If you have one less child, the next generation would use one person's less worth of resources, and would therefore be able to support its population in marginally more comfort. Everyone else will have marginally more children because their children can be supported in marginally more comfort, which will wipe out most of the change from having one less child yourself.

The example that comes to mind to show the how the sex thing isn't a problem is that of a robot car with a goal to drive as many miles as possible. Every day it will burn through all its fuel and fuel up. Right after it fuels up, it will have no desire for further fuel - more fuel simply does not help it go further at this point, and forcing it can be detrimental. Clearly not contradictory

You could have a similar situation with a couple wanting sex iff they haven't had sex in a day, or wanting an orange if you've just eaten an apple but wanting an apple if you've just eaten an orange.

To strictly show that something violates vNM axioms, you'd have to show that this behavior (in context) can't be fulfilling any preferences better than other options that the agent is aware of - or at least be able to argue that the revealed utility function is contrived and unlikely to hold up in other situations (not what the agent "really wants").

Constantly wanting what one doesn't have can have this defect. If I keep paying you to switch my apple for your orange and back (without actually eating either), then you have a decent case, if you're pretty confident I'm not actually fulfilling my desire to troll you ;)

The "want's a relationship when single" and "wants to be single when not" thing does look like such a violation to me. If you let him flip flop as often as he desires, he's not going to end up happily endorsing his past actions. If you offered him a pill that would prevent him from flip flopping, he very well may take it. So there's a contradiction there.

To bring human-specific psychology into it, its not that his inherent desires are contradictory, but that he wants something like "freedom", which he doesn't know how to get in a relationship and something like "intimacy", which he doesn't know how to get while single. It's not that he want's intimacy when single and freedom when not, it's that he wants both always, but the unfulfilled need is the salient one.

Picture me standing on your left foot. "Oww! Get off my left foot!". Then I switch to the right "Ahh! Get off my right foot!". If you're not very quick and/or the pain is overwhelming, it might take you a few iterations to realize the situation you're in and to put the pain aside while you think of a way to get me off both feet (intimacy when single/freedom in a relationship). Or if you can't have that, it's another challenge to figure out what you want to do about it.

I wouldn't model you as "just VNM-irrational", even if your external behaviors are ineffective for everything you might want. I'd model you as "not knowing how to be VNM-rational in presence of strong pain(s)", and would expect you to start behaving more effectively when shown how.

(and that is what I find, although showing someone how to be more rational is not trivial and "here's a proof of the inconsistency of your actions now pick a side and stop feeling the desire for the other side" is almost never sufficient. You have to be able to model the specific way that they're stuck and meet them there)

tl;dr: We're not VNM-rational because we don't know how to be, not because it's not something we're trying to do.

Can you give me an example of this in reality? The math works, but I notice I am still confused, in that values should not just be a variable in utility function... they should in fact change the utility function itself.

If they're relegated to a variable, that seems to go against the original stated goal of wanting moral progress. in which case the utility function originally was constructed wrong.

How are you changing the values you optimize for without changing your utility function? This now seems even more handwavey to me.

There's a new subreddit for GIFs of cute baby elephants: http://www.reddit.com/r/babyelephantgifs

1. Oh, I see. You're taking wage to be determined by production, which in turn is determined by population according to the Cobb-Douglas formula, and then asking "what's the optimal population?". Got it.

2. Yup, better now.

So, anyway, now that I understand your argument better, there's something that looks both important and wrong, but maybe I'm misunderstanding. You're assuming that A -- the constant factor in the Cobb-Douglas formula -- is the same for all countries. But surely it isn't, and surely this accounts for a large amount of the variation in productivity and wealth between countries. It seems like this would lead to big differences in w between countries even if they're all close to optimal population.

There are several things here I fail to understand.

1. Why d/dN? If you're looking for optimal income per capita, you need d/dw=0 not d/dN=0.

2. The result you've allegedly reached is that w = w0 exp(alpha-1) where alpha<1, which means w<w0, which means you're not actually in the regime where net utility equals N[U(w)-U(w0)], so you've been doing calculus on the wrong formulae.

3. Clearly utility is not only a function of income. (Even considering only money, you need to consider assets as well as income.) Of course considering only income is a handy simplification that may turn something impossibly complicated into something susceptible to analysis, but I think you should be explicit about making that simplification because the importance of things other than money is actually a pretty big deal.

4. This all seems like a more complicated but still minor variation on simple and familiar observations like these: (a) simple versions of utilitarianism say well-off people should give almost all they have to poorer people; (b) simple versions of average utilitarianism say we should kill all the least happy people; (c) simple versions of total utilitarianism say we should prefer an enormous population of people with just-better-than-nothing lives to a normal-sized population of very happy people. I would expect solutions to (or bullet-biting on) these problems to deal with the more complicated but similarly counterintuitive conclusions presented here (assuming for the sake of argument that either my objections above are wrong or else the conclusions remain when the errors are repaired).

The two links you give to discussions of "the statistical value of a life" are discussing very different things. Thing One: An extrapolation (from infinitesimal changes to the change from p=0 to p=1) of the dollar-value to a given individual of their survival. Thing Two: An estimate of the dollar-value placed by society on a person's survival.

