Maximising expected utility follows from self-evident premises?

Vasco Grilo🔸

This is a linkpost for https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Morgenstern_utility_theorem

This is a linkpost for Von Neumann–Morgenstern utility theorem, which shows that one accepts 4 premises if and only if one maximises expected utility. In my mind, all the 4 premises are self-evident. So I do not see how one can reject maximising expected utility in principle. Relatedly, I think the Repugnant Conclusion follows from 3 self-evident premises.

In 1947, John von Neumann and Oskar Morgenstern proved that any individual whose preferences satisfied four axioms [premises] has a utility function, where such an individual's preferences can be represented on an interval scale [which "allows for defining the degree of difference between measurements"] and the individual will always prefer actions that maximize expected utility.^[1] That is, they proved that an agent is (VNM-)rational [has preferences satisfying the 4 axioms] if and only if there exists a real-valued function u defined by possible outcomes such that every preference of the agent is characterized by maximizing the expected value of u, which can then be defined as the agent's VNM-utility (it is unique up to affine transformations i.e. adding a constant and multiplying by a positive scalar). No claim is made that the agent has a "conscious desire" to maximize u, only that u exists.
[...]
Completeness assumes that an individual has well defined preferences:
Axiom 1 (Completeness) For any lotteries and $M$ , either $L ⪰ M$ or $M ⪰ L$ .
(the individual must express some preference or indifference^[4]). Note that this implies reflexivity.
Transitivity assumes that preferences are consistent across any three options:
Axiom 2 (Transitivity) If $L ⪰ M$ and $M ⪰ N$ , then $L ⪰ N$ .
Continuity assumes that there is a "tipping point" between being better than and worse than a given middle option:
Axiom 3 (Continuity): If $L ⪯ M ⪯ N$ , then there exists a probability $p \in [0, 1]$ such that $p L + (1 - p) N \sim M$
where the notation on the left side refers to a situation in which $L$ is received with probability $p$ and $N$ is received with probability $(1 - p)$ .
[...]
Independence assumes that a preference holds independently of the probability of another outcome.
Axiom 4 (Independence): For any $M$ and $p \in [0, 1)$ (with the "irrelevant" part of the lottery underlined):
$L ⪯ N$ if and only if $(1 - p) L + p M - -- - ⪯ (1 - p) N + p M - -- -$
In other words, the probabilities involving $M$ cancel out and don't affect our decision, because the probability of $M$ is the same in both lotteries.

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Anthony DiGiovanniJan 1913

Why do you consider completeness self-evident? (Or continuity, although I'm more sympathetic to that one.)

Also, it's important not to conflate "given these axioms, your preferences can be represented as maximizing expected utility w.r.t. some utility function" with "given these axioms [and a precise probability distribution representing your beliefs], you ought to make decisions by maximizing expected value, where 'value' is given by the axiology you actually endorse." I'd recommend this paper on the topic (especially Sec. 4), and Sec. 2.2 here.

Vasco Grilo🔸Jan 192

Hi Anthony,

I think completeness is self-evident because "the individual must express some preference or indifference". Reality forces them to do so. For example, if they donate to organisation A over B, at least implicitly, they imply donating to A is as good or better than donating to B. If they decide to keep the money for personal consumption, at least implicitly, they imply that is as good or better than donating.

I believe continuity is self-evident because rejecting it implies seemingly non-sensical decisions. For example, if one prefers 100 $ over 10 $, and this over 1 $, continuity says there is a probability p such that one is indifferent between 10 $ and a lottery involving a probability p of winning 1 $, and 1 - p of winning 100 $. One would prefer the lottery with p = 0 over 10 $, because then one would be certain to win 100 $. One would prefer 10 $ over the lottery with p = 1, because then one would be certain to win 1 $. If there was not a tipping point between preferring the lottery or 10 $, one would have to be insensitive to an increased probability of an outcome better than 10 $ (100 $), and a decreased probability of an outcome worse than 10 $ (1 $), which I see as non-sensical.

Anthony DiGiovanniJan 1911

Thanks! I'll just respond re: completeness for now.

When we ask "why should we maximize EV," we're interested in the reasons for our choices. Recognizing that I'm forced by reality to either donate or not-donate doesn't help me answer whether it's rational to strictly prefer donating, strictly prefer not-donating, be precisely indifferent, or none of the above.
Incomplete preferences have at least one qualitatively different property from complete ones, described here, and reality doesn't force you to violate this property.
Not that you're claiming this directly, but just to flag, because in my experience people often conflate these things: Even if in some sense your all-things-considered preferences need to be complete, this doesn't mean your preferences w.r.t. your first-order axiology need to be complete. For example, take the donation case. You might be very sympathetic to a total utilitarian axiology, but when deciding whether to donate, your evaluation of the total utilitarian betterness-under-uncertainty of one option vs. another doesn't need to be complete. You might, say, just rule out options that are stochastically dominated w.r.t. total utility, and then decide among the remaining options based on non-consequentialist considerations. (More on this idea here.)

