Arguments for utilitarianism are impossibility arguments under unbounded prospects

[-]Garrett Baker2y51

However, either the axioms themselves (e.g. the continuity/Archimedean axiom, or general versions of Independence or the Sure-Thing Principle) rule out expectational total utilitarianism, or the kinds of arguments used to defend the axioms (Russell and Isaacs, 2021).

I don't understand this part of your argument. Can you explain how you imagine this proof working?

Otherwise, it seems like most of your arguments come down to showing that lots of paradoxes happen when you do math to infinite ethics.

There are many arguments on LessWrong for, and against infinite ethics. I don't think any, including this one, actually show "utilitarianism is irrational or self-undermining". For example, as you came close to saying in your responses, you could just have bounded utility functions! That ends up being rational, and seems not self-undermining because after looking at many of these arguments it seems like maybe you're kinda forced to.

I think there's also some work on using hyper-reals or other generalizations to quantify infinities, and solving various problems that way.

Overall, I wish you'd explain the arguments in the papers you linked better. The one argument you actually wrote in this post was interesting, you should have done more of that!

[-]MichaelStJules2y*50

Thanks for the comment!

I don't understand this part of your argument. Can you explain how you imagine this proof working?

St Petersburg-like prospects (finite actual utility for each possible outcome, but infinite expected utility, or generalizations of them) violate extensions of each of these axioms to countably many possible outcomes:

The continuity/Archimedean axiom: if A and B have finite expected utility, and A < B, there's no strict mixture of A and an infinite expected utility St Petersburg prospect, like , $0 < p < 1$ , that's equivalent to B, because all such strict mixtures will have infinite expected utility. Now, you might not have defined expected utility yet, but this kind of argument would generalize: you can pick A and B to be outcomes of the St Petersburg prospect, and any strict mixture with A will be better than B.
The Independence axiom: see the following footnote.^[2]
The Sure-Thing Principle: in the money pump argument in my post, B-$100 is strictly better than each outcome of A, but A is strictly better than B-$100. EDIT: Actually, you can just compare A with B.

I think these axioms are usually stated only for prospects for finitely many possible outcomes, but the arguments for the finitary versions, like specific money pump arguments, would apply equally (possibly with tiny modifications that wouldn't undermine them) to the countable versions. Or, at least, that's the claim of Russell and Isaacs, 2021, which they illustrate with a few arguments and briefly describe some others that would generalize. I reproduced their money pump argument in the post.

For example, as you came close to saying in your responses, you could just have bounded utility functions! That ends up being rational, and seems not self-undermining because after looking at many of these arguments it seems like maybe you're kinda forced to.

Ya, I agree that would be rational. I don't think having a bounded utility function is in itself self-undermining (and I don't say so), but it would undermine utilitarianism, because it wouldn't satisfy Impartiality + (Separability or Goodsell, 2021's version of Anteriority). If you have to give up Impartiality + (Separability or Goodsell, 2021's version of Anteriority) and the arguments that support them, then there doesn't seem to be much reason left to be a utilitarian of any kind in the first place. You'll have to give up the formal proofs of utilitarianism that depend on these principles or restrictions of them that are motivated in the same ways.

You can try to make utilitarianism rational by approximating it with a bounded utility function, or applying a bounded function to total welfare and taking that as your utility function, and then maximizing expected utility, but then you undermine the main arguments for utilitarianism in the first place.

Hence, utilitarianism is irrational or self-undermining.

Overall, I wish you'd explain the arguments in the papers you linked better. The one argument you actually wrote in this post was interesting, you should have done more of that!

I did consider doing that, but the post is already pretty long and I didn't want to spend much more on it. Goodsell, 2021's proof is simple enough, so you could check out the paper. The proof for Theorem 4 from Russell, 2023 looks trickier. I didn't get it on my first read, and I haven't spent the time to actually understand it. EDIT: Also, the proofs aren't as nice/intuitive/fun or flow as naturally as the money pump argument. They present a sequence of prospects constructed in very specific ways, and give a contradiction (violating of transitivity) when you apply all of the assumptions in the theorem. You just have to check the logic.

