Suppose that we are in a speculative very far future setting and we somehow have managed to abolish suffering everywhere. And suppose that some utilitarian scientists have discovered a system of maximum spaciotemporal utility density - imagine some kind of super quantum computer simulating the best MMORPG (the O may also stand for orgy here ) that can possibly exist, and also simulating the players who play that game (and those players can't escape the simulation, but they think the game is all there is in the universe, so they don't want to (and probably wouldn't want to even if they knew the truth, because everything is less interesting than their game)). That system, call it U0 for convenience, also has a finite spacial and temporal volume. The game is over after a maximum time is reached. Then everything is reverted to the initial state. If that's not cool enough, imagine that the game is cyclical and you can't say when it starts and when it ends, but there's some ingenious mechanism that enforces periodicity, so that there's a maximum time after some state of the game is exactly the same as a previous state.

Then, the inhabitants of our world happily and willingly decide to plaster the cosmos with those U0 systems (giving up their own lives in the process) and maintain only a minimal infrastructure to keep everything running perfectly.

Is there something wrong with that happening?

Problem 0: Intuitive discomfort.
I don't see objections like that as valid, except you can be more specific about what's wrong about that scenario.

Problem 1: It would seem that the redundancy of the U0 system looks kinda boring. The same system spread all over the cosmos isn't better subjectively than a single instance of the game (at least as seen from the inside of the system). You could argue that the system also involves quantum randomness, so that you can't predict the exact evolution of the system, but it still maintains its optimality when averaged over all possible evolutions (here it also needs to be assumed that any system that doesn't involve quantum randomness is not optimal). But if you really have a full quantum computer, it will simulate all possible evolutions of the system at once, so you still don't have real "added value" from other instances of the system. So, you would still end up with some kind of monotonity throughout space.

Problem 2: The scientists could have made an error and only found a possibly optimal system. In reality, they would need to run an infinite number of similar systems to find out which is the best one. So, we build systems U0, U1, U2,... and so on and successively close down those systems who turned out to be worse than others (but still no suffering occurs in any of those systems, so it's a "safe" evolutionary process). What's problematic in this case is that shutting down a whole simulated world might look kinda nasty, but I guess utilitarians can live with that if those worlds are replaced with absolutely better ones.

Problem 3: Whether the system is really optimal or not critically depends on the utility measure you use. Perhaps there's no clearly canonical one, but an infinite variety of plausible and equally valid measures. That's another case where we could see a need to build different systems U0, U1, U2,... and so on - one for each utility measure. Other than with problem 2, here there's no need to shut down any of those systems. The problem with this approach is that over a greater spaciotemporal volume the utility density for each measure is lower than it would be if you only used the system that's optimal for the specific measure. It's also far from clear which measures have priority. That would probably give rise to lots of political debates (or something worse).

Problem 4: Such a system might not be optimal, if utility density is a function of the size of the system and larger systems allow for higher utility density. That would imply that you can't compartmentalize the cosmos that way and still get maximum utility. Either you optimize utility holistically across the whole cosmos or not at all, if you want to do things right. And doing that looks very problematic if the cosmos turns out to be infinite (see Nick Bostrom's paper The Infinitarian Challenge To Aggregative Ethics).

Actually, problem 3 could be even worse, so that for different utility measures you run into different problems.

Do you see more problems? What do you think about this thought experiment? Personally, I think the idea is interesting, but the problems I mentioned are likely enough to make the whole approach really questionable. Are there any good alternative ideas to get to an ideal state of the cosmos?

For three, if you have a probability distribution for what utility is, it's equivalent to just having a more complex utility function. Just tile the universe with what has the highest expected value.

As a variation of three, it might turn out that you vastly overestimated the sentience of the system, so it isn't really generating much utility at all. The fact that we are not currently in the simulation can be taken as very strong evidence of this, as almost everyone will be if it is sentient.

Infinite utility would be a problem no matter what you do. There's no reason to mention it here specifically.

What if the highest net utility density includes suffering? I would find it hard to believe it doesn't.

Ah, great reply, DanielLC! It seems that problem 3 needs to be dissected further. We need to differentiate here between following situations:

I: There is a true utility measure (in the sense of moral realism).
II: There is no true utility measure and everything we have are just more or less convenient tools for guiding our behavior.

