My Current Solution to the Repugnant Conclusion

A relatively simple way of making the repugnant conclusion more intuitive to me is to recognise that individual selves are largely an illusion, i.e. that empty individualism (or, its better marketed/more spiritual sounding but functional equivalent open individualism) is correct.

Suppose you've set up your parameters such that pinpricks actually involve negative utility - because (trigger warning of slightly graphical image for the second of these links) in many cases it obviously isn't actually negative, which muddies our intuitions. Then for empty individualism a tiny amount of torture either is actually closely analogous to a large number of pinpricks (that just happen to be locally clustered). The OI equivalent is that a small amount of torture is analogous to a pinprick on the cosmic entity

Even in the case of humans, we can imagine how pinpricks could add up. A single superficial pinprick isn't that bad - and the difference between a 1mm and 2mm insertion would be very slight. But if you insert hundreds or thousands of pins 10+mm deep, gradually you move towards an experience that seems as bad as any other torture. 

To put it simply, sufficiently many pinpricks, of sufficient depth to be unpleasant, are torture. And they can be torture at a degree of virtually any level of pain a human is capable of experiencing - so I don't see a need to introduce dramatic discontinuities in our moral evaluation to explain why other tortures are somehow still morally worse.

I certainly share your intuition that there is no number of pinpricks such that it would ever be worse than 1 billion people in horrific torture. But if pinpricks can be aggregated to begin with, it seems arbitrary to assign an upper limit to the aggregate disvalue of pinpricks. So here's weird solution that I just thought of (and probably has all kinds of obvious problems that I don't have time to consider): Maybe there are some harms that are so small that they can't aggregate to begin with. For me, this is an easier bullet to bite than the proposal that miniscule harms can be added together but only up to a certain point.

Thanks Alfredo, I enjoyed this post. 

My favourite response to the repugnant conclusion is David Benatar's asymmetry argument. The idea that the absence of suffering in potential beings is good even when there's no one to enjoy that good, whereas the absence of pleasure isn't bad unless there's someone who's deprived of it. So adding more sentient beings whose lives contain any suffering isn't an improvement - so the repugnant conclusion never gets off the ground. It's the same intuition behind why it isn't a tragedy that Mars has no sentient life.

I think this is basically the same theory as: https://forum.effectivealtruism.org/posts/je5TiYESSv53tWHC9/utilitarians-should-accept-that-some-suffering-cannot-be-1

Nit: I don't think the welfare of the people you describe in Population A is only 100 times greater than the welfare of the people you describe in Population B. More like a million. (Of course, given the population ratio you hypothesize this correction makes no difference.)

(For the record I endorse the repugnant conclusion; I think Z really is better than A.)

Thank you so much for sharing this article; I really enjoy reading about social ethics in general.

To begin with, I think one should relativize "Repugnant Conclusion" to a system of population ethics: here, sum utilitarianism (classical, unweighted, etc.)
Moreover, there are several possible repugnant conclusions, even for a single system of population ethics.

I agree with this (as can be seen in what I develop further below with the $S_d$):

Now, I should very clear on what, exactly, my favored solution entails. My favored solution doesn’t exactly entail that there is some pair of worlds <w1, w2> in the sequence where, no matter how small the difference in intensity between w1 and w2, adding more people to w2 could never outweigh w1. What it entails is that, for any real number d, there has to be a pair of worlds <w1, w2> in the sequence where the difference in intensity between w1 and w2 is d, and no matter how many more people are in w2, w2 could never outweigh w1. That’s still really weird though!

But I would say that I do not see a reason (I may be overlooking an aspect, of course) to conclude this (if the definition is absolute) :

What this means is there is some population, say, K, where there are lots of people living pretty-good lives, and if you were to lower the level of well-being in K by even a little bit (like, 0.0000000000000000000000001%) no matter how many more people you added, that would never be better than K.

and same :

To put it in terms of pain: if K is a world where 10000000000 people are experiencing a pain of level p, K is worse than a world where 50 TRILLION TIMES as many people are experiencing a pain that is 99.999999999% as intense as p. That’s pretty nuts!! If the level of pain is only slightly less intense, you presumably shouldn’t allow 50 trillion times as many people to experience pain, even if it is a slightly less intense pain. But that’s what my favored solution entails.

