical predictions about the future.Next, we calibrate two leading members of the Variable-Value family -namely, Number-Dampened Generalized Utilitarianism (NDGU) and Rank-Discounted Generalized Utilitarianism (RDGU) -to this assumption.Informally, the weak Mere Addition principle that we assume ensures that merely adding people who are very well-off by modern standards, such as happy professors in the developed world, does not make the population worse.The empirical prediction is that the future of humanity is long and prosperous, such that, in particular, the average welfare of the future population is even higher than the welfare of someone who is very well-off by modern 4 Our aim is not to examine all Variable-Value views.In particular, because we are principally investigating the usefulness of weakening Mere Addition in response to the Repugnant Conclusion, we shall not be concerned with those Variable-Value views that satisfy the strong version of Mere Addition (i.e., the version entailed by Total Utilitarianism), such as the theory examined in Sider (1991).Instead, the aim is to examine those views that (unlike Average Utilitarianism) imply some weak instance of Mere Addition, without implying the strong version of Mere Addition.
We note also that a normative reason for excluding from our examination the view in Sider (1991) is that it implies what Arrhenius's (ming) calls "The Very Anti Egalitarian Conclusion", and which states that for any perfectly equal population of at least two persons with positive welfare, there is a population which has the same number of people, lower average (and thus lower total) welfare and inequality, which is better.In fact, Sider himself rejects the view due to implications like this (Sider, 1991: 270).standards.
We should stress two things from the start.First, for our argument to succeed we do not have to assume that the future must (or even will) be prosperous and populous.We only need the conditional claim that if the future is prosperous and populous, you would nevertheless support a mere addition of a happy present-day professor.Second, from the point of view of the universe there is of course nothing special about relatively well-off professors in 2023 when evaluating the full intertemporal population.We write of this case merely because we expect that both you, the reader, and we, the authors, have some intuition about it.
Our main observation is that, when combined with the above two assumptions, these two leading members of the Variable-Value family calibrate to be nearly totalist, in the sense that they come quite close to agreeing with Total Utilitarianism.In particular, they imply countless instances of the Repugnant Conclusion.(By an "instance" of the Repugnant Conclusion, we mean the judgement that some particular population consisting only of lives that are barely worth living is better than some particular perfectly equal population of very well-off people.)Of course, these Variable-Value views do not imply the qualitative Repugnant Conclusion stated above -which holds for all populations of very well-off people.But the aforementioned implications, we argue, should nevertheless seem repugnant to those who oppose the Repugnant Conclusion.5 So, these results seem problematic for any argument that the Repugnant Conclusion requires us to reject Total Utilitarianism in favor of either NDGU or RDGU.

Variable-Value axiologies and their implications
Before discussing in detail the previously mentioned implications of Variable-Value axiologies, let us explain what types of axiologies we have in mind.
An infinity of population axiologies could have value vary, for instance, by whether population size is odd or even.Here we follow the population ethics literature which has understood the term "Variable-Value axiologies" to refer to a particular structure of well-behaved families of social welfare functions that are designed to respond to the tension between, on the one hand, satisfying (some version of) Mere Addition and, on the other hand, avoiding the Repugnant Conclusion.In his first paragraph on Variable-Value approaches, Arrhenius (ming) summarizes: "These principles are sometimes called 'compromise theories' since a Variable-Value Principle can be said to be a compromise between Total and Average Utilitarianism.With small populations enjoying high welfare, a Variable Value Principle behaves like Total Utilitarianism and assigns most of the value to the total sum of welfare.For large populations with low welfare, the principle mimics Average Utilitarianism and assigns most of the value to average welfare" (88).In the context of Ng's (1989) trilemma among Mere Addition, Non-Antiegalitarianism, 6 and avoiding the Repugnant Conclusion, we interpret the core of the Variable-Value idea to be a principled approach to rejecting Mere Addition in favor of the other two. 7So, we are interested in complete and transitive families of social welfare functions that: • are well-behaved in the sense of satisfying (ex post) Pareto, Extended Egalitarian Dominance, Non-Antiegalitarianism, and other non-controversial principles in the 6 Non-Antiegalitarianism says that a perfectly equal population is better than a population with the same number of people, inequality, and lower average welfare.(See, e.g., Arrhenius 2000.) 7For this reason, we disregard Sider's (1991) example of Geometrism, which rejects Non-Antiegalitarianism, and any other candidate axiology that does so; we are unaware of any author in the population ethics literature (including Sider in fact) who defends any Antiegalitarian Variable-Value proposal as plausibly the true population axiology.literature; 8 • avoid the Repugnant Conclusion by rejecting Mere Addition; • compromise between Total Utilitarianism for small populations and a populationsize-insensitive alternative for large populations; 9 and • can be calibrated by a parameter that quantifies the distance between Total Utilitarianism and the size-insensitive alternative as two convergent endpoints.