^{^}
You could refuse to define the expected utilility, but the argument generalizes
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Russell and Isaacs, 2021 define Countable Independence as follows:
For any prospects $X_{1}, X_{2}, \dots$ , and $Y_{1}, Y_{2}, \dots$ , and any probabilities $p_{1}, p_{2}, \dots$ that sum to one, if $X_{1} ≲ Y_{1}, X_{2} ≲ Y_{2}, \dots$ , then
$\sum i p_{i} X_{i} ≲ \sum i p_{i} Y_{i}$
If furthermore $X_{j} < Y_{j}$ for some $j$ such that $p_{j} > 0$ , then
$\sum i p_{i} X_{i} < \sum i p_{i} Y_{i}$
Then they write:
Improper prospects clash directly with Countable Independence. Suppose $X$ is a prospect that assigns probabilities $p_{1}, p_{2}, \dots$ to outcomes $x_{1}, x_{2}, \dots$ . We can think of $X$ as a countable mixture in two different ways. First, it is a mixture of the one-outcome prospects $x_{1}, x_{2}, \dots$ in the obvious way. Second, it is also a mixture of infinitely many copies of X itself. If $X$ is improper, this means that $X$ is strictly better than each outcome $x_{i}$ . But then Countable Independence would require that X is strictly better than X. (The argument proceeds the same way if X is strictly worse than each outcome xi instead.)

[-]MichaelDickens2y10

Based on your explanation in this comment, it seems to me that St. Petersburg-like prospects don't actually invalidate utilitarian ethics as it would have been understood by e.g. Bentham, but it does contradict the existence of a real-valued utility function. It can still be true that welfare is the only thing that matters, and that the value of welfare aggregates linearly. It's not clear how to choose when a decision has multiple options with infinite expected utility (or an option that has infinite positive EV plus infinite negative EV), but I don't think these theorems imply that there cannot be any decision criterion that's consistent with the principles of utilitarianism. (At the same time, I don't know what the decision criterion would actually be.) Perhaps you could have a version of Bentham-esque utilitarianism that uses a real-valued utility function for finite values, and uses some other decision procedure for infinite values.

[-]MichaelStJules2y10

Ya, I don't think utilitarian ethics is invalidated, it's just that we don't really have much reason to be utilitarian specifically anymore (not that there are necessarily much more compelling reasons for other views). Why sum welfare and not combine them some other way? I guess there's still direct intuition: two of a good thing is twice as good as just one of them. But I don't see how we could defend that or utilitarianism in general any further in a way that isn't question-begging and doesn't depend on arguments that undermine utilitarianism when generalized.

You could just take your utility function to be where $σ$ is any bounded increasing function, say arctan, and maximize the expected value of that. This doesn't work with actual infinities, but it can handle arbitrary prospects over finite populations. Or, you could just rank prospects by stochastic dominance with respect to the sum of utilities, like Tarsney, 2020.

You can't extend it the naive way, though, i.e. just maximize $E [\sum_{i} U_{i}]$ whenever that's finite and then do something else when it's infinite or undefined, though. One of the following would happen: the money pump argument goes through again, you give up stochastic dominance or you give up transitivity, each of which seems irrational. This was my 4th response to Infinities are generally too problematic.

[-]MichaelStJules2y10

Also, I'd say what I'm considering here isn't really "infinite ethics", or at least not what I understand infinite ethics to be, which is concerned with actual infinities, e.g. an infinite universe, infinitely long lives or infinite value. None of the arguments here assume such infinities, only infinitely many possible outcomes with finite (but unbounded) value.

[-]Garrett Baker2y30

The argument you made that I understood seemed to rest on allowing for an infinite expectation to occur, which seems pretty related to me to infinite ethics, though I'm no ethicist.

[-]MichaelStJules2y*30

The argument can be generalized without using infinite expectations, and instead using violations of Limitedness in Russell and Isaacs, 2021 or reckless preferences in Beckstead and Thomas, 2023. However, intuitively, it involves prospects that look like they should be infinitely valuable or undefinably valuable relative to the things they're made up of. Any violation of (the countable extension of) the Archimedean Property/continuity is going to look like you have some kind of infinity.

The issue could just be a categorization thing. I don't think philosophers would normally include this in "infinite ethics", because it involves no actual infinities out there in the world.

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An individual’s welfare can be the value of their own utility function, although preferences or utility functions defined in terms of each other can lead to contradictions through indirect self-reference (Bergstrom, 1989, Bergstrom, 1999, Vadasz, 2005, Yann, 2005 and Dave and Dodds, 2012). I set aside this issue here.

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This argument works with a step size that's bounded below, even by a tiny value, like 1 millionth of a second or 1 millionth more (counterfactual) utility. If the step sizes have to keep getting smaller and smaller and converge to 0, then we may never reach $K$ .

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Although there are stronger arguments that's actually infinite. It's one of the simplest and most natural models that fits with our observations of global flatness. See the Wikipedia article Shape of the Universe.

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For generalizations without actual utility values, see violations of Limitedness in Russell and Isaacs, 2021 and reckless preferences in Beckstead and Thomas, 2023.