To make my point of view clearer I will try to distinguish several things by taking up the definitions given (generalizing a few points slightly):

Framework and definitions

Let $W$ be a set of possible worlds, equipped with a preference relation $\succ$ ("is better than", or "is less bad than" in the case of suffering). Each world $X\in W$ is characterized by a number of people $n\left(X\right)\in N^{\ast }$ and a welfare level $l\left(X\right)\in R_{>0}$ (or a suffering level $s\left(X\right)\in R_{>0}$, where a larger value means more intense suffering).

Trade-off principles (inspired by what Dagher says)

Definition. For $d\in (0,1)$:

Trade Small-Quality for Big-Quantity (welfare) for d ($\text{TradeWelfare}\left(d\right)$): For any population $X$, and any natural $n$, real $l$, such that $X$ has $n$ people each enjoying welfare level $l$, there is an all-things-considered better population $X^{\ast }$ of $m>n$ people each enjoying welfare level $(1-d)l$.

Formally: $\text{TradeWelfare}\left(d\right):\forall X\in W,\exists X^{\ast }\in W\text{such that}n\left(X^{\ast }\right)>n\left(X\right),l\left(X^{\ast }\right)\le (1-d)\cdot l\left(X\right),\text{and}{X}^{\ast }\succ X$.

Definition. For $d\in (0,1)$:

Trade Big-Quantity for Big-Intensity (suffering) for d ($\text{TradeSuffering}\left(d\right)$): For any population $X$, and any natural $n$, real $s$, such that $X$ has $n$ people each suffering at level $s$, there is an all-things-considered better population $X^{\ast }$ of $m<n$ people each suffering at level $(1+d)s$.

Formally:$\text{TradeSuffering}\left(d\right):\forall X\in W,\exists X^{\ast }\in W\text{such that}n\left(X^{\ast }\right)<n\left(X\right),s\left(X^{\ast }\right)\ge (1+d)\cdot s\left(X\right),\text{and}{X}^{\ast }\succ X$

.

Blocking sets

Definition. For $d\in (0,1)$, one defines the sets of worlds that block the trade-offs of $d$: 

$S_d^{\text{wel}}=\{X\in W:\forall X^{\ast }\text{with}n(X^{\ast })>n(X\left)\text{and}l\right(X^{\ast })\le (1-d)\cdot l(X),X^{\ast }\nsucc X\}$

$S_d^{\text{suf}}=\{X\in W:\forall X^{\ast }\text{with}n(X^{\ast })<n(X\left)\text{and}s\right(X^{\ast })\ge (1+d)\cdot s(X),X^{\ast }\nsucc X\}$

$S_d^{\text{wel}}$ is the set of worlds such that, for the reduction factor d, no increase in population produces a better world. $S_d^{\text{suf}}$ is the analogue for suffering: the worlds for which concentrating the suffering on fewer people (with an increase in intensity of $d$) does not produce a less bad world.

One thus has : 

$\neg \text{TradeWelfare}\left(d\right)⟺S_d^{\text{wel}}\ne \varnothing$

 $\neg \text{TradeSuffering}\left(d\right)⟺S_d^{\text{suf}}\ne \varnothing$

Repugnant Conclusions

Definition. For the classical Repugnant Conclusion (${\text{RC}}_{\text{wel}}$), I propose the following definition :
For any world $X_0$ of $n_0$ people at welfare level $l_0$, there exists a world $X^{\ast }$ with an arbitrarily large number of people at welfare level arbitrarily close to zero, such that $X^{\ast }\succ X_0$.

