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A Solution to the Mere Addition Paradox


by Socrethics   First version 2014   Last version 2018






Table of Contents


1.  Introduction

2.  Incommensurability

3.  Indifference Curves 

4.  Sympathy

5.  Population Ethics under Uncertainty






1.  Introduction


The Mere Addition Paradox was identified by Derek Parfit [Parfit 1984, Chapter 19].

For a description and analysis see On the Buddhist Truths and the Paradoxes in Population Ethics.


Above article ends with the following conclusion:

         Population ethics using a two-parameter model (welfare and population size) can be characterized by a conflict of interest.

         Conflicting interests shape conflicting intuitions.

         Conflicting intuitions make it impossible to find a coherent normative theory.

In order to find a coherent theory one would have to find a universal interest and a corresponding universal intuition. The obvious candidate to meet this request is sympathy [Contestabile 2010, 111].




2.  Incommensurability


The aggregation of incommensurable preferences is the most general approach to solve conflicts of interest. The “repugnancy” of a population can be considered to be a matter of individual preference. A majority-definition of “repugnancy” is then derived by aggregating the individual preferences for each combination of (population size) and (average welfare). If the individuals are well informed about the consequences of a certain population policy and if they can freely express their attitude towards risk, the aggregation of their preferences delivers an intersubjective criterion to decide between policies. The focus shifts to forecasting and educational advertising.


There is a theoretical hurdle in the process of aggregation. Incommensurable preferences lead to Arrowian impossibility theorems [Arrhenius 2000, 264]. No voting system can convert the ranked preferences of individuals into a community-wide ranking while also meeting a certain set of reasonable criteria [Arrow]. In practice this theoretical obstacle can be bypassed. The Arrowian impossibility theorem only becomes effective if three or more options are at stake and not if the voters have to confirm or decline a specific population policy.


Besides the promotion of democracy there is no normative claim in this approach. Democracy, however, does not solve the Mere Addition Paradox, because it cannot prevent the Repugnant Conclusion. In democratic polls a majority can decide to indefinitely expand the population at the cost of the average welfare. In such a case the expansionist majority completely overrules the quality-conscious minority. A solution, which is based on sympathy, has to protect minorities. In the following we phrase this demand in the language of indifference curves:




3.  Indifference Curves



Distributive justice

How would an axiology look like, which is based on sympathy and which solves the Mere Addition Paradox?

The guiding idea for the sought-after axiology can be found in Fig.1:




File:Social indifference curves small.png



Picture from Welfare Economics, Wikipedia



The Mere Addition Paradox can be seen as a conflict of interest between perfectionists (person #1 in Fig.1) and expansionists (person #2 in Fig.1):

         For the expansionists utility is defined by the population size, as long as there remains a minimally positive average welfare.

         For the perfectionists utility is defined by the average welfare, as long as there remains a minimal population size

The term welfare stands for economic welfare in this paper, because in practice the focus of population ethics is on economic welfare. But the analysis is also valid for happiness, quality of life and life satisfaction, as far as these terms correlate with economic welfare.


In our context indifference curves define the distributive (in)justice between perfectionists and expansionists.

It is assumed that the available resources can be distributed in such a way that the result is

         either a large population with low average welfare (the preference of the expansionists)

         or a small population with high average welfare (the preference of the perfectionists)

         or a compromise.


-        With the Utilitarian Indifference Curve there is no distributive justice, i.e. one of the two interests can completely dominate the other (in practice classical utilitarianism favors expansionists because it is easier to increase the population size than the average welfare)

-        The Max-Min Indifference Curve is egalitarian, i.e. the two interests have the same weight.

-        The Intermediate Indifference Curve is prioritarian, i.e. it prioritizes the expansionist interest or the perfectionist interest, depending on the segment of the curve.


The following description is adapted from Welfare economics, Wikipedia:



Utilitarian Indifference Curves

The Utilitarian Indifference Curves on the left hand side of Fig.1 correspond to different levels of total utility, where

total utility = utility of person #1  +  utility of person #2. Since a Utilitarian Indifference Curve confines an isosceles triangle, each total (represented by a point on the curve) results in the same value.



Max-Min Indifference Curves

The Max-Min (respectively Maximin) Indifference Curves in the middle of Fig.1 correspond to different levels of total utility, where total utility is measured by the utility of the worst-off. The total utility of a Max-Min Indifference Curve is depicted by the vertex of the curve. The vertex is defined by an average welfare and a population size, which have the same utility for both person #1 and person #2. Solely increasing the utility of person 1 (no matter how much) doesn’t increase the total, because the total is measured by the utility of the worst-off (in this case the utility of person #2). Analogously the total cannot be increased by solely increasing the utility of person #2.



Intermediate Indifference Curves

The Intermediate Indifference Curves on the right hand side of Fig.1 can be interpreted as showing that – with increasing inequality – a larger increase in the utility of person #2  (i.e. a larger expansion of the population size) is needed to compensate the loss of utility of person #1 (and vice-versa). The Intermediate Indifference Curve can be constructed in such a way, that – at a certain point – it becomes virtually impossible to increase the total by an increase in the population size, i.e. the curve turns into a vertical line (as in the Max-Min Curve).



Repugnant conclusions

The Mere Addition Paradox is characterized by

-        the Repugnant Conclusion, which says that it is repugnant to minimize the average welfare in order to maximize the population size.

