Logarithms and Total Utilitarianism

Looking at the math of dividing a fixed pool of resources among a non-fixed number of people, a feature of log(r) that matters a lot is that log(0)<0. The first unit of resources that you give to a person is essentially wasted, because it just gets them up to 0 utility (which is no better than just having 1 fewer person around).

That favors having fewer people, so that you don't have to keep wasting that first unit of resource on each person. If the utility function for a person in terms of their resources was f(r)=r-1 you would similarly find that it is best not to have too many people (in that case having exactly 1 person would work best).

Whereas if it was f(r)=sqrt(r) then it would be best to have as many people as possible, because you're starting from 0 utility at 0 resources and sqrt is steepest right near 0. Doing the calculation... if you have R units of resources divided equally among N people, the total utility is sqrt(RN). log(1+r) is similar to sqrt - it increases as N increases - but it is bounded if R is fixed and just approaches that bound (if we use natural log, that bound is just R).

To sum up: diminishing marginal utility favors having more people each with fewer resources (in addition to favoring equal distribution of resources), f(0)<0 favors having fewer people each with more resources (to avoid "wasting" the bit of resources that get a person up to 0 utility), and functions with both features like log(r) favor some intermediate solution with a moderate population size.

Note, that the key feature of log function used here is not its slow growth, but the fact that it takes negative values on small inputs. For example, if we take the function u(r)=log (r+1), so that u(0)=0, then RC holds.

Although there are also solutions that prevent RC without taking negative values, e.g u(r) = exp{-1/r}.

In your linear model, the hypotheses "the utility of a person living on 1 resource is 0" and "doubling resources doubles utility" imply that utility is always 0. Maybe you meant the second hypothesis to be "doubling the number of resources beyond the first doubles utility", so that a person's utility is 0.1 times the number of resources beyond the first. In this version of the linear model, the total utility of population A is 9 (0.9 utility per person), and the total utility of population Z is 0.1 (approximately 0.001 utility per person).

Nice example! I still think the most reasonable objection to something like total utilitarianism is that population ethics is a matter of preferences, and my preferences are complicated. If humans prefer to set aside part of the universe as a nature preserve rather than devoting those resources to more humans, then so be it - human preferences are the only preferences we've got.

This has shifted my views very positively in favor of total log utilitarianism, as it dissolves quite cleanly the Repugnant Conclussion. Great post!

The repugnant conclusion just says "a sufficiently large number of lives barely worth living is preferable to a smaller number of good lives". It says nothing about resources; e.g., it doesn't say that the sufficiently large number can be attained by redistributing a fixed supply.

Firstly, excellent post! A cool idea, well-written, and very thought-provoking. Some thoughts on the robustness of the result:

Suppose that every individual were able to produce at least 1 unit of resources throughout their life. Then total utility is monotonically increasing in the number of people, and you have the repugnant conclusion again. How likely is this supposition? Assuming we have arbitrarily advanced technology, including AI, humans will be pretty irrelevant to the production of resources like food or compute (if we're in simulation). But plausibly humans could still produce "goods" which are valuable to other humans, like friendship. Let's plug this into your model above and see what happens. I'll assume that humans need at least 1/K physical resources to survive, but otherwise their utility is logarithmic in the amount of physical resources + friendship that they get. Also, assume that every person receives as much friendship as they produce. So

, with an upper bound of N = KR. When F >= 1, then the optimal value of N is in fact KR, and so utility per person is

, which can be arbitrarily close to 0 (depending on K). When 0 <= F < 1, then I think you get something like your original result again, but I'm not sure. Empirically, I expect that the best friends can produce F >> 1, i.e. if you had nothing except just enough food/water to keep yourself alive, but also you were the sole focus of their friendship, then you'd consider your life well worth living. Idk about average production, but hopefully that'll improve in the future too. In summary, friendship may make things repugnant :P

Here's another version of the repugnant conclusion and your argument. Suppose that the amount of resources used by each person is roughly fixed per unit time (because, say, we're living in simulation), but that there's a period of infancy and early childhood which uses up resources and isn't morally valuable. Then the resources used up by one person are I + L, where I is the length of their infancy and L is the length of the rest of their life, but the utility gained from their life is a function of L alone. What function of L? Perhaps you think that it's linear in L - for example, If you're a hedonic utilitarian, it's plausible that people will be just as happy later in their life as earlier. (In fact, right now, old people tend to be happiest). If so, you must endorse the anti-repugnant conclusion, where you'd prefer a population with very few very long-lived people, to minimise the fixed cost of infancy. If you're a preference utilitarian, maybe you think that there's diminishing marginal utility to having your preferences satisfied. It then follows that there's an optimal point at which to kill people, which isn't too soon (otherwise you're incurring high fixed costs) and isn't too late (otherwise people's marginal utility diminishes too much) - a conclusion which is analogous to your result.

Check my post on Nonlinear perception of happiness [LW(p) · GW(p)] - the logarithm is assumed to be in a different place, but the part about implication to ethics contains a version of this argument.

I don't know to what extent have others explored the connection between total utilitarianism and equality

Diminishing marginal utility is one of the standard arguments for redistribution.

This feels like one of those cases where there ought to be a mistake somewhere, given how many eyes have been on the problem, and how simple this example is. Yet I cannot find any errors. All it takes for the repugnant conclusion to be avoided is the logarithmic (or slower, like log(log(R/N))) dependence of utility on the available resources. I'm impressed. Maybe someone can update the relevant wiki entry.

One line of attack against the idea that we should reject the repugnant conclusion is to ask why the lives are barely worth living. If it's because the many people have the same good lives but they're p-zombies 99.9999% of the time, I can easily believe that increasing the population until there's more total conscious experiences makes the tradeoff worthwhile.

I'm pretty sure that the result N = R/e is not in fact arbitrary due to the choice of logarithmic model, but that it would hold true for any logarithmic relationship between per-person resources and utility. This has the really interesting implication that regardless of your exact utility function, or total available resources, the optimal world always gives each person about 2.7x the amount of resources they would need to lead a utility-neutral life.