Ghatanathoah

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How do bounded utility functions work if you are uncertain how close to the bound your utility is?

My main objection for the simplified utility functions is that they are presented as depending only upon the current external state of the world in some vaguely linear and stable way. Every adjective in there corresponds to discarding a lot of useful information about preferences that people actually have.

 

The main argument I've heard for this kind of simplification is that your altruistic, morality-type preferences ought to be about the state of the external world because their subject is the wellbeing of other people, and the external world is where other people live.  The linearity part is sort of an extension of the principle of treating people equally. I might be steelmanning it a little, a lot of times the argument is less that and more that having preferences that are in any way weird or complex is "arbitrary." I think this is based on the mistaken notion that "arbitrary" is a synonym for "picky" or "complicated."

I find this argument unpersuasive because altruism is also about respecting the preferences of others, and the preferences of others are, as you point out, extremely complicated and about all sorts of things other than the current state of the external world.  I am also not sure that having nonlinear altruistic preferences is the same thing as not valuing people equally. And I think that our preferences about the welfare of others are often some of the most path-dependent preferences that we have.

EDIT: I have sense found this post, which discusses some similar arguments and refutes them more coherently than I do.

Second EDIT: I still find myself haunted by the "scary situation" I linked to and find myself wishing there was a way to tweak a utility function a little to avoid it, or at least get a better "exchange rate" than "double tiny good thing and more-than doubling horrible thing while keeping probability the same."  I suppose there must be a way since the article I linked to said it would not work on all bounded utility functions.

How do bounded utility functions work if you are uncertain how close to the bound your utility is?

In general though, this consideration is likely to be irrelevant. Most universes will be nowhere near the upper or lower bounds, and the chance of any individual's decision being single-handedly responsible for doing a universe scale shifts toward a utility bound is so tiny that even estimating orders of magnitude of the unlikelihood is difficult. These are angels-on-head-of-pin quibbles.

 

That makes sense.  So it sounds like the Egyptology Objection is almost a form of Pascal's Mugging in and of itself. If you are confronted by a Mugger (or some other, slightly less stupid scenario where there is a tiny probability of vast utility or disutility) the odds that you are at a "place" on the utility function that would affect the credibility threshold for the Mugger one way or another are just as astronomical as the odds that the Mugger is giving you.  So an agent with a bounded utility function is never obligated to research how much utility the rest of the universe has before rejecting the mugger's offer.  They can just dismiss it as not credible and move on.

And Mugging-type scenarios are the only scenarios where this Egyptology stuff would really come up, because in normal situations with normal probabilities of normal amounts of (dis)utility, the rescaling and reshifting effect makes your "proximity to the bound" irrelevant to your behavior.  That makes sense!

I also wanted to ask about something you said in an earlier comment:

I suspect most of the "scary situations" in these sorts of theories are artefacts of trying to formulate simplified situations to test specific principles, but accidentally throw out all the things that make utility functions a reasonable approximation to preference ordering. The quoted example definitely fits that description.

I am not sure I understand exactly what you mean by that. How do simplified hypotheticals for testing specific principles make utility functions fail to approximate preference ordering? I have a lot of difficulty with this, where I worry that if I do not have the perfect answer to various simplified hypotheticals it means that I do not understand anything about anything. But I also understand that simplified hypotheticals often causes errors like removing important details and reifying concepts.

How do bounded utility functions work if you are uncertain how close to the bound your utility is?

The main "protection" of bounded utility is that at every point on the curve, the marginal utility of money is nonzero, and the threat of disutility is bounded. So there always exists some threshold credibility below which no threat (no matter how bad) makes expected utility positive for paying them.

 

That makes sense. What I am trying to figure out is, does that threshold credibility change depending on "where you are on the curve."  To illustrate this, imagine two altruistic agents, A and B,  who have the same bounded utility function.  A lives in a horrifying hell world full of misery.  B lives in a happy utopia.  So A is a lot "closer" to the lower bound than B. Both  A and B are confronted by a Pascal's Mugger who threatens them with an arbitrarily huge disutility.

Does the fact that agent B is "farther" from lower bound than agent A mean that the two agents have different credibility thresholds for rejecting the mugger? Because the amount of disutility that  B needs to receive to get close to the lower bound is larger than the amount that A needs to receive?  Or will their utility functions have the same credibility threshold because they have the same lower and upper bounds, regardless of "how much" utility or disutility they happen to "possess" at the moment? Again, I do not know if this is a coherent question or if it is born out of confusion about how utility functions work.

It seems to me that an agent with a bounded utility function shouldn't need to do any research about the state of the rest of the universe before dismissing Pascal's Mugging and other tiny probabilities of vast utilities as bad deals.  That is why this question concerns me.