Thing One (which is what your VL is measuring) is inevitably going to be very sensitive to the person's wealth. Thing Two needn't be, and in fact isn't (most modern societies are willing to go to about as much trouble to save a poor person's life as a rich person's). I think the $6M figure you cite is more a Thing Two than a Thing One.

If we take your calculations at face value, here is what they tell us. We start with a broadly-plausible estimate that in some sense a life is worth about $6M. We suppose that a "typical" life corresponds to an income of about $50k/year. We do some calculations. And we arrive at the conclusion that the life of a very poor person -- someone whose income is your y0 of $300/year -- is worth something on the order of $250. (!!!)

First reaction: This is a reductio ad absurdum: something must be desperately wrong here. Second reaction: Well, maybe not so much; this is not really about assigning different values to rich and poor people's lives, but about how they, in their very different financial situations, convert between utility and money. Third reaction: No, wait, this really is about assigning different values to these people's lives; in particular there is an income level (not very far from y0, in this particular model) at which the utility reaches zero, and no talk of conversion factors will change that.

So I think you either need to bite the bullet and say that very poor people's lives aren't worth saving, or reconsider some assumptions. (Fiddling with the details of the utility function, etc., as in your closing comments, might move the value assigned to a life-at-income-y0 from, say, $250 to, say, $5k, which -- taken as an indication of how desperately important money is to someone so poor, rather than of the absolute value of their life -- is at least semi-reasonable. But it won't do anything to change the fact that someone sufficiently poor will get zero or negative utility.)

The assumption I would suggest revisiting is the one that says, roughly, that death is like merely not-having-lived in terms of utility.

It seems to me entirely possible, and in fact probably right, that (1) quite a lot of people's lives are bad enough that if we were choosing, godlike, between two possible worlds that differ simply in the addition or subtraction of some of those lives, we could reasonably prefer there to be fewer rather than more of them, but also that (2) once one of those lives is there, ending it is a very bad thing. A life just barely bad enough that the person living it considers death an improvement is probably quite a lot worse than a life just barely bad enough that adding another to the world is neutral.

(Of course quality of life isn't the same thing as income, but that's just a matter of the toy model being used here.)

So this would leave us with the following state of affairs: The life of a rather miserably-off person (for which very low income is a kinda-passable proxy) is bad enough that having more such lives in the world doesn't, as such, improve the world. (So they would have U=0 or even U<0.) But, once that life is there, taking it away or failing to save it is still a very bad thing (because of that person's preferences, and the impact on other people). That seems fair enough. But at this point it's worth noting that those value-of-life estimates are all concerned with the value of saving the life, rather than that of having it exist in the first place. Which probably means that there's still something wrong with the calculation.

It's nearly 2am local time so I'll leave my thoughts in that rather half-baked form.

At least one of us is very confused about pretty much everything here.

Since von Neumann–Morgenstern utility functions are invariant under affine transformations, we can [...]

Not if you're serious about total utilitarianism, which needs to be able to add up utilities and therefore looks quite different (at least when the number of lives can vary) as the constant term varies.

[EDITED to add: The issues below are because the mathematical typesetting got messed up in a way that made + signs disappear; they are not real mistakes and the error has now been fixed in the original post.]

This condition ensures that s is the utility at the minimum income.

I must be misunderstanding. You've written U = psf(y) where the condition in question is f(y0)=0. But this implies U=p.s.0=0 when y=y0. So s is not the utility at minimum income. In fact it looks to me as if s is simply a scaling factor applied to utilities, and as such is perfectly arbitrary.

Then, a little later, you go from VL = s f(y)/f'(y) to s = VL.f'(y) - f(y) but that's completely wrong; it should be s = VL.f'(y)/f(y), which in the log case says s = VL / y log(y/y0); in the case y=y0 this just says that VL=0 whatever s may be, which is not surprising since you deliberately chose to rescale your utilities to make it so ("we can define the utility of being dead as 0").

I'm not entirely sure whether you

• propose QALYs as a means to optimize a total utilitarians goals or
• discuss what you as a total utilitarian could do best to optimize this goal or
• something else related to both

I assume the former because you wrote

I chose them because I wanted the social welfare function to be relatively easy to calculate.

In general if you choose any oversimplified scheme to optimize for you will not earn what you want. What gets measured gets optimized.

The following quotes are from Wikipedia: http://en.wikipedia.org/wiki/Quality-adjusted_life_year

QALYs are empirically known to be oversimplified and more a theoretical economists tool to derive general optimization potential that a precise tool.

The four theoretical assumptions underlying QALYs are invalid (quality of life should be measured in consistent intervals; life years and QOL should be independent; people should be neutral about risk; and willingness to sacrifice life years should be constant over time).

They are neither recommended for individual health care decisions where they

place[s] disproportionate importance on physical pain or disability over mental health. The effects of a patient's health on the quality of life of others [..] do not figure into these calculations.

nor on the population as an aggregate where

the weight assigned to a particular condition can vary greatly, depending on the population being surveyed.

Also if you want to use it as a tool to personally rate some means you should consider that

those who do not suffer from the affliction in question will, on average, overestimate the detrimental effect on quality of life, compared to those who are afflicted.

So I propose that you choose a more elaborate tool set if you want to optimize a complex goal. Otherwise you fall into the same trap as you wat to avoid from UFAI: Overoptimizing oversimple goals.