I guess your argument fails because it still contains too much rigidity. For example: the choice of critical level can depend on the choice set: the set of all situations that we can choose. I have added a section in my original blog post, which I copy here.
<0. However, suppose another situation S2 is available for us (i.e. we can choose situation S2), in which that person i will not exist, but everyone else is maximally happy, with maximum positive utilities. Although person i in situation S1 will have a positive utility, that person can still prefer the situation where he or she does not exist and everyone else is maximally happy. It is as if that person is a bit altruistic and prefers his or her non-existence in order to improve the well-being of others. That means his or her critical level C(i,S1) can be higher than the utility U(i,S1), such that his or her relative utility becomes negative in situation S1. In that case, it is better to choose situation S2 and not let the extra person be born.
If instead of situation S2, another situation S2’ becomes available, where the extra person does not exist and everyone else has the same utility levels as in situation S1, then the extra person in situation S1 could prefer situation S1 above S2’, which means that his or her new critical level C(i,S1)’ remains lower than the utility U(i,S1).
In other words: the choice of the critical level can depend on the possible situations that are eligible or available to the people who must make the choice about who will exist. If situations S1 and S2 are available, the chosen critical level will be C(i,S1), but if situations S1 and S2’ are available, the critical level can change into another value C(i,S1)’. Each person is free to decide whether or not his or her own critical level depends on the choice set.>>
So suppose we can choose between two situations. In situation A, one person has utility 0 and another person has utility 30. In situation Bq, the first person has utility −10 and instead of a second person there are now a huge number of q persons, with very low but still positive utilities (i.e. low levels of k). If the extra people think that preferring Bq is sadistic/repugnant, they can choose higher critical levels such that in this choice set between A and Bq, situation A should be chosen. If instead of situation A we can choose situations B or C, the critical levels may change again. In the end, what this means is something like: let’s present to all (potential) people the choice set of all possible (electable) situations that we can choose. Now we let them choose their preferred situation, and let them then determine their own critical levels to obtain that preferred situation given that choice set.
I’m not entirely sure what you mean by ‘rigidity’, but if it’s something like ‘having strong requirements on critical levels’, then I don’t think my argument is very rigid at all. I’m allowing for agents to choose a wide range of critical levels. The point is though, that given the well-being of all agents and critical levels of all agents except one, there is a unique critical level that the last agent has to choose, if they want to avoid the sadistic repugnant conclusion (or something very similar). At any point in my argument, feel free to let agents choose a different critical level to the one I have suggested, but note that doing so leaves you open to the sadistic repugnant conclusion. That is, I have suggested the critical levels that agents would choose, given the same choice set and given that they have preferences to avoid the sadistic repugnant conclusion.
Sure, if k is very low, you can claim that A is better than Bq, even if q is really really big. But, keeping q fixed, there’s a k (e.g. 10^10^10) such that Bq is better than A (feel free to deny this, but then your theory is lexical). Then at some point (assuming something like the continuity), there’s a k such that A and Bq are equally good. Call this k’. If k’ is very low, then you get the sadistic repugnant conclusion. If k’ is very high, you face the same problems as lexical theories. If k’ not too high or low, you strike a compromise that makes the conclusions of each less bad, but you face both of them, so it’s not clear this is preferable. I should note that I thought of and wrote up my argument fairly quickly and quite late last night, so it could be wrong and is worth checking carefully, but I don’t see how what you’ve said so far refutes it.
My earlier points relate to the strangeness of the choice set dependence of relative utility. We agree that well-being should be choice set independent. But by letting the critical level be choice set dependent, you make relative utility choice set dependent. I guess you’re OK with that, but I find that undesirable.
I honestly don’t see yet how setting a high critical level to avoid the repugnant sadistic conclusion would automatically result in counter-intuitive problems with lexicality of a quasi-negative utilitarianism. Why would striking a compromise be less preferable than going all the way to a sadistic conclusion? (for me your example and calculations are still unclear: what is the choice set? What is the distribution of utilities in each possible situation?)
With rigidity I indeed mean having strong requirements on critical levels. Allowing to choose critical levels dependent on the choice set is an example that introduces much more flexibility. But again, I’ll leave it up to everyone to decide for themselves how rigidly they prefer to choose their own critical levels. If you find the choice set dependence of critical levels and relative utilities undesirable, you are allowed to pick your critical level independently from the choice set. That’s fine, but we should accept the freedom of others not to do so.