One continuous example of this is an exponential discounter, where the decisions are time-invariant but from a global view the space of potential future utility is exponentially shrinking.

Thanks, that example made it a lot easier to get my head around the idea! I think understand it better now.  This might not be technically accurate, but to me having a uniform rescaling and reshifting of utility that preserves future decisions like that doesn't even feel like I am truly "valuing" future utility less. I know that in some sense I am, but it feels more like I am merely adjusting and recalibrating some technical details of my utility function in order to avoid "bugs" like Pascal's Mugging. It feels similar to making sure that all my preferences are transitive to avoid money pumps, the goal is to have a functional decision theory, rather to to change my fundamental values.

How do bounded utility functions work if you are uncertain how close to the bound your utility is?

TLDR: What I really want to know is: 

1. Is an agent with a bounded utility function justified (because of their bounded function) in rejecting any "Pascal's Mugging" type scenario with tiny probabilities of vast utilities, regardless of how much utility or disutility they happen to "have" at the moment? Does everything just rescale so that the Mugging is an equally bad deal no matter what the relative scale of future utility is?

2. If you have a bounded utility function, are your choices going to be the same regardless of how much utility various unchangeable events in the past generated for you? Does everything just rescale when you gain or lose a lot of utility so that the relative value of everything is the same? I expect the answer is going to be "yes" based on our previous discussion, but am a little uncertain because of the various confused thoughts on the subject that I have been having lately.

Full length Comment:

I don't think I explained my issue clearly. Those arguments about Pascal's Mugging are addressing it from the perspective of its unlikeliness, rather than using a bounded utility function against it.

I am trying to understand bounded utility functions and I think I am still very confused.  What I am confused about right now is how a bounded utility function protects from Pascal's Mugging at different "points" along the function. 

Imagine we have a bounded utility function that has a "S" curve shape.  The function goes up and down from 0 and flattens as it approaches the upper and lower bounds. 

If someone has utility at around 0, I see how they resist Pascal's Mugging.  Regardless of whether the Mugging is a threat or a reward, it approaches their upper or lower bound and then diminishes. So utility can never "outrace" probability. 

But what if they have a level of utility that is close to the upper bound and a Mugger offers a horrible threat? If the Mugger offered a threat that would reduce their utility to 0, would they respond differently than they would to one that would send it all the way to the lower bound?  Would the threat get worse as the utility being cancelled out by the disutility got further from the bound and closer to 0? Or is the idea that in order for a threat/reward to qualify as a Pascal's Mugging it has to be so huge that it goes all the way down to a bound?

And if someone has a level of utility or disutility close to the bound, does that mean disutility matters more so they become a negative utilitarian close to the upper bound and a positive utilitarian close to the lower one? I don't think that is the case, I think that, as you said, "the relative scale of future utility makes no difference in short-term decisions." But I am confused about how.

I think I am probably just very confused in general about utility functions and about bounded utility functions. While some people have criticized bounded utility functions, I have never come across this specific type of criticism before. It seems far more likely that I am confused than that I am the first person to notice an obvious flaw.

How do bounded utility functions work if you are uncertain how close to the bound your utility is?

Hi, one other problem occurred to me in regards to short term decisions and bounded utility.

Suppose you are in a situation where you have a bounded utility function, plus a truly tremendous amount of utility.  Maybe you're an immortal altruist who has helped quadrillions of people, maybe you're an immortal egoist who has lived an immensely long and happy life. You are very certain that all of that was real, and it is in the past and can't be changed.

You then confront a Pascal's Mugger who threatens to inflict a tremendous amount of disutility unless you give the $5.  If you're an altruist they threaten to torture quintillions of people, if you are an egoist they threaten to torture you for a quintillion years, something like that. As with standard Pascal's mugging, the odds of them be able to carry this threat out are astronomically unlikely. 

In this case, it still fells like you ought to ignore the mugger. Does that make sense considering that, even though your bounded utility function assigns less disvalue to such a threat, it also assigns less value to the $5 because you have so much utility already? Plus, if they are able to carry out their threat, they would be able to significantly lower your utility so that it is much "further away from the bound" than it was before. Does it matter that as they push your utility further and further "down" away from the bound, utility becomes "more valuable."  

Or am I completely misunderstanding how bounded utility is calculated? I've never seen this specific criticism of bounded utility functions before, and much smarter people than me have studied this issue, so I imagine that I must be? I am not sure exactly how adding utility and subtracting disutility is calculated.  It seems like if the immortal altruist whose helped quadrillions of people has a choice between gaining 3 utilons, or inflicting 2 disutilons to gain 5 utilitons, that they should be indifferent between the two, even if they have a ton of utility and very little disutility in their past. 