Karthik TadepalliJan 1912

Maximizing expected utility is not the same as maximizing expected value. The latter assumes risk neutrality, but vNM is totally consistent with maximizing expected utility under arbitrary levels of risk aversion, meaning that it doesn't provide support for your view expressed elsewhere that risk aversion is inconsistent with vNM.

The key point is that there is a subtle difference between maximizing a linear combination of outcomes, vs maximizing a linear combination of some transformation of outcomes. That transformation can be arbitrarily concave, such that we would end up making a risk averse decision.

Vasco Grilo🔸Jan 192

Thanks for the comment, Karthik! I strongly upvoted it. I have changed "expected value" to "expected utility" in this post, and updated to the following the last paragraph of the comment of mine you linked to.

I reject risk aversion with respect to impartial welfare (although it makes all sense to be risk averse with respect to money), as I do not see why the value of additional welfare would decrease with welfare.

NunoSempereJan 204

I am extremely sympathetic to vNM, but think it's not constructive. I think the world is too high-dimensional, and in some sense we are low compute agents in a high compute world. See here for a bit more background.

For example, there are lotteries L and M which are complex enough that a) I would express a strong preference if given enough time to parse it, b) the best option is not to actually choose between them but do something else.
For continuity, you can't necessarily know which p it is.
If you want to extract someone's utility function, this is an ~nlogn operation (using mergesort where each ordering step ellicits a numerical comparison). This line of research is interesting to me, but because of the expense it only works with enough buy in, which one may not have.

In practice, I think vNM works as an idealization of the values of a high or infinite compute agent, but because making it constructive is very expensive, sometimes the best action is not to go through with that but to fall back on heuristics or shortcuts, heuristics which you won't be sure of either (again, as low compute agents in a higher complexity world).

Vasco Grilo🔸Jan 202

Thanks, Nuño. I strongly endorse maximising expected welfare, but I very much agree with using heuristics. At the same time, I would like to see more cost-effectiveness analyses.

MichaelStJulesJan 195

I don't think any of the axioms are self-evident. FWIW, I don't really think anything is self-evident, maybe other than direct logical deductions and applications of definitions.

I have some sympathy for rejecting each of them, except maybe transitivity, which I'm pretty strongly inclined not to give up. I give weight to views that violate the other axioms, under normative uncertainty.

Some ways you might reject them:

Continuity: Continuity rules out infinities and prospects with finite value but infinite expected value, like St Petersburg lotteries. If continuity is meant to apply to all logically coherent prospects (as usually assumed), then this implies your utility function must be bounded. This rules out expectational total utilitarianism as a general view.
Continuity: You might think some harms are infinitely worse than others, e.g. when suffering reaches the threshold of unbearability. It could also be that this threshold is imprecise/vague/fuzzy, and we would also reject completeness to accommodate that.
Completeness: Some types of values/goods/bads may be incomparable. Or, you might think interpersonal welfare comparisons, e.g. across very different kinds of minds, are not always possible. Tradeoffs between incomparable values would often be indeterminate. Or, you might think they are comparable in principle, but only vaguely so, leaving gaps of incomparability when the tradeoffs seem too close.
Independence: Different accounts of risk aversion or difference-making risk aversion (not just decreasing marginal utility, which is consistent with Independence).

Karthik TadepalliJan 204

Continuity doesn't imply your utility function is bounded, just that it never takes on the value "infinity", ie for any value it takes on, there are higher and lower values that can be averaged to reach that value.

Vasco Grilo🔸Jan 202

Thanks, Michael.

1. Continuity: Continuity rules out infinities and prospects with finite value but infinite expected value, like St Petersburg lotteries. If continuity is meant to apply to all logically coherent prospects (as usually assumed), then this implies your utility function must be bounded. This rules out expectational total utilitarianism as a general view.
2. Continuity: You might think some harms are infinitely worse than others, e.g. when suffering reaches the threshold of unbearability. It could also be that this threshold is imprecise/vague/fuzzy, and we would also reject completeness to accommodate that.

In practice, I think the effects of one's actions decay to practically 0 after 100 years or so. In principle, I am open one's actions having effects which are arbitrarily large, but not infinite, and continuity does not rule out arbitrarily large effects.

3. Completeness: Some types of values/goods/bads may be incomparable. Or, you might think interpersonal welfare comparisons, e.g. across very different kinds of minds, are not always possible. Tradeoffs between incomparable values would often be indeterminate. Or, you might think they are comparable in principle, but only vaguely so, leaving gaps of incomparability when the tradeoffs seem too close.

Reality forces us to compare outcomes, at least implicitly.

4. Independence: Different accounts of risk aversion or difference-making risk aversion (not just decreasing marginal utility, which is consistent with Independence).

I just do not see how adding the same possibility to each of 2 lotteries can change my assessment of these.