I now believe the statement of mine you quote was incorrect, and Iâve updated the optimizerâs curse section, primarily to remove the sentence you quoted (as I think itâs unnecessary in any case) and to alter an earlier part where I made a very similar claim so that it now says:
As best I can tell:
*The optimizerâs curse is likely to be a pervasive problem and is worth taking seriously.
*In many situations, the curse will just indicate that weâre probably overestimating how much better (compared to the alternatives) the option we estimate is best isâit wonât indicate that we should actually change what option we pick.
*But the curse can indicate that we should pick an option other than that which we estimate is best, if we have reason to believe that our estimate of the value of the best option is especially uncertain, and we donât model that information.
(I think I already knew this but just previously didnât explain it properly, leaving the conditions I had in mind as assumed, even though they often wonât hold in practice.)
But I think this updated version doesnât address the points you make. From âif weâre dealing with a bounded budgetâin which case, one might prefer a suboptimal option with low varianceâ, it sounds to me like maybe what youâre getting at is risk-aversion and/âor diminishing returns to a particular thing?
For example, letâs say I can choose either A, which gives me $1 thousand in expectation, or B, which gives me $1 million in expectation. So far, B obviously seems way better. But what if B is way higher uncertainty (or way higher risk, if one prefers that phrasing)? Then maybe Iâd prefer A.
Iâd personally consider this biased if itâs pure risk-aversion, and the dollar values perfectly correspond to my âutilityâ from this. But in reality, each additional dollar is less valuable. For example, perhaps Iâm broke, and by far the most important thing is that I get $1000 to get myself out of a real holeâa quite low chance of much higher payoffs isnât worth it, because I get far less than 1000 times as much value out of 1000 times as much money.
If thatâs what you were getting at, I think thatâs all valid, and I think the optimizerâs curse does probably magnify those reasons to sometimes not go with what you estimate will give you, in expectation, the most of some thing you value. But I think really that doesnât depend on the optimizerâs curse, and is more about uncertainty in general. Also, I think itâs really important to distinguish âmaximising expected utilityâ from âmaximising expected amount of some particular thing I valueâ. My understanding is that ârisk-aversionâ based on diminishing returns to dollars, for example, can 100% make sense within expected utility maximisationâitâs only pure risk-aversion (in terms of utility itself) that canât.
(Let me know if I was totally misunderstanding you.)
I am very satisfied with the new text. I think you understood me pretty well; the problem is, I was a little bit unclear and ambiguous.
Iâm not sure if this impacts your argument: I think diminishing returns accounts pretty well for saturation (ie., gaining $1 is not as important as losing $1); but itâs plausible to complement subjective expected utility theory with pure risk-aversion, like Lara Buchak does.
But what I actually had in mind is something like, in the extreme for unbounded utility, St. Petersburg paradox: if youâre willing to constantly bet all your budget, youâll sure end up with $0 and bankrupt. In real life, I guess that if you were constantly updating your marginal utility per dollar, this wouldnât be a problem (so I agree with youâthis is not a challenge to expected utility maximisation).
Yeah, Iâve seen mentions of Buchakâs work and one talk from her, but didnât really get it, and currently (with maybe medium confidence?) still think that, when talking about utility itself, and thus having accounted for diminishing returns and all that, one should be risk-neutral.
I hadnât heard of martingales, and have relatively limited knowledge of the St Petersburg paradox. It seems to me (low confidence) that:
things like the St Petersburg paradox and Pascalâs mugging are plausible candidates for reasons to reject standard expected utility maximisation, at least in certain edge cases, and maybe also expected value reasoning
Recognising that there are diminishing returns to many (most?) things at least somewhat blunts the force of those weird cases
Things like accepting risk aversion or rounding infinitemal probabilities to 0 may solve the problems without us having to get rid of expected value reasoning or entirely get rid of expected utility maximisation (just augment it substantially)
There are some arguments for just accepting as rational what expected utility maximisation says in these edge casesâitâs not totally clear that our aversion to the ânaive probabilisticâ answer here is valid; maybe that aversion just reflects scope neglect, or the fact that, in the St Petersburg case, thereâs the overlooked cost of it potentially taking months of continual play to earn substantial sums
I donât think these reveal problems with using EPs specifically. It seems like the same problems could occur if you talked in qualitative terms about probabilities (e.g., âat least possibleâ, âfairly good oddsâ), and in either case the âfixâ might look the same (e.g., rounding down either a quantitative or qualitative probability to 0 or to impossibility).