-        its reversal, which says that it is repugnant to minimize the population size in order to maximize the average welfare.

If these repugnant conclusions disappear, the paradox is solved.


The Stanford Encyclopedia lists eight different ways of dealing with the Repugnant Conclusion [Stanford], but none of them is based on indifference curves:

         The Utilitarian Indifference Curve allows completely overruling the opposite interest, i.e. it is vulnerable to both the Repugnant Conclusion and its reversal.

         The Max-Min Indifference Curve does not allow overruling the opposite interest and is therefore immune to both kinds of repugnant conclusions.

         The Intermediate Indifference Curve allows approximating the Min-Max Indifference Curve for high average welfare and for large populations, so that the repugnant conclusions can be avoided as well.

Distributive justice is the result of sympathy, law of reciprocity or – as Kant called it – the categorical imperative.




4.  Sympathy



Degrees of sympathy

The indifference curves can be assigned to degrees of sympathy as follows:

1.      Utilitarian Indifference Curve:

There is no sympathy between the two conflicting interests.

2.      Max-Min Indifference Curve:

Perfect sympathy, both interests are equally considered.

3.      Intermediate Indifference Curve:

Moderate sympathy, the interests are unequally considered, but it is impossible to completely overrule the opposing interest.


Since the targeted solution emphasizes sympathy, we exclude the Utilitarian Indifference Curve and declare the Max-Min Indifference Curve to be the ethical ideal. This has the following consequences:

         If resources are at disposition then the expansionists – which like to have plenty of children – would use them for expanding the population, whereas the perfectionists would use them for increasing the average welfare. The Min-Max Indifference Curve says that only a population policy which considers both interests equally, improves the state of affairs.

         If, in contrast, if there is a lack of resources then the expansionists would rather decrease the average welfare than having less children, whereas the perfectionists would rather stop having children than decreasing the average welfare. The Min-Max Indifference Curve says that population size and average welfare have to be equally reduced.



Ordering by degree of sympathy

What makes a population better in a diagram with the parameters average welfare and population size?

The crucial point consists in measuring the equal weight, given to the conflicting interests. In order to quantify equality we have to define (under the side-constraint of pre-defined available resources)

-        the maximal population size that can be realized with a minimal average welfare (expansionist interest)

-        the maximal average welfare that can be realized with a minimal population size (perfectionist interest)


If we consider now any population, which satisfies the side-constraint, then we can assign

-        a percentage for serving the expansionist interest (actual relative to maximal population size)

-        a percentage for serving the perfectionist interest (actual relative to maximal average welfare)

A population which serves both interests with the same percentage corresponds to perfect sympathy. All other populations can be ordered according to their degree of sympathy (deviation from equality).




5.  Population Ethics under Uncertainty


A different normative approach consists in looking at the situation from the perspective of an impartial empathic observer (see Ideal Observer Theory). The solution of the Mere Addition Paradox, which is described in chapter 4, is a solution of the conflict between quantity and quality. From the perspective of an impartial empathic observer, however, the major theoretical problem is not quantity versus quality, but positive versus negative total welfare. We do not know with certainty, if total (national, global) welfare is positive or negative [Contestabile 2016]. Given this uncertainty, it makes sense to disregard the population size and focus on the average welfare. Average utilitarianism is the most popular axiology among welfare economists [Arrhenius, 53]. A change in the population size

-        is ethically good, if the average increases

-        is ethically bad, if the average decreases


Average utilitarianism has several theoretical deficiencies [Arrhenius, 54-57], but they are all based on the assumption that the sign of total welfare is known. The crucial point remains the metric, which is used to compare happiness with suffering. As long as the sign of total welfare is uncertain, we should actually construct the hedonistic scale in such a way that total welfare becomes zero. From then on we can observe in which direction the average moves. But no matter, if the tendency is positive or negative, the uncertainty about the sign remains, so that it is still justified to neglect the population size.


One could argue that – as long as the task is only to observe the changes of the average – one could as well operate with current indices like the OECD Better Life Index or the Satisfaction with Life Index. But there is a major difference with regard to ethical priorities. The current indices suggest that the traumatic suffering of a minority can easily be compensated by the happiness of the majority. The revised index, in contrast, operates with an asymmetric hedonistic scale, which makes it hard to compensate (see Negative Utilitarianism and Justice). Under these premises the most efficient way to improve total welfare is to reduce the worst cases of suffering with the highest priority.






1.      Arrhenius Gustav (2000), Future Generations, A Challenge for Moral Theory, FD-Diss., Uppsala University, Dept. of Philosopy, Uppsala: University Printers

2.      Arrow Kenneth J. (1966), Social Choice and Individual Values, Wiley, New York

3.      Contestabile Bruno (2010), On the Buddhist Truths and the Paradoxes in Population Ethics, Contemporary Buddhism Vol.11, No.1, 103-113, Routledge, London

4.      Contestabile, Bruno (2016), The Denial of the World from an Impartial View, Contemporary Buddhism Vol.17, No.1, 49-61, Routledge, London

5.      Parfit Derek (1984), Reasons and Persons, Clarendon Press, Oxford

6.      Stanford Encyclopedia of Philosophy (2016), The Repugnant Conclusion