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A Solution to the Mere Addition Paradox


B.Contestabile       admin@socrethics.com        First version 2014   Last version 2018






Table of Contents


1.  Introduction

2.  Indifference Curves 

3.  Incommensurability

4.  Theory and Practice






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.  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) and expansionists (person #2):

         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

The term (positive) welfare is used here as a synonym for happiness and quality of life.


There are two different strategies to increase the average welfare, which correspond to different kinds of happiness [Contestabile 2010, 107] and accordingly different types of perfectionists:

(1)   Satisfy as many desires as possible

(2)   Eliminate desires in order to avoid frustrations


These two strategies correspond to different axiologies:

(1)   Hospital axiology or classical utilitarianism

(2)   Perfectionist Buddhism or negative preference utilitarianism

For information on these axiologies see Negative Utilitarianism and Buddhist Intuition.


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 one of the two interests to some extent (but not completely).

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).




1.      Utilitarian Indifference Curve:

No matter how unequal the distribution, an increase in the utility of person #2 always compensates the loss of utility of person #1 (and vice-versa). There is no sympathy between the two conflicting interests.


2.      Intermediate Indifference Curve:

The opposite interest is partially considered. This is the characteristics of a moderate degree of sympathy.


3.      Max-Min Indifference Curve:

The definition of the vertex presupposes a high degree of readiness to make concessions. The population size cannot be increased without at the same time increasing the average welfare (and vice-versa). These are the characteristics of a high degree of sympathy.



Repugnant Conclusions

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

         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.




3.  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. The interest of the majority could be far from the interest of an empathic impartial observer.




4.  Theory and Practice


How could the theory of chapter 3 be implemented in practice?

As long as there is no consensus on the desirable degree of sympathy, we do not know which indifference curve has to be applied.


Is there something like an “objective” criterion for the desirable degree of sympathy?

An objective manifestation of sympathy corresponds to the interest of an empathic impartial observer. Unfortunately, under these premises, we do not even know, if the sign of total welfare is positive or negative [Contestabile, 2016]. It makes therefore sense to disregard the population size and focus on the average welfare.






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