Situation I can be split up into different cases:

Case A: We know the true utility measure exactly.
Case B: We don't know it exactly, but can approximate it successively (that's the situation of problem 2).
Case C: We don't know the true utility measure, but have a probability distribution for what it really is.
Case D: We don't know neither true utility measure, nor the suitable probability distribution, but we can approximate a suitable probability distribution successively.
Case E: We really have no clue what the true utility measure could be, even approximately or in terms of probability.

Cases A,B,C and D are relatively benign, because you can approach the truth with some degree of confidence. Case E is what I thought about when I've written problem 3.

Is there a practical difference between the cases IE and II? What's the difference between a truth you can't know and a truth that doesn't exist?

As a variation of three, it might turn out that you vastly overestimated the sentience of the system, so it isn't really generating much utility at all. The fact that we are not currently in the simulation can be taken as very strong evidence of this, as almost everyone will be if it is sentient.

I interpret that as failure to determine the real utility of the system. That's a very real problem, if you start with a really abstract model of an ideal system. Even this problem can be split up into at least two elements:
1: Our ability to predict the utility of a given system might be really bad.
2: Our ability to measure the utility of a system might be really bad.

Regarding the argument that we don't seem to be in such a simulation: I think it's really important to state that observation. On the other hand, it may be wrong to argue from our anecdotal evidence of living in this world that doesn't seem to be a simulation. Even if you could argue that living in a simulation would be much more probable, it can still be argued whether such probability considerations are really applicable to this situation. After all, exceptions still exist (we might just accidentally live in a real world, although living in a simulation might be more likely), and it might not be possible to apply probability theory to the class of all worlds or something like that.
Apart from that, our world could be a simulation that is run for less benign reasons, or for reasons that are benign but extremely counter-intuitive - or for benign reasons, but extreme practical failure to accomplish those benign objectives.

Infinite utility would be a problem no matter what you do. There's no reason to mention it here specifically.

Really? What would be problematic about the situation that our cosmos was infinite but we succeeded in (starting to) tiling it with systems of really maximal utility density? Is there anything that can be better than that?

What if the highest net utility density includes suffering? I would find it hard to believe it doesn't.

Right. That it doesn't include suffering was just an assumption to make the scenario a bit less objectionable. But it's not a really essential assumption. I just didn't want to give negative utilitarians some obvious aspect to complain about (as I'm more interested in other aspects of this thought experiment). Unfortunately, I think you are right to think it's quite unlikely that the highest net utility density doesn't include any suffering.

We really have no clue what the true utility measure could be, even approximately or in terms of probability.

You can't have no probability distribution. Find the maximum entropy prior.

On the other hand, it may be wrong to argue from our anecdotal evidence of living in this world that doesn't seem to be a simulation.

If we are already in a utility-maximizing simulation, we can hardly building a utility-maximizing simulation within it.

After all, exceptions still exist (we might just accidentally live in a real world, although living in a simulation might be more likely),...

Yes. That's how probability works.

... and it might not be possible to apply probability theory to the class of all worlds or something like that.

There are four probability axioms:

• P(A) > 0
• P(True) = 1
• P(A) + P(B) + P(C) + ... = P(A or B or C or ...) if A, B, C,... are mutually exclusive
• P(A|B) = P(A and B)/P(B)

Which doesn't apply? It's also possible to create a Dutch book scenario for any violation of those axioms. Can you give me an example of a combination of bets you'd be willing to take, such that you will always lose more than you win?

Apart from that, our world could be a simulation that is run for less benign reasons

If we are in a different simulation, that's irrelevant. If we make a simulation within that simulation, we'd be more likely to be in the second level than the first.

Really? What would be problematic about the situation that our cosmos was infinite but we succeeded in (starting to) tiling it with systems of really maximal utility density? Is there anything that can be better than that?

If there is an infinite amount of positive utility, and an infinite amount of negative utility, the total utility depends on how you add it. If there's an infinite universe, there is almost definitely infinite of both of those to begin with, so all choices will result in the same net utility.

If it's even possible to have infinite positive and negative utility, or, for that matter, if it's impossible but the probability doesn't decrease with the amount fast enough, expected utility will not converge, and there's no clear way to make a decision.

