# Logarithms and Total Utilitarianism

_Epistemic status: I might be reinventing the wheel here_

A common cause for rejection of total utilitarianism is that it implies the so-called [Repugnant Conclusion](https://plato.stanford.edu/entries/repugnant-conclusion/), of which a lot has been written elsewhere. I will argue that while this implication is solid in theory, it does not apply in our current known universe. My view is similar to the one expressed [here](https://www.lesswrong.com/posts/prEZkHYawwnhzswyf/the-mere-cable-channel-addition-paradox), but I try to give more details.

The Repugnant Conclusion IRL
----------------------------

The greatest relevance of the RC in practice arises in situations of scarce resources and Malthusian [population traps](https://en.wikipedia.org/wiki/Malthusian_trap)¹: We compare population A, where there are few people with each one having plentiful resources, and population Z, which has grown from A until the average person lives in near-subsistence conditions.

Let's formalize this a bit: suppose each person requires 1 unit of resources for living, so that the utility of a person living on 1 resources is exactly 0: a completely neutral life. Furthermore, suppose utility is linear w.r.t. resources: doubling resources means doubling utility and 10 resources correspond to 1 utility. If there are 100 resources in the world, population A might contain 10 people with 10 resources each and total utility 10; population Z might contain 99 people with 100/99 resources each and total utility also 10.

So in this model, we are indifferent between A and Z even as everyone in Z is barely subsisting, and this would be the Repugnant Conclusion². But this conclusion depends crucially on the relationship between resources and utility which we have assumed to be linear. What if our assumption is wrong? What is this relationship in the actual world? Note that this is an empirical question³.

It is well known that self-reported happiness varies logarithmically with income⁴, both between countries and for individuals within each country, so it seems reasonable to assume that the utility-resources relation is logarithmic: exponential increases in resources bring linear increases in utility.

Back to our model, assuming log utility, how do we now compare A and Z? If utility per person is < refresh to render LaTeX > where < refresh to render LaTeX > are the resources available to that person, then total utility is < refresh to render LaTeX > . Assuming equality in the population (see the Equality section), if < refresh to render LaTeX > are total resources and < refresh to render LaTeX > is population size, each person has resources < refresh to render LaTeX > and so we have

<refresh to render LaTeX>

We can plot total utility (vertical axis) as a function of _N_ (horizontal axis) for < refresh to render LaTeX >

![](https://i.imgur.com/D1vgOhF.png)

Here we can see two extremes of cero utility: at < refresh to render LaTeX > where there are no persons and at < refresh to render LaTeX > where each person lives with 1 resources, at subsistence level. In the middle there is a sweet spot, and the maximum M lies at around 37 people⁵.

Now we can answer our question! Population A, where < refresh to render LaTeX > is better than population Z where < refresh to render LaTeX > , but M is a superior alternative to both.

So I have shown that there is a population M greater and better than A where everyone is worse off, how is that different from the RC? Well, the difference is that this does not happen for every population, but only for those where average well being is relatively high. Furthermore, the average individual in M is far above subsistence.

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Equality
--------

In my model I assumed an equal distribution of resources over the population, mainly to simplify the calculations, but also because under the log relationship and if the population is held constant, total utilitarianism endorses equality. I will try to give an intuition for this and then a formal proof.

![](https://i.imgur.com/hFHm5G7.png)

This graph represents individual utility (vertical axis) vs individual resources (horizontal axis). If there are two people, A and B, each having 2.5 and 7.5 resources respectively, we can reallocate resources so that both now are at point M, with 5 each. Note that the increase in utility for A is 3, while the decrease for B is a bit less than 2, so total utility increases by more than 1.

This happens no matter where in the graph are A and B due to the properties of the log function. As long as there is a difference in wealth you can increase total utility by redistributing resources equally.

For a formal proof, see ⁶.

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Implications
------------

The main conclusion I get from this is that although total utilitarianism is far from perfect, it might give good results in practice. The Repugnant Conclusion is not dead, however. We can certainly imagine some sentient aliens, AIs or animals whose utility function is such that greater, worse-average-utility populations end up being better. But in this case, should we really call it repugnant? Could our intuition be fine-tuned for thinking about humans, and thus not applicable to those hypothetical beings?

I don't know to what extent have others explored the connection between total utilitarianism and equality, but I was surprised when I realized that the former could imply the latter. Of course, even if total utility is all that matters, it might not be possible to reshuffle it among individuals with complete liberty, which is the case in my model.

* * *

Footnotes
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1: One might consider other ways of controlling individual utility in a population besides resources (e.g. mind design, torture...) but these seem less relevant to me.

2: Actually, in the original formulation Z is shown to be _better_ than A, not just equally good.

3: As long as utility is well defined, that is. Here I will use self-reported happiness as a proxy for utility.

4: See the charts [here](https://ourworldindata.org/happiness-and-life-satisfaction#income)

5: We can find the exact maximum for any R with a bit of calculus:

<refresh to render LaTeX>

<refresh to render LaTeX>

A nice property of this is that the ratio < refresh to render LaTeX > that maximizes < refresh to render LaTeX > is constant for al < refresh to render LaTeX > (the exact constant < refresh to render LaTeX > obtained here is just due to the arbitrary choice of base 10 for the logarithms)

  
6: For a population of < refresh to render LaTeX > individuals the distribution of < refresh to render LaTeX > resources which maximizes total utility < refresh to render LaTeX > is that where < refresh to render LaTeX > for all < refresh to render LaTeX > . The proof goes by induction on < refresh to render LaTeX > .

This is obvious in the case < refresh to render LaTeX > . For the induction step, we can separate a population of < refresh to render LaTeX > into two sets of < refresh to render LaTeX > and 1 individuals respectively so that total utility is < refresh to render LaTeX > . Suppose we allocate < refresh to render LaTeX > resources to the group of < refresh to render LaTeX > , and < refresh to render LaTeX > to the last person. By hypothesis, each of the < refresh to render LaTeX > people must receive < refresh to render LaTeX > resources to maximize their total utility so < refresh to render LaTeX >

Now we have to decide how much should < refresh to render LaTeX > be.

<refresh to render LaTeX>

Solving for < refresh to render LaTeX > :

<refresh to render LaTeX>

Therefore, for each of the < refresh to render LaTeX > first individuals < refresh to render LaTeX > and for the last one < refresh to render LaTeX >