For clarity, we restrict our attention to the very large set of axiologies that, for perfectly equal populations of size n, reduce to g(x)f (n) for some increasing g and some concave, bounded f , where x is the average welfare in the population.(In section 4, we however illustrate these axiologies' implications for a highly unequal intertemporal population.)In what follows, we ask quantitative questions about f .In particular, we use a calibration method (informally described in the next section) which determines, for instance, how quickly f approaches its bound as population size increases.A totalist f would be linear.
We ask quantitatively: how totalist does a plausible Variable-Value view have to be?The answer, we argue, turns out to be rather totalist.
Two well-known examples of Variable-Value axiologies that satisfy all of the above discussed properties are Number-Dampened Generalized Utilitarianism (with an appropriate choice of functional form), and Rank-Discounted Generalized Utilitarianism.We will investigate these views in detail in section 4 after first informally presenting our general argument in the next section.
8 Since we will not make direct use of these principles, it suffices to define them informally.Pareto says that if every person is at least as well-off in population A as in population B, then A is at least as good as B. Extended Egalitarian Dominance says that if there is perfect equality in population A which is of greater size than population B, and every person in A has higher positive welfare than every person in B, then A is better than B. (See, e.g., Arrhenius ming.)Non-Antiegalitarianism is defined in footnote 6. 9 Recall the previous citation to Arrhenius (ming: 88).

Our calibration argument, informally
To clarify: it is not our own view that the Repugnant Conclusion is indeed repugnant or must be avoided by the correct theory of population ethics (Zuber et al., 2021).But many population ethicists have held that view, and we respond to them here.The population ethics literature is unclear on what exactly it is that is supposed to make the Repugnant Conclusion repugnant.We do not intend to take a stance on this question.In the rest of this paper, we will informally describe a judgement as being "repugnant" when it involves preferring a much larger, much worse-off population over a large but much smaller, much better-off perfectly equal population.We mean "repugnant in the sense of the Repugnant Here is why.We conjecture that you, the reader, have a strong conviction that the mere deletion of your life, or ours, or that of any other very well-off person by present-day standards, would not make the intertemporal world population better (even if it could be accomplished by magic, the "mere" deletion having no effect on the welfare of anyone else) than the actual intertemporal world population in which this very well-off person by present-day standards in fact exists.A great life judged by the standards of our times does not in and of itself make the world worse. 10evaluating the actual intertemporal population, after all, from the point of view of timeless population ethics.From the point of view of the universe, why is today special?The answer -as briefly mentioned in the introduction -is that of course it is not.But we are writing to you, the reader.And you live in the present and come to our argument with beliefs and intuitions and, in the case of your life, hedonic experiences.We think lives like yours and ours are easy, readily available, and informative test cases for the present purposes.In fact, we think that test cases like these illustrate that there is at least one population and one welfare level such that adding a life at that welfare level to that population does not make matters worse: namely, adding a good life, like a professor's in our times, to the actual intertemporal world population.
So why is agreement on this case important?Because it disciplines the calibration of any Variable-Value axiology, the combination of totalism on the one hand and, on the other hand, the number-insensitive counterpart (such as AU or leximin).We propose that two facts are probably true of the actual intertemporal world population.But more importantly, we assume that even if these two facts are not actually true, you would find that, if these facts were true, then the mere addition of a very well-off person by present-day standards would still not make the intertemporal world population worse.The two facts are: • The future is vast: the actual intertemporal world population is enormous.
• The future is splendid: the actual intertemporal world population is full of lives much better than ours, i.e., much better than that of a happy present-day professor.
These facts, plus our judgement about the mere addition of a very well-off person by present-day standards, bound our calibration of quantitatively how non-totalist a Variable-Value axiology can be.In steps: 1.Because the population is enormous, we will be making decisions like an averagist (or otherwise like a non-totalist, depending on the details of the particular Variable-Value view), unless the tuning parameter is calibrated to move away from totalism only very slowly.
2. Because someone like us is relatively badly-off compared to the splendid full distribution, adding such a person pulls down the average, disadvantages the lexical ladder, or otherwise looks undesirable to the non-totalist part of the axiology.