But it does seem that, in practice, people not using EPs are more likely to round down low probabilities to 0. This could be seen as good, for avoiding Pascalâs mugging, and/âor as bad, for a whole host of other reasons (e.g., ignoring many x-risks).
Maybe a fuller version of this post would include edge cases like that, but I know less about them, and I think they could create âissuesâ (arguably) even when one isnât using explicit probabilities anyway.
I mostly agree with you. I subtracted the reference to martingales from my previous comment because: a) not my expertise, b) this discussion doesnât need additional complexity.
Iâm sorry for having raised issues about paradoxes (perhaps there should be a Godwinâs Law about them); I donât think we should mix edge cases like St. Petersburg (and problems with unbounded utility in general) with the optimizerâs curse â itâs already hard to analyze them separately.
when talking about utility itself, and thus having accounted for diminishing returns and all that, one should be risk-neutral.
Pace Buchak, I agree with that, but I wouldnât say it aloud without adding caveats: in the real world, our problems are often of dynamic choice (and so one may have to think about optimal stopping and strategies, information gathering, etc.), we donât observe utility-functions, we have limited cognitive resources, and we are evaluated and have to cooperate with others, etc. So I guess some âpureâ risk-aversion might be a workable satisficing heuristics to [signal you] try to avoid the worst outcomes when you canât account for all that. But thatâs not talking about utility itselfâand certainly not talking probability /â uncertainty itself.
I subtracted the reference to martingales from my previous comment because: a) not my expertise, b) this discussion doesnât need additional complexity.
Iâm sorry for having raised issues about paradoxes (perhaps there should be a Godwinâs Law about them); I donât think we should mix edge cases like St. Petersburg (and problems with unbounded utility in general) with the optimizerâs curse â itâs already hard to analyze them separately.
In line with the spirit of your comment, I believe, I think that itâs useful to recognise that not all discussions related to pros and cons of probabilities or how to use them or that sort of thing can or should address all potential issues. And I think that itâs good to recognise/âacknowledge when a certain issue or edge case actually applies more broadly than just to the particular matter at hand (e.g., how St Petersburg is relevant even aside from the optimizerâs curse). An example of roughly the sort of reasoning I mean with that second sentence, from Tarsney writing on moral uncertainty:
The third worry suggests a broader objection, that content-based normalization approach in general is vulnerable to fanaticism. Suppose we conclude that a pluralistic hybrid of Kantianism and contractarianism would give lexical priority to Kantianism, and on this basis conclude that an agent who has positive credence in Kantianism, contractarianism, and this pluralistic hybrid ought to give lexical priority to Kantianism as well. [...]
I am willing to bite the bullet on this objection, up to a point: Some value claims may simply be more intrinsically weighty than others, and in some cases absolutely so. In cases where the agentâs credence in the lexically prioritized value claim approaches zero, however, the situation begins to resemble Pascalâs Wager (Pascal, 1669), the St. Petersburg Lottery (Bernoulli, 1738), and similar cases of extreme probabilities and magnitudes that bedevil decision theory in the context of merely empirical uncertainty. It is reasonable to hope, then, that the correct decision-theoretic solution to these problems (e.g. a dismissal of ârationally negligible probabilitiesâ (Smith, 2014, 2016) or general rational permission for non-neutral risk attitudes (Buchak, 2013)) will blunt the force of the fanaticism objection.
But I certainly donât think you need to apologise for raising those issues! They are relevant and very worthy of discussionâI just donât know if theyâre in the top 7 issues Iâd discuss in this particular post, given its intended aims and my current knowledge base.
Oh, I only apologised because, well, if we start discussing about catchy paradoxes, weâll soon lose the track of our original point.
But if you enjoy it, and since it is a relevant subject, I think people use 3 broad âstrategiesâ to tackle St. Petersburg paradoxes and the like:
[epistemic status: low, but it kind makes sense]
a) âeconomistâ: âif you use a bounded version, or takes time into account, the paradox disappears: just apply a logarithmic function for diminishing returns...â
b) âphilosopherâ: âunbounded utility is weirdâ or âbeware, itâs Pascalâs Wager with objective probabilities!â
c) âstatisticianâ: âthe problem is this probability distribution, you canât apply central limit /â other theorem, or the indifference principle, or etc., and calculate its expectationâ
I now believe the statement of mine you quote was incorrect, and Iâve updated the optimizerâs curse section, primarily to remove the sentence you quoted (as I think itâs unnecessary in any case) and to alter an earlier part where I made a very similar claim so that it now says:
(I think I already knew this but just previously didnât explain it properly, leaving the conditions I had in mind as assumed, even though they often wonât hold in practice.)