These problems are highly non-trivial. It might be worth writing a doctoral thesis about them. Anyway, I guess the main point of that would be to demonstrate how difficult those problems really are.

DanielLC wrote:

We really have no clue what the true utility measure could be, even approximately or in terms of probability.

You can't have no probability distribution. Find the maximum entropy prior.

I am running into several problems here (the notion of a probability space is very relevant here):
1. What's the base space (=sample space) of the measure? In other words: What do you want to measure exactly? The utility of what? Sentient minds in physical reality? Sentient minds as abstractions in some kind of mind space? Actions (how do you define an action?)? World states? World states integrated over time? "Everything"?
2. Once there's a suitable base space, you need a suitable sigma algebra on it. For topological spaces the default candidate is the Borel sigma algebra. So, you either need a suitable topology on your base space or a good idea for defining a suitable sigma algebra directly. This step is important, because when taking the wrong sigma algebra you can run into paradoxa like the Banach-Tarski-Paradox.
3. Ok, let's assume that we solved the first two steps. Then we can consider the class of all (signed) measures on that measure space (base space + sigma algebra). That class would be the base space of candidate measures. It looks like it would be considerably more complex than the initial base space for a single candidate measure. Finding a suitable sigma algebra on that might be an even more complicated problem than 2.
4. Now we are looking for a probability distribution on that weird measure space. Is there a suitable class of candidate probability distributions? Let's say for simplicity we consider the class of all probability distributions. Then, the problem might be that a probability with maximum entropy might not exist, because the entropy is unbounded. The same problem could appear for every "sensible" choice of a class of candidate probability distributions.

Wikipedia wrote:Note that not all classes of distributions contain a maximum entropy distribution. It is possible that a class contain distributions of arbitrarily large entropy (e.g. the class of all continuous distributions on R with mean 0), or that the entropies are bounded above but there is no distribution which attains the maximal entropy (e.g. the class of all continuous distributions X on R with E(X) = 0 and E(X2) = E(X3) = 1[citation needed]).

Even worse: It might not be clear how to define entropy in such an obscure setting.

DanielLC wrote:

On the other hand, it may be wrong to argue from our anecdotal evidence of living in this world that doesn't seem to be a simulation.

If we are already in a utility-maximizing simulation, we can hardly building a utility-maximizing simulation within it.

It could also be argued that we need to build a utility-maximizing simulation within it. For example, that would be the case if the simulation was a game about building the best utility-maximizing simulation that's possible to build within the game simulation.

DanielLC wrote:

... and it might not be possible to apply probability theory to the class of all worlds or something like that.

There are four probability axioms:

• P(A) > 0
• P(True) = 1
• P(A) + P(B) + P(C) + ... = P(A or B or C or ...) if A, B, C,... are mutually exclusive
• P(A|B) = P(A and B)/P(B)

Which doesn't apply? It's also possible to create a Dutch book scenario for any violation of those axioms. Can you give me an example of a combination of bets you'd be willing to take, such that you will always lose more than you win?

The problem lies on a deeper level here: What is the class of all worlds? What qualifies as world? How many worlds are there (see Max Tegmarks Multiverses for example)? That class might be a proper class. It is often possible to work around that kind of problem, but it makes things more esoteric at the very least.
And even if you have nailed down that class, you still need to find a suitable sigma-algebra on it. Additionally, defining meaningful probability distributions on such a measure space might be rather messy.

DanielLC wrote:

Apart from that, our world could be a simulation that is run for less benign reasons

If we are in a different simulation, that's irrelevant. If we make a simulation within that simulation, we'd be more likely to be in the second level than the first.

I don't see why it should really matter on which level of simulation we are. Nested simulations are a complicated separate topic anyway.

DanielLC wrote:

Really? What would be problematic about the situation that our cosmos was infinite but we succeeded in (starting to) tiling it with systems of really maximal utility density? Is there anything that can be better than that?

If there is an infinite amount of positive utility, and an infinite amount of negative utility, the total utility depends on how you add it. If there's an infinite universe, there is almost definitely infinite of both of those to begin with, so all choices will result in the same net utility.

This is only a problem in the case that the systems contain any suffering. Having infinite positive utility and no negative utility doesn't look bad from my point of view.

Anyway, having infinite negative utility is a very valid assumption.