Definition. For the dual Repugnant Conclusion (${\text{RC}}_{\text{suf}}$), I propose the following definition :
For any world $X_0$ of $n_0$ people suffering at intensity $s_0$, there exists a world $X^{\ast }$ with a very small number of people suffering at arbitrarily large intensity, such that $X^{\ast }\succ X_0$ (i.e., $X^{\ast }$ is "less bad" than $X_0$). In other words, a small number of people in extreme torture is preferable to an astronomical number of people undergoing slight suffering.

${\text{RC}}_{\text{wel}}$ as a consequence of $\text{TradeWelfare}\left(d\right)$

Proposition. If $\text{TradeWelfare}\left(d\right)$ is true for a certain $d\in (0,1)$ (that is, if $S_d^{\text{wel}}=\varnothing$) and if $\succ$ is transitive, then ${\text{RC}}_{\text{wel}}$ follows.

Proof. Let $X_0$ be a world of $n_0$ people at welfare level $l_0$. By iterated application of $\text{TradeWelfare}\left(d\right)$, one constructs a sequence $(X_k{)}_{k\ge 0}$ such that:

• $n\left(X_k\right)$ is strictly increasing in $N^{\ast }$, hence $n\left(X_k\right)\to +\infty$;
• $l\left(X_k\right)=(1-d{)}^k\cdot l_0\to 0$ as $k\to +\infty$;
• $X_0\prec X_1\prec X_2\prec \cdots$ by transitivity.

One thus obtains a world $X_N$ (for $N$ sufficiently large) with an arbitrarily large number of people at welfare level arbitrarily close to zero, and $X_N\succ X_0$. This is ${\text{RC}}_{\text{wel}}$. $\square$

The factor $4$ and the value ${10}^{-6}$ used by Dagher are illustrative choices.
The argument works for any $d\in (0,1)$ and any population growth factor, since a strictly increasing sequence in $N$ necessarily diverges.

${\text{RC}}_{\text{suf}}$ as a consequence of $\text{TradeSuffering}\left(d\right)$

Proposition. If $\text{TradeSuffering}\left(d\right)$is true for a certain $d\in (0,1)$ and if $\succ$ is transitive, then ${\text{RC}}_{\text{suf}}$ follows.

Proof. Let $X_0$ be a world of $n_0$ people suffering at intensity $s_0$. 
By iterated application of $\text{TradeSuffering}\left(d\right)$, one constructs a sequence $(X_k{)}_{0\le k\le n_0-1}$such that:

• $n\left(X_k\right)$ is strictly decreasing in $N^{\ast }$, until reaching 1;
• $s\left(X_k\right)=(1+d{)}^k{s}_0$, which grows exponentially;
• $X_0\prec X_1\prec X_2\prec \cdots$ by transitivity.

Contrary to the welfare case, the sequence is necessarily finite (a strictly decreasing sequence in $N^{\ast }$ reaches 1 in at most $n_0-1$ steps).
But this suffices: if $n_0$ is very large and $s_0$ very small (an astronomical number of people undergoing pinpricks), the final world has a single person suffering at intensity $(1+d{)}^{n_0-1}{s}_0$, which can be arbitrarily large.
${\text{RC}}_{\text{suf}}$ asserts that this world of a single person (or very few people) in extreme torture is "less bad" than the initial world of billions of "pinpricks" (very little pain). $\square$

Blocking ${\text{RC}}_{\text{wel}}$ and ${\text{RC}}_{\text{suf}}$: ($\star$) vs ($\star \star$)

To avoid ${\text{RC}}_{\text{wel}}$ and ${\text{RC}}_{\text{suf}}$, it is necessary that $S_d^{\text{wel}}\ne \varnothing$ and $S_d^{\text{suf}}\ne \varnothing$ for all $d$.
These are the necessary conditions: 

$\forall d\in (0,1),S_d^{\text{wel}}\ne \varnothing \left({\star }_{\text{wel}}\right)$

 $\forall d\in (0,1),S_d^{\text{suf}}\ne \varnothing \left({\star }_{\text{suf}}\right)$