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Independence: For any prospects $X$ , $Y$ and $Z$ , and probability $p, 0 < p < 1$ , if $X < Y$ , then $p X + (1 - p) Z < p Y + (1 - p) Z$ ,

where $p X + (1 - p) Z$ is the prospect that's $X$ with probability $p$ , and $Z$ with probability $1 - p$ .

Russell and Isaacs, 2021 define Countable Independence as follows:

For any prospects $X_{1}, X_{2}, \dots$ , and $Y_{1}, Y_{2}, \dots$ , and any probabilities $p_{1}, p_{2}, \dots$ that sum to one, if $X_{1} ≲ Y_{1}, X_{2} ≲ Y_{2}, \dots$ , then

\sum i p_{i} X_{i} ≲ \sum i p_{i} Y_{i}

If furthermore $X_{j} < Y_{j}$ for some $j$ such that $p_{j} > 0$ , then

\sum i p_{i} X_{i} < \sum i p_{i} Y_{i}

The standard finitary Independence axiom is a special case.

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The Sure Thing Principle can be defined as follows:

Let $A$ and $B$ be prospects, and let $E$ be some event with probability neither 0 nor 1. If $A ≲ B$ conditional on each of $E$ and $n o t E$ , then $A ≲ B$ . If furthermore, $A < B$ conditional on $E$ or A < B conditional on $n o t E$ , then $A < B$ .

In other words, if we weakly prefer $B$ either way, then we should just weakly prefer $B$ . And if, furthermore, we strictly prefer $B$ on one of the two possibilities, then we should just strictly prefer $B$ .

Russell and Isaacs, 2021 define the Countable Sure Thing Principle as follows:

Let $A$ and $B$ be prospects, and let $E$ be a (countable) set of mutually exclusive and exhaustive events, each with non-zero probability. If $A ≲ B$ conditional on each $E \in E$ , then $A ≲ B$ . If furthermore, $A < B$ conditional on some $E \in E$ , then $A < B$ .

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See also Christiano, 2022. Both depend on St Petersburg game-like prospects with infinitely many possible outcomes and, when defined, infinite expected utility. For more on the St Petersburg paradox, see Peterson, 2023. Some other foreseeable sure loss arguments require a finite but possibly unbounded number of choices, like McGee, 1999 and Pruss, 2022.

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Or, as in Russell and Isaacs, 2021, each of the countably many prospects used to construct it.

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Note that the probabilities sum to 1, because $\sum_{n = 1}^{\infty} 1 / 2^{n} = 1$ , so this is in fact a proper probability distribution.

The expected value is $\sum_{n = 1}^{\infty} 2^{n} \frac{1}{2^{n}} = {lim}_{N \to \infty} \sum_{n = 1}^{N} 1 = {lim}_{N \to \infty} N = \infty$

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From $u (x_{n + 1}) \geq 2 u (x_{n})$ for each $n > 1$ , we have, by induction, $u (x_{n + 1}) \geq 2^{n} u (x_{n})$ . Then, for each $N \geq 1$ ,

$E [u (X)] = \sum_{n = 1}^{\infty} u (x_{n}) p (x_{n}) = \sum_{n = 1}^{\infty} u (x_{n}) \frac{1}{2^{n}} \geq \sum_{n = 1}^{N} u (x_{n}) / 2^{n} \geq \sum_{n = 1}^{N} 2^{n - 1} u (x_{1}) / 2^{n} = N u (x_{1}) / 2$ ,

which can be arbitrarily large, so $E [u (X)] = \infty$ .

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This would follow either by extension to expected utilities over countable prospects, or assuming we respect Statewise Dominance and transitivity.

For the latter, we can modify the prospect to a truncated one with finitely many outcomes $X_{N}$ for each $N > 1$ , by defining $X_{N} = X$ if $X < x_{N}$ , and $X_{N} = x_{N}$ (or $x_{1}$ ) otherwise. Then $E [u (X_{N})]$ is finite for each N, but ${lim}_{N \to \infty} E [u (X_{N})] = \infty$ . Furthermore, for each N, not only is it the case that $E [u (X)] = \infty > E [u (X_{N})]$ , but $X$ also strictly statewise dominates $X_{N}$ , i.e. X is with certainty at least as good as $X_{N}$ , and is, with nonzero probability, strictly better. So, given any prospect $Y$ with finite (expected) utility, there’s an $N$ such that $E [u (X_{N})] > E [u (Y)]$ , so $X_{N} ≻ Y$ , but since $X ≻ X_{N}$ , by transitivity, $X ≻ Y$ .