DanielLC wrote:If it's even possible to have infinite positive and negative utility, or, for that matter, if it's impossible but the probability doesn't decrease with the amount fast enough, expected utility will not converge, and there's no clear way to make a decision.

I guess I know what you mean (the Riemann Series Theorem is relevant in this context), but that's exactly the reason why I settled for a utility density approach. While the total utility of the whole universe can be computed in an arbitrary fashion and thus, strictly spoken, is not well-defined, the utility of any sufficiently large spherical region of the universe will be maximal. In the limit of infinite radius the utility would be infinite. I'd say that's pretty much the best common sense definition of the utility of such a universe. It doesn't need to concern us that there are less common sense approaches that result in arbitrary or not defined utilities. A utilitarian calculus needs to be made to work. You can always come up with something that breaks it down, but as utilitarians we need to calculate, so we use a calculus that works fine. It's not possible to use utilitarianism in a meaningful way if you don't make convenient assumptions, like the assumption that a construct like "utility" actually makes sense in some way. Making less assumptions might be preferable, but we need to make at least as many assumptions as are necessary to enable a meaningful way of calculation in a given situation. The less arbitrary and unnatural those assumptions are, the better.

In the limit of infinite radius the utility would be infinite.

Reality isn't a limit. I don't like the idea of treating it as such. In addition, this doesn't help with the expected utility problem. There's no clear order to add possibilities in. The only way I can make sense of it is to use a convergent prior, under which it is impossible to have infinite utility.

Ok, if you insist on the position that these considerations make no sense for you in a total utilitarian setting, then what about average utilitarianism?

Let's define the average utility of a region as the total utility of that region divided by the volume of that region. This is always a finite number (unless you find some miraculous way to generate infinite utility in a finite region - a Tiplerian Omega Point maybe?).

To keep it simple, let's take spheres with varying radius. If we take any sphere and make it very large, the average utility of that region will tend towards a constant finite quantity, if the universe is tiled sufficiently homogenously with maximum utility density systems. In this situation, we have a finite average utility in the limit of infinite radius. This tells us much more than whether the "total utility" I defined previously is infinity, a finite quantity, or minus infinity. Using systems with maximum utility density will maximize the global average utility.

If the universe isn't totally homogenously tiled with these systems, there might be non-vanishing variations in the average density as the radius of a sphere increases. In this case the utility density won't converge. Nevertheless, the average density might still have a lower bound. In this case, we could still speak about a lower total average utility density bound. Granted, this is a weird concept. It just tells you that once you consider a sufficiently large volume, the average density of that region should at least get arbitrarily close to that lower bound.

Still, the problem of the dependence of the order of summation remains. If you don't take spheres, but pick a large patch with positive utility and then add up so many patches with negative utility that the total sum or average will be negative, and you iterate this process so that you get the whole volume of the universe, you will get a negative total sum or average utility of the whole universe, or something that doesn't even converge in any sense.

The concept of an average utility of the whole universe depends critically on the "shapes" you use to take your average - it's just a conditional limit. But I'd say that taking everything other than spheres in the spacial case or future light cones (because they represent the part of the universe we can influence causally) in the spacio-temporal case is unnatural.

In any case, I think these considerations show that average utilitarianism makes slightly more sense than total utilitarianism in a setting in which the universe has infinite spacio-temporal volume. A word of caution is necessary in this place: This model of average utilitarianism is not the same as the mainstream version of it. Here, we are taking an average over utility as "commodity" in space-time, whereas the mainstream version would consider the average utility of all individuals.

That both versions come to different results can easily be seen when considering two situations which are absolutely identical, except for individuals living further apart in one version. The version with more "empty" space will have lower average spacio-temporal utility, but the same average utility when only considering individuals. On one hand, the spacio-temporal version makes sense, because it would be quite a waste to leave lots of space unused, if you can colonize it with people of identical average utility. On the other hand, it would be a bad idea to fill up that space with people of lower average utility (perhaps due to resource constraints), because that would lower to total average utility of all individuals.

Anyway, I actually think it's the subjective experiences of individuals that really matter and not their arrangement in space-time. Perhaps that's the best argument against any "utility density" considerations. So far, the only advantage of the spacio-temporal version of average utilitarianism is that in this scenario it enables at least some more or less meaningful calculations.

Utilitarianism seems to get very weird once you deal with infinities.