It seems to me that Dagher asserts that denying ${\text{RC}}_{\text{wel}}$ and ${\text{RC}}_{\text{suf}}$ necessarily leads to stronger conditions : the existence of worlds that belong to all the $S_d^{\text{wel}}$ (resp. $S_d^{\text{suf}}$) simultaneously:

${\bigcap }_{d\in (0,1)}{S}_d^{\text{wel}}\ne \varnothing (\star {\star }_{\text{wel}})$

 ${\bigcap }_{d\in (0,1)}{S}_d^{\text{suf}}\ne \varnothing (\star {\star }_{\text{suf}})$

A world in ${\bigcap }_d{S}_d^{\text{wel}}$ is a world whose welfare level cannot be reduced by any amount, however small, without the result being recoverable by adding people. 
A world in ${\bigcap }_d{S}_d^{\text{suf}}$ is a world such that, however small the increase in intensity, no reduction in the number of people produces a less bad world. 
These are the worlds that Dagher calls "K", it seems to me (whether it concerns wel or suf).

$(\star )$ does not imply $(\star \star )$

I would be inclined to say this: $(\star \star )\Rightarrow (\star )$, but $(\star )\nRightarrow (\star \star )$.

For the implication $(\star \star )\Rightarrow (\star )$ :
If $K\in {\bigcap }_d{S}_d^{\text{wel}}$, then in particular $K\in S_d^{\text{wel}}$ for each $d$, hence $S_d^{\text{wel}}\ne \varnothing$.

For the non-implication $(\star )\nRightarrow (\star \star )$:
A decreasing intersection of non-empty sets can be empty (a priori).
Perhaps an additional argument makes the implication true, but I have not found one for the moment.

Consequence: the results about an absolute K do not seem necessary

The consequences that Dagher presents as inevitable, the existence of a world $K\in {\bigcap }_d{S}_d^{\text{wel}}$ such that a reduction in welfare of 0.0000000000000000000000001% (implicitly, arbitrarily small) can never be compensated by any number of people, rest on $(\star \star )$, not on $(\star )$. (From my current perspective)

Note: maybe from the start Dagher meant a $K_d$ in $S_d$ (for a specific $d$)? But the wording really seems to set an arbitrary threshold after defining K, so it gives the impression that it’s absolute?

Now, to block ${\text{RC}}_{\text{wel}}$, only $\left({\star }_{\text{wel}}\right)$ is necessary.

 $\left({\star }_{\text{wel}}\right)$ admits models in which:

• For each fixed $d$, $S_d^{\text{wel}}\ne \varnothing$: there exist worlds that block the transitive chain for that d;
• But $S_d^{\text{wel}}$ "shrinks" as $d\to 0$, and ${\bigcap }_{d>0}{S}_d^{\text{wel}}=\varnothing$: no world plays the role of an absolute threshold.

In other words, the "absolute K" result is the price to pay for $(\star {\star }_{\text{wel}})$, not for the rejection of ${\text{RC}}_{\text{wel}}$ itself.
One can reject ${\text{RC}}_{\text{wel}}$ by accepting only $\left({\star }_{\text{wel}}\right)$, which is logically weaker and does not seem to lead to the same counterintuitive consequences. (I might be wrong)
The same reasoning applies to ${\text{RC}}_{\text{suf}}$ with $\left({\star }_{\text{suf}}\right)$ and $(\star {\star }_{\text{suf}})$.

Remark on $\text{TradeSuffering}\left(d\right)$ and $\text{TradeWelfare}\left(d\right)$:

It seems to me that what leads to the different $RC$ outcomes is that $d$ is ‘fixed’, chosen in advance for the $X$s in question.
However, varying $d$ at each stage, and thus potentially accepting the proposal in each world but for a $d$ specific to that world, might not result in the $RC$ outcomes. 
A position where the trade-off is accepted at each step, but with a non-uniform $d_k$ that depends on the step $k$ (or on the world $X_k$). 
In that case, the total distance traversed in the intensity spectrum after $N$ steps is governed by the product ${\prod }_{k=1}^N(1+d_k)$. This product converges to a finite strictly positive limit if and only if $\sum \ln (1+d_k)$ converges, which (for small $d_k$) is equivalent to $\sum d_k<\infty$. 
If this condition is satisfied, the sequence of worlds never traverses the entire spectrum, and neither ${\text{RC}}_{\text{wel}}$ nor ${\text{RC}}_{\text{suf}}$ follows, even though each individual step is accepted.