How do bounded utility functions work if you are uncertain how close to the bound your utility is?

Thanks, again for your help :) That makes me feel a lot better. I have the twin difficulties of having severe OCD-related anxiety about weird decision theory problems, and being rather poor at the math required to understand them.

The case of the immortal who becomes uncertain of the reality of their experiences is I think what that "Pascal's Mugging for Bounded Utilities" article I linked to the the OP was getting at. But it's a relief to see that it's just a subset of decisions under uncertainty, rather than a special weird problem. 

How do bounded utility functions work if you are uncertain how close to the bound your utility is?

the importance to the immortal of the welfare of one particular region of any randomly selected planet of those 10^30 might be less than that of Ancient Egypt. Even if they're very altruistic.

 

Ok, thanks, I get that now, I appreciate your help. The thing I am really wondering is, does this make any difference at all to how that immortal would make decisions once Ancient Egypt is in the past and cannot be changed? Assuming that they have one of those bounded utility functions where their utility is asymptotic to the bound, but never actually reaches it, I don't feel like it necessarily would.

If Ancient Egypt is in the past and can't be changed, the immortal might, in some kind of abstract sense, value that randomly selected planet of those 10^30 worlds less than they valued Egypt. But if they are actually in a situation where they are on that random planet, and need to make altruistic decisions about helping the people on that planet, then their decisions shouldn't really be affected.  Even if the welfare of that planet is less valuable to them than the welfare of Ancient Egypt, that shouldn't matter if their decisions don't affect Ancient Egypt and only affect the planet. They would be trading less valuable welfare off against other less valuable welfare, so it would even out. Since their utility function is asymptotic to the bound, they would still act to increase their utility, even if the amount of utility they can generate is very small. 

I am totally willing to accept the Egyptology argument if all it is saying is that past events that cannot be changed might affect the value of present-day events in some abstract sense (at least if you have a bounded utility function).  Where I have trouble accepting it is if those same unchangeable past events might significantly affect what choices you have to make about future events that you can change.  If future welfare is only 0.1x as valuable as past welfare, that doesn't really matter, because future welfare is the only welfare you are able to affect. If it's only possible to make a tiny difference, then you might as well try, because a tiny difference is better than no difference. The only time when the tininess seems relevant to decisions is Pascal's Mugging type scenarios where one decision can generate tiny possibilities of huge utility.

How do bounded utility functions work if you are uncertain how close to the bound your utility is?

The phrasing here seems to be a confused form of decision making under uncertainty. Instead of the agent saying "I don't know what the distribution of outcomes will be", it's phrased as "I don't know what my utility function is".

I think part of it is that I am conflating two different parts of the Egyptology problem. One part is uncertainty: it isn't possible to know certain facts about the welfare of Ancient Egyptians that might affect how "close to the bound" you are. The other part is that most people have a strong intuition that those facts aren't relevant to our decisions, whether we are certain of them or not. But there's this argument that those facts are relevant if you have an altruistic bounded utility function because they affect how much diminishing returns your function has.

For example, I can imagine that if I was an altruistic immortal who was alive during ancient Egypt, I might be unwilling to trade a certainty of a good outcome in ancient Egypt for an uncertain amazingly terrific outcome in the far future because of my bounded utility function. That's all good, it should help me avoid Pascal's Mugging.  But once I've lived until the present day, it feels like I should continue acting the same way I did in the past, continue to be altruistic, but in a bounded fashion.  It doesn't feel like I should conclude that, because of my achievements as an altruist in Ancient Egypt, that there is less value to being an altruist in the present day.

In the case of the immortal, I do have all the facts about Ancient Egypt, but they don't seem relevant to what I am doing now.  But in the past, in Egypt, I was unwilling to trade certain good outcomes for uncertain terrific ones because my bounded utility function meant I didn't value the larger ones linearly.  Now that the events of Egypt are in the past and can't be changed, does that mean I value everything less?  Does it matter if I do, if the decrease in value is proportionate?  If I treat altruism in the present day as valuable, does that contradict the fact that I discounted that same value back in Ancient Egypt? 

I think that's why I'm phrasing it as being uncertain of what my utility function is. It feels like if I have a bounded utility function, I should be unwilling (within limits) to trade a sure thing for a small possibility of vast utility, thereby avoiding Pascal's Mugging and similar problems. But it also feels like, once I have that sure thing, and the fact that I have it cannot be changed, I should be able to continue seeking more utility, and how many sure things I have accumulated in the past should not change that.

How do bounded utility functions work if you are uncertain how close to the bound your utility is?