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For Countable Independence: We defined $X = \sum_{n = 1}^{\infty} \frac{1}{2^{n}} x_{n}$ . We can let $X_{n} = x_{n}$ in the definition of Countable Independence. However, it's also the case that $X = \sum_{n = 1}^{\infty} \frac{1}{2^{n}} X$ , so we can let $Y_{n} = X$ in the definition of Countable Independence. But $Y_{n} = X > x_{n} = X_{n}$ for each $n$ , so by Countable Independence, $X > X$ , contrary to reflexivity.

For the Countable Sure-Thing Principle: define $Y$ to be identically distributed to $X$ but independent from $X$ . Let $E = {X = x_{n} | n \geq 1}$ . $Y > x_{n}$ for each $n$ , so $Y > X$ conditional on $X = x_{n}$ , for each $n$ . By the Countable Sure-Thing Principle, this would imply $Y > X$ . However, doing the same with $E = {Y = x_{n} | n \geq 1}$ also gives us $X > Y$ , violating transitivity.

These arguments extend to the more general kinds of improper prospects in Russell and Isaacs, 2021.

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In practice, you should give weight to the possibility that it has infinite or undefined value. However, the argument that follows can be generalized to this case using stochastic dominance reasoning or, if you do break ties between actual infinities, any reasonable way of doing so.

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Or give you an accurate finite upper bound on how it will turn out.

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And the genie isn’t going to do anything good with it.

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Interestingly, if expectational total utilitarianism is consistent with Harsanyi’s theorem, then it is not the only way for total utilitarianism to be consistent with Harsanyith’s theorem. Say individual welfare takes values in the interval $[2, 3]$ . Then the utility functions $\sum_{n = 1}^{N} u_{i} + N$ and $N \sum_{n = 1}^{N} u_{i}$ agree with both Harsanyi’s theorem and total utilitarianism. According to them, a larger population is always better than a smaller population, regardless of the welfare levels in each. However, some further modest assumptions give us expectational total utilitarianism, e.g. each individual can welfare level 0.

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So, assuming reflexivity, transitivity, the Independence of Irrelevant Alternatives. Also, we need the set of prospects to be rich enough to include some of the kinds of prospects used in the proofs.

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Exceptions include average utilitarianism, symmetric person-affecting views, maximin and maximax.

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Russell, 2023 writes:

Stochastic Dominance is a fairly uncontroversial principle of decision theory—even among those who reject other parts of standard expectational decision theory (such as Quiggin, 1993; Broome, 2004), and even in settings where other parts of standard expectational decision theory give out (see for example Easwaran, 2014).¹⁰We should not utterly foreclose giving up Stochastic Dominance—we are facing paradoxes, so some plausible principles will have to go—but I do not think this is a very promising direction. In what follows, I will take Stochastic Dominance for granted.

and in footnote 10:

For other defenses of Stochastic Dominance, on which I here have drawn, see Tarsney (2020, 8); Wilkinson (2022, 10); Bader (2018).

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There is some controversy here, because we might instead say that A statewise dominates $B$ if and only if $A$ is at least as good as $B$ under every possibility, including each possibility with probability 0. Russell, 2023 writes:

For example, the probability of an ideally sharp dart hitting a particular point may be zero—but the prospect of sparing a child from malaria if the dart hits that point (and otherwise nothing) may still be better than the prospect of getting nothing no matter what. But these two prospects are stochastically equivalent. Perhaps what is best depends on what features of its outcomes are sure—where in general this can come apart from what is almost sure—that is, has probability one.

However, I don’t think this undermines the results of Russell, 2023, because the prospects considered don’t disagree on any outcomes of probability 0.

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Insofar as it isn’t evidence for how well off moral patients today and in the future can or will be, and ignoring acausal influence.

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The same objection is raised earlier in McMahan, 1981, p. 115 referring to past generations more generally. See also discussion of it and similar objections in Huemer, 2008, Wilkinson, 2022, Beckstead and Thomas, 2023, Wilkinson, 2023 and Russell, 2023.

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And a single preorder over prospects, so transitivity, reflexivity and the independence of irrelevant alternatives, and a rich enough set of possible prospects.

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These can be extended to satisfy Stochastic Dominance by making stochastically equivalent prospects equivalent and taking the transitive closure to get a new preorder.

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Or, while keeping everyone at least as well off, in cases of incomparability.

LESSWRONG
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LESSWRONG
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7

Arguments for utilitarianism are impossibility arguments under unbounded prospects

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Summary

Basic terminology

Motivation and outline

Unbounded utility functions are irrational

A money pump argument

Anti-utilitarian theorems

Summary so far

Responses

Infinities are generally too problematic

Accept irrational behaviour or deny its irrationality

Sacrifice or weaken utilitarian principles

It's not just utilitarianism

Acknowledgements