An Issue with the Repugnant Conclusion

I shall argue against an outline of the argument from my perspective as a statistician and not a philosopher,

A straightforward way of capturing the No-Difference View is total utilitarianism according to which the best outcome is the one in which there would be the greatest quantity of whatever makes life worth living (Parfit 1984 p. 387). However, this view implies that any loss in the quality of lives in a population can be compensated for by a sufficient gain in the quantity of a population; that is, it leads to the Repugnant Conclusion.

plato.stanford.edu (emphasis mine)

We will argue that taken in a general sense this is false due to unstated but required assumptions. Let’s define total utility as:

$$T\left(N\right)=\sum _{i=1}^{N}u\left(X_N\right)$$

Where $u$ is some utility function and $X_N$  is some (possibly complicated object I won’t bother trying to define adequately here) allocation of resources in the world with $N$ inhabitants.

Let’s open with the counterexample to the general claim. 

Counterexample to general claim

Suppose $u\left(X_N\right)=1/N^2$. Then:

$$T\left(N\right)=\sum _{i=1}^{N}u\left(X_N\right)=\sum _{i=1}^{N}1/N^2=1/N$$

From which we can see that 

$$T\left(1\right)>\underset{N\to \infty }{lim}T\left(N\right)$$

Hence there is no increase in population compensates the loss of utility to ‘existing’ inhabitants. 

The implication here is that any kind of ‘Repugnant Conclusion’ argument is heavily dependent on the form of the assumed utility function. It cannot be a general argument with respect to the utility function involved.

In particular the Repugnant Conclusion argument can only work where utility is, roughly speaking, linear or better in population size. Although in the latter case the Conclusion would end up being Happy rather than Repugnant.

Motivated version

While the counterexample above is unmotivated, one could easily produce a version in narrative. For instance, suppose the wellbeing of a being was dependent on three square meals a day.  How do might we model this? Here’s a quick shot:

$$u\left(X_N\right)=I(R/N>t)$$

Where $R$ is some notion of total resources, $t$ is some ‘three meals’ threshold, and $I$ is the inidicator function. So everyone has utility 1 so long as they get at least three square meals a day and 0 otherwise. For fixed $R$, again we would see for $R>t$ that 

$$T\left(1\right)=1>\underset{N\to \infty }{lim}T\left(N\right)=0$$

One could add a bunch of ‘smoothing’ to this format or other bells and whistles, but the essence would remain.

Comments & Questions

This criticism is most similar to that of the ‘Variable value principles’ of the Plato article. The difference here is that we are not trying to find a ‘modification’ of total utilitarianism. Instead we argue that the Conclusion doesn’t follow from the premises in the general case, even if we are total utilitarians.

The general cause of the failure mode is, approximately, not considering utility as a function of the distribution of resources across the population.

An implication is that we should not see arguments that assume bigger population is better population without clear statements about the used assumptions on the form of utility functions. On pain of such arguments leading to anti-optimal decisions if they get it wrong.