I really still don't know what you mean by "knowing how close to the bound you are".

 

What I mean is, if I have a bounded utility function where there is some value, X, and (because the function is bounded) X diminishes in value the more of it there is, what if I don't know how much X there is? 

For example, suppose I have a strong altruistic preference that the universe have lots of happy people. This preference is not restricted  by time and space, it counts the existence of happy people as a good thing regardless of where or when they exist.  This preference is also agent neutral, it does not matter whether I, personally, am responsible for those people existing and being happy, it is good regardless. This preference is part of a bounded utility function, so adding more happy people starts to have diminishing returns the closer one gets to a certain bound. This allows me to avoid Pascal's Mugging.

However, if adding more people has diminishing returns because the function is bounded, and my preference is not restricted by time, space, or agency, that means that I have no way of knowing what those diminishing returns are unless I know how many happy people have ever existed in the universe.  If there are diminishing returns based on how many people there are, total, in the universe, then the value of adding more people in the future might change depending on how many people existed in the past.

That is what I mean by "knowing how close to the bound" I am. If I value some "X", what if it isn't possible to know how much X there is? (like I said before, a version of this for egoistic preferences might be if the X is happiness over your lifetime, and you don't know how much X there is because you have amnesia or something).

I was hoping that I might be able to fix this issue by making a bounded utility function where X diminishes in value smoothly and proportionately.  So a million happy people in ancient Egypt has proportional diminishing returns to a billion and so on.  So when I am making choices about  maximizing X in the present, the amount of X I get is diminished in value, but it is proportionately diminished, so the decisions that I make remain the same.  If there was a vast population in the past, the amount of X I can generate has very small value according to a bounded utility function. But that doesn't matter because it's all that I can do.

That way, even if X decreases in value the more of it there is, it will not effect any choices I make where I need to choose between different probabilities of getting different amounts of X in the future.  

I suppose I could also solve it by making all of my preferences agent-relative instead of agent-neutral, but I would like to avoid that. Like most people I have a strong moral intuition that my altruistic preferences should be agent-neutral.  I suppose it might also get me into conflict with other agents with bounded agent-relative utility functions if we value the same act differently.

If I am explaining this idea poorly, let me try directing you to some of the papers I am referencing. Besides the one I mentioned in the OP, there is this one by Beckstead and Thomas (pages 16, 17, and 18 are where it discusses it). 

How do bounded utility functions work if you are uncertain how close to the bound your utility is?

REA doesn't help at all there, though. You're still computing U(2X days of torture) - U(X days of torture)

I think I see my mistake now, I was treating a bounded utility function using REA as subtracting the "unbounded" utilities of the two choices and then comparing the post-subtraction results using the bounded utility function. It looks like you are supposed to judge each one's utility by the bounded function before subtracting them.

Unfortunately REA doesn't change anything at all for bounded utility functions. It only makes any difference for unbounded ones.

That's unfortunate. I was really hoping that it could deal with the Egyptology scenario by subtracting the unknown utility value of Ancient Egypt and only comparing the difference in utility between the two scenarios.  That way the total utilitarian (or some other type of altruist) with a bounded utility function would not need to research how much utility the people of Ancient Egypt had in order to know how good adding happy people to the present day world is.  That just seems insanely counterintuitive.

I suppose there might be some other way around the Egyptology issue. Maybe if you have a bounded or nonlinear utility function that is sloped at the correct rate it will give the same answer regardless of how happy the Ancient Egyptians were. If they were super happy then the value of whatever good you do in the present is in some sense reduced. But the value of whatever resources you would sacrifice in order to do good is reduced as well, so it all evens out.  Similarly, if they weren't that happy, the value of the good you do is increased, but the value of whatever you sacrifice in order to do that good is increased proportionately.  So a utilitarian can go ahead and ignore how happy the ancient Egyptians were when doing their calculations. 

It seems like this might work if the bounded function has adding happy lives have diminishing returns at a reasonably steady and proportional rate (but not so steady that it is effectively unbounded and can be Pascal's Mugged).

With the "long lived egoist" example I was trying to come up with a personal equivalent to the Egyptology problem. In the Egyptology problem, a utilitarian does not know how close they are to the "bound" of their bounded utility function because they do not know how happy the ancient Egyptians were.  In the long lived egoist example, they do not know how close to the bound they are because they don't know exactly how happy and long lived their past self was.  It also seems insanely counterintuitive to say that, if you have a bounded utility function, you need to figure out exactly how happy you were as a child in order to figure out how good it is for you to be happy in the future.  Again, I wonder if a solution might be to have a bounded utility function with returns that diminish at a steady and proportional rate.

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