2. 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, Arrhenius (Reference ArrheniusForthcoming: 88) 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.’ In the context of Ng’s (Reference Ng1989) trilemma among Mere Addition, Non-Antiegalitarianism ^{Footnote 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 favour of the other two. ^{Footnote 7} So, 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 literature; ^{Footnote 8}

• avoid the Repugnant Conclusion by rejecting Mere Addition;

• compromise between Total Utilitarianism for small populations and a population-size-insensitive alternative for large populations; ^{Footnote 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 $g\left( {\overline x} \right)f\left( n \right)$ for some increasing $g$ and some concave, bounded $f$ , where $\overline x$ 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.

3. 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. Reference Zuber, Spears and Gustafsson2021). 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 Conclusion’. Perhaps no such judgement is actually repugnant at all, but if repugnance is to be found in the Repugnant Conclusion, we propose that it needs to be found consistently in any similarly ‘repugnant’ judgements (and hereafter without the quotation marks).

But let’s say that you do find the Repugnant Conclusion repugnant; that you find Non-Antiegalitarianism unrejectable; and that you therefore abandon the strong Mere Addition principle in favour of a Variable-Value axiology. Have you then escaped repugnance? We suggest that it depends upon the calibration of the resulting partially totalist view.

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. ^{Footnote 10}

A natural question is why we are emphasizing our times, your life, or ours. We are evaluating 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. 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. 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. 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 (Reference Rabin2000) 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 $2000 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. ^{Footnote 11}

4. Calibrated Examples Using Two Leading Variable-Value Axiologies

To proceed, let us stipulate that there is a welfare level beyond which lives at that level are excellent by the standards of 21st-century developed countries. For concreteness, we shall occasionally assume that a typical happy professor in a present-day developed country is at that level – we assume that most readers of this paper will have familiarity with such a life. When we state two Variable-Value axiologies algebraically, we will assume that the lifetime welfare of such excellent lives can be represented by some positive real number. The particular number we use to represent the welfare of excellent lives does not matter, so long as we can assume that these lives are indeed excellent; that excellent lives are clearly better than lives barely worth living; and that there could be lives that the axiology values as even better than these excellent lives (such as in a long, prosperous future). 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. ^{Footnote 12}

We adopt the notation that finite vectors of real numbers are populations, where these number represent each person’s lifetime welfare. For instance, ${\bf x} = \left( {{x_1},\ldots, {x_n}} \right)$ is a population. ${\bf X} = \left\{ {{\bf x,y} \ldots } \right\}$ is the set of all populations. The natural-number length of the vector (e.g. ${\cal N}\left( {\bf x} \right)$ ) is the size of the population. ${\bf x} \mathbin{\lower.3ex\hbox{$\buildrel\prec\over{\smash{\scriptstyle\sim}\vphantom{_x}}$}} {\bf y}$ means that $\bf y$ is at least as good as $\bf x$ . The first view in the Variable-Value family that we consider can be stated as follows: ^{Footnote 13}

Number-Dampened Generalized Utilitarianism (NDGU). There is a concave (and increasing) real-valued function $f$ such that for any $\boldsymbol x,y \in X$ :

$${\boldsymbol x} \mathbin{\lower.3ex\hbox{$\buildrel\prec\over{\smash{\scriptstyle\sim}\vphantom{_x}}$}} {\boldsymbol y} \Leftrightarrow \overline x{\rm{\;}}f\left( {{\cal N}\left( {\boldsymbol x} \right)} \right) \le \overline y{\rm{\;}}f\left( {{\cal N}\left( {\boldsymbol y} \right)} \right)$$

To our knowledge, this family was introduced to the literature implicitly in the diagrams of Hurka (Reference Hurka1983). 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

for some $\alpha \gt 0$ . The parameter $\alpha $ 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 $\alpha $ is more totalist; ^{Footnote 14} $\alpha $ nearer to zero is more averagist.

Now let ${{\bf x}_{[]}} = \left( {{x_{\left[ 1 \right]}},\ldots, {x_{\left[ r \right]}},\ldots, {x_{\left[ m \right]}}} \right)$ be the nondecreasing reordering of x, that is, an ordering of the welfare levels in x such that for any $i$ , ${x_{\left[ i \right]}} \le {x_{\left[ {i + 1} \right]}}$ . The second Variable-Value view that we shall consider can then be stated as follows:

Rank-Discounted Generalized Utilitarianism (RDGU). There is a $\beta \in \left( {0,1} \right)$ such that for any $\boldsymbol x,y \in X$ :

$${\boldsymbol x} \mathbin{\lower.3ex\hbox{$\buildrel\prec\over{\smash{\scriptstyle\sim}\vphantom{_x}}$}} {\boldsymbol y} \Leftrightarrow \mathop \sum \limits_r {\beta ^r}g\left( {{x_{\left[ r \right]}}} \right) \le \mathop \sum \limits_r {\beta ^r}g\left( {{y_{\left[ r \right]}}} \right)$$

where $g$ is increasing and weakly concave.

This view was introduced and characterized by Asheim and Zuber (Reference Asheim and Zuber2014). It avoids the (universally quantified) Repugnant Conclusion because ${\beta ^1} + {\beta ^2} + {\beta ^3}$ … 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.

Like $\alpha $ tunes NDGU, $\beta $ tunes RDGU. Asheim and Zuber (Reference Asheim and Zuber2014) prove that as $\beta $ approaches 1, RDGU approaches totalism, ^{Footnote 15} and as $\beta $ approaches 0, RDGU approaches a variable-population version of leximin.

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 otherwise-existing 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 $\beta \gt 0$ , which means that some mere additions are valuable. And, in fact, the closer $\beta $ 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 $\alpha $ and RDGU’s $\beta $ 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. Table 1 describes two populations differentiated 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.

Table 1. Two hypothetical populations: a case of weak mere addition

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 it, then NDGU’s $\alpha \ge2.34639 \times {10^{12}}$ and RDGU’s $\beta \ge0.999999999999777$ . In other words, we have now calibrated NDGU and 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 Reference Parfit1984: 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 judgement 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 judgements. So, NDGU and RDGU, calibrated to a plausible, real-world judgement, substantively agree with Total Utilitarianism in this theoretically prominent (but practically unrealistic) choice between large and small populations.

5. 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. ^{Footnote 16} The results of this paper could thus be interpreted as suggesting, along with a growing literature – including Spears and Budolfson (Reference Spears and Budolfson2021) on additions to an unaffected population and Arrhenius and Stefánsson (Reference Arrhenius and StefánssonForthcoming) 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. ^{Footnote 17} In 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.