(I haven't really explored this idea much, but that's what I'm thinking at the moment)

Just a quick note: 

It seems to me that, broadly speaking, for every element $K_d^{suf}$ in $S_d^{suf}$, we could (not necessarily, but it's still possible) choose a $d-\epsilon$ with $\epsilon$ arbitrary small and accept the trade (because the suffering is lower), which means there is a sort of “threshold” imposed by $S_d^{suf}$.

This “threshold” seems reasonable in itself ; I had a hunch about this threshold effect even before thinking it through (for suffering).

Questions

The fact that the threshold can be arbitrarily small is indeed surprising, but it doesn’t seem fundamental to me ; but actually, even if it doesn't seem "fundamental", I get the impression that's wrong, wouldn't we accept just any percentage threshold? 
Then again, I don't necessarily have a clear idea of what context we're talking about, so I might change my mind once I have a better understanding of the examples.

Executive summary: The author proposes that the Repugnant Conclusion can be avoided by rejecting the principle that small quality losses can always be compensated by large quantity gains, arguing instead that populations with sufficiently low welfare levels have a hard upper limit on how much value they can contribute.

Key points:

1. The Repugnant Conclusion follows from two seemingly plausible principles: that small welfare losses can always be offset by sufficiently large population increases, and that the "better than" relation is transitive.
2. The author's solution rejects the first principle, holding that there is an upper limit to how good a population of lives barely worth living can be — a limit the author argues is less than the goodness in a high-welfare population like A.
3. The author illustrates this with a "pinprick" case: no number of pinpricks, however large, can aggregate to a level of disvalue exceeding that of horrific torture, suggesting that low-intensity harms have a hard ceiling on total disvalue.
4. The entailed consequence is that, at some point in the sequence, even a 0.0000000000000000000000001% reduction in welfare level means that no increase in population size — including 50 trillion times as many people — could make the resulting world better.
5. The author argues this is less strange than it appears because quantity has a decreasing marginal ability to compensate for losses in pain intensity as intensity approaches the "pinprick" range.
6. The author acknowledges the solution remains "quite weird" but notes this is true of every proposed solution to the puzzle, and considers accepting the Repugnant Conclusion only the second most plausible alternative.

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I may be misunderstanding something in the argument, but this potential solution is I think already discussed in detail in Reasons+Persons, where Parfit introduces the repugnant conclusion? He calls it the 'lexical value' solution, I think? And I don't think this write up addresses any of the strong arguments that Parfit makes against it there?

In Parfit's original argument, he doesn't just rely on an appeal to intuition that B is better than A. Instead we move to B from A in two steps: first we go to A+ (more people having lives just worth living, but original population unaffected). Parfit claims that A+ is not worse than A. And then we move to world B, where we are now significantly increasing the welfare of already existing people, with only a small drop in the welfare of the original population. Parfit claims that B is better than A+.

If we reject that A+ is not worse than A, then we have to say it is sometimes wrong to create new people, even if their lives are worth living. This seems very strange. If we reject that B is better than A, then we have to believe that a small drop in welfare for people whose welfare is already very high can outweigh a large increase in welfare for people whose welfare is very low. This is the bullet that I think the lexical value solution bites?

But this is a tough bullet to bite! It is the opposite of egalitarian. It says we should prioritise changes in the welfare of already existing high welfare people (whose welfare is above the lexical level) over changes in the welfare of also already existing low welfare people (welfare below the lexical level).