3. Because we judge that adding someone like us is nevertheless not a worsening, it must be the case that the tuning parameter is calibrated to move away from totalism only very slowly, so that the totalist benefits of the addition outweigh (in this case) the non-totalist costs of adding a relatively badly-off person.
And this brings us to the implications for the Repugnant Conclusion.If the tuning parameter is such that the Variable-Value axiology is, in the end, calibrated to be quite close to totalism, then it will often agree with totalism about how to rank populations.
And that means that it will make many repugnant judgements where it prefers larger, worse-off populations to smaller, better-off ones, agreeing with totalism even in many quantitavely extreme cases.The universally quantified Repugnant Conclusion is escaped, but repugnance is not.So the spectre of repugnance seems hardly a reason to choose a Variable-Value approach.The next section makes this general argument quantitatively precise.
Some readers will have recognized that our argument presents an application of a familiar logic in decision theory: calibration of variable-value objective functions to reveal tensions between intuitions for large-quantity decisions and intuitions for small-quantity decisions.The leading result in this literature is Rabin's (2000) celebrated argument about expected utility theory.Formally, we extend Rabin's argument about choice under risk to analogous functional forms in population ethics.
Rabin established that an expected utility maximizer can only be moderately risk averse when relatively small sums of money are involved -e.g.always turning down 50-50 gambles between losing $100 and winning $105 -if she is extremely risk averse when larger sums of money are involved -e.g.turning down 50-50 gambles between losing $2,000 and winning any (including infinite) amount of money.So, the lesson of Rabin's argument is that an expected utility maximizer is either surprisingly risk averse when stakes are large or surprisingly risk neutral when stakes are small.Similarly, the lesson of our calibration exercise is that Variable-Value views are either surprisingly totalist or surprisingly strongly anti-natalist, assuming a vast and splendid future. 11future).This last stipulation prevents the social evaluation of lifetime welfare from being too sharply bounded above, an assumption which some might deny.(We briefly return to this issue in the next section.) For precision and reproducibility, we have posted at the journal website an editable Excel spreadsheet that conducts all of the computations in this calibration exercise. 12e adopt the notation that finite vectors of real numbers are populations, where these number represent each person's lifetime welfare.For instance, x = (x 1 , ..., x n ) is a population.X = {x, y . ..} is the set of all populations.The natural-number length of the vector (e.g., N (x)) is the size of the population.x y means that y is at least as good as x.
The first view in the Variable-Value family that we consider can be stated as follows:13

Number-Dampened Generalized Utilitarianism (NDGU).
There is a concave (and increasing) real-valued function f such that for any x, y ∈ X: To our knowledge, this family was introduced to the literature implicitly in the diagrams of Hurka (1983).The concavity of f means that NDGU reduces the value of additions to the population as population size grows.Moreover, if (but only if) f is bounded, then NDGU avoids the universally quantified Repugnant Conclusion.
For illustrative purposes, we assume in our example that α for some α > 0. The parameter α is the crucial parameter of calibration that tunes how quickly, as population size increases, f transitions from a TU-like gradient for small populations to an AU-like nearly constant 1 for large populations.Larger α is more totalist;14 α nearer to zero is more averagist.

Now let x
) be the nondecreasing reordering of x, that is, an ordering of the welfare levels in x such that for any i, . The second Variable- Value view that we shall consider can then be stated as follows: Rank-Discounted Generalized Utilitarianism (RDGU).There is a β ∈ (0, 1) such that for any x, y ∈ X: where g is increasing and weakly concave.
This view was introduced and characterized by Asheim and Zuber (2014).It avoids the (universally quantified) Repugnant Conclusion because β 1 + β 2 + β 3 ... is a convergent series, which ensures that the aggregated value of a perfectly equal population is bounded and remains finite, no matter how large it becomes.In other words, the (universally quantified) Repugnant Conclusion does not follow from RDGU.
RDGU does not satisfy the strong Mere Addition principle that Total Utilitarianism entails.This is because adding a life to a population lowers the weights of any otherwiseexisting better-off lives, which may worsen the population by more than the additional life improves it.However, RDGU must satisfy some weakened Mere Addition principle, since β > 0, which means that some mere additions are valuable.And, in fact, the closer β is to 1, the closer RDGU comes to agreeing with totalism.We want to examine what RDGU implies if we assume that it satisfies a particular, plausible and very weak instance of the Mere Addition principle.
With these definitions, we are ready to calibrate.What does it imply for NDGU's α and RDGU's β if adding a present-day excellent life to the intertemporal world population does not make the population worse?
The first step is to state a plausible case for the mere addition of an excellent life not making the intertemporal world population worse.RDGU to what we propose is a plausible judgement about a particular mere addition, given the above realistic (but of course not unquestionable) empirical assumptions.
Finally, we can apply these calibrations from a real-world case to the theoretical case of repugnant-style conclusions.In Parfit's canonical formulation of the Repugnant Conclusion, the small population consists of 10 billion people, "all with very high quality of life" (Parfit 1984: 388).With the above calibrations, how large would a large population have to be in order to be judged as better, if (potentially scaled) per-person welfare were only 1% as great as in the small population?
For Total Utilitarianism, the answer is immediate: At least a trillion people, that is, 100 times as large as the small population.NDGU, calibrated as above, however recommends the large population if it contains at least 1.30 trillion people.In other words, according to this calibrated version of NDGU, a perfectly equal population of 10 billion very well-off people is worse than a population of 1.30 trillion people where each person's (potentially scaled) welfare is only 1% as great as in the small population.RDGU however recommends the small population if it contains at least 1.13 trillion people.In other words, according to calibrated RDGU, a perfectly equal population of 10 billion very well-off people is worse than a population of 1.13 trillion people where each person's welfare is only 1% as great as in the small population.
So, these views, once calibrated to recommend the mere addition in Table 1, differ from Total Utilitarianism in this repugnant-like judgment by respectively 30% and 13%.
These are not radical, qualitative differences.We propose that anyone worried about the repugnance of totalism should find the same repugnance in these judgments.So, NDGU and RDGU, calibrated to a plausible, real-world judgment, substantively agree with Total Utilitarianism in this theoretically prominent (but practically unrealistic) choice between large and small populations.

Lesson and concluding remarks
Recall that the intuitive appeal of Variable-Values views was supposed to be that they could avoid the Repugnant Conclusion while satisfying at least some weak instances of the Mere Addition principle.We have now seen, however, that if two leading members of the family of Variable-Values views satisfy what we take to be a very plausible, and certainly weak, instance of Mere Addition -and moreover do so when we make plausible empirical assumptions about the intertemporal world population but don't assume that the contributive value of welfare is bounded above -then these two members have implications that, we suspect, those who oppose Total Utilitarianism due to the Repugnant Conclusion will find repugnant. 16The results of this paper could thus be interpreted as suggesting, along with a growing literature -including Spears and Budolfson (2021) on additions to an unaffected population and Arrhenius and Stefánsson (ming) on risky choice between uncertain populations -that the repugnance of Total Utilitarianism can be found in even those theories that were designed to avoid such repugnance.
Our results in this paper are merely suggestive.They are offered to provide perspective based on quantitative examples, not qualitative proofs.
Here is what they suggest to us: First, a wish to avoid repugnance may not be a good reason to favour the aforementioned members of the Variable-Value family over Total Utilitarianism.Second, perhaps it is time to learn to live with such repugnance.
Before concluding, it is worth acknowledging that those who are sufficiently attracted to the Variable-Value views we have discussed might draw a radically different conclusion from our results, namely, that they show that one should put an upper bound on the contributive value of a person's welfare. 17In our result for RDGU, we assumed that the g function is concave but not bounded; whereas in our NDGU result we assume that, for any given population size, the contributive value of a person's welfare is proportional to their (potentially transformed or scaled but unbounded) level of welfare.Those who are sufficiently committed to either of these views may thus take our paper as an argument that the contributive value of a person's welfare is bounded above.Such bounding will of course have implications that many will find unattractive.In particular, it implies that some arbitrarily great welfare gains -enjoyed by people who are already very well off -result in no gain in moral value.Still, we acknowledge that some may find this to be a bullet worth biting, and others may even find this to be independently plausible.
Table 1 describes two populationsdifferentiated by one excellent life.The past population size of 120 billion is chosen to match demographers' estimates; the future population size is chosen (as a conditional hypothetical) to reflect optimistic longtermist hopes.If either NDGU or RDGU are used in a scaled or transformed version, then "average utility" refers to those transformed values; otherwise it is simply average lifetime welfare.The next step is to find the parameter values.If the population with the added excellent life is at least as good as the population without, then NDGU's α ≥ 2.34639 × 10 12 and RDGU's β ≥ 0.999999999999777.In other words, we have now calibrated NDGU and

Table 1 :
Two hypothetical populations: A case of weak mere addition