But I think this updated version doesnât address the points you make. From âif weâre dealing with a bounded budgetâin which case, one might prefer a suboptimal option with low varianceâ, it sounds to me like maybe what youâre getting at is risk-aversion and/âor diminishing returns to a particular thing?
For example, letâs say I can choose either A, which gives me $1 thousand in expectation, or B, which gives me $1 million in expectation. So far, B obviously seems way better. But what if B is way higher uncertainty (or way higher risk, if one prefers that phrasing)? Then maybe Iâd prefer A.
Iâd personally consider this biased if itâs pure risk-aversion, and the dollar values perfectly correspond to my âutilityâ from this. But in reality, each additional dollar is less valuable. For example, perhaps Iâm broke, and by far the most important thing is that I get $1000 to get myself out of a real holeâa quite low chance of much higher payoffs isnât worth it, because I get far less than 1000 times as much value out of 1000 times as much money.
If thatâs what you were getting at, I think thatâs all valid, and I think the optimizerâs curse does probably magnify those reasons to sometimes not go with what you estimate will give you, in expectation, the most of some thing you value. But I think really that doesnât depend on the optimizerâs curse, and is more about uncertainty in general. Also, I think itâs really important to distinguish âmaximising expected utilityâ from âmaximising expected amount of some particular thing I valueâ. My understanding is that ârisk-aversionâ based on diminishing returns to dollars, for example, can 100% make sense within expected utility maximisationâitâs only pure risk-aversion (in terms of utility itself) that canât.
(Let me know if I was totally misunderstanding you.)
I am very satisfied with the new text. I think you understood me pretty well; the problem is, I was a little bit unclear and ambiguous.
Iâm not sure if this impacts your argument: I think diminishing returns accounts pretty well for saturation (ie., gaining $1 is not as important as losing $1); but itâs plausible to complement subjective expected utility theory with pure risk-aversion, like Lara Buchak does.
But what I actually had in mind is something like, in the extreme for unbounded utility, St. Petersburg paradox: if youâre willing to constantly bet all your budget, youâll sure end up with $0 and bankrupt. In real life, I guess that if you were constantly updating your marginal utility per dollar, this wouldnât be a problem (so I agree with youâthis is not a challenge to expected utility maximisation).
Yeah, Iâve seen mentions of Buchakâs work and one talk from her, but didnât really get it, and currently (with maybe medium confidence?) still think that, when talking about utility itself, and thus having accounted for diminishing returns and all that, one should be risk-neutral.
I hadnât heard of martingales, and have relatively limited knowledge of the St Petersburg paradox. It seems to me (low confidence) that:
things like the St Petersburg paradox and Pascalâs mugging are plausible candidates for reasons to reject standard expected utility maximisation, at least in certain edge cases, and maybe also expected value reasoning
Recognising that there are diminishing returns to many (most?) things at least somewhat blunts the force of those weird cases
Things like accepting risk aversion or rounding infinitemal probabilities to 0 may solve the problems without us having to get rid of expected value reasoning or entirely get rid of expected utility maximisation (just augment it substantially)
There are some arguments for just accepting as rational what expected utility maximisation says in these edge casesâitâs not totally clear that our aversion to the ânaive probabilisticâ answer here is valid; maybe that aversion just reflects scope neglect, or the fact that, in the St Petersburg case, thereâs the overlooked cost of it potentially taking months of continual play to earn substantial sums
I donât think these reveal problems with using EPs specifically. It seems like the same problems could occur if you talked in qualitative terms about probabilities (e.g., âat least possibleâ, âfairly good oddsâ), and in either case the âfixâ might look the same (e.g., rounding down either a quantitative or qualitative probability to 0 or to impossibility).
But it does seem that, in practice, people not using EPs are more likely to round down low probabilities to 0. This could be seen as good, for avoiding Pascalâs mugging, and/âor as bad, for a whole host of other reasons (e.g., ignoring many x-risks).
Maybe a fuller version of this post would include edge cases like that, but I know less about them, and I think they could create âissuesâ (arguably) even when one isnât using explicit probabilities anyway.
I mostly agree with you. I subtracted the reference to martingales from my previous comment because: a) not my expertise, b) this discussion doesnât need additional complexity.
Iâm sorry for having raised issues about paradoxes (perhaps there should be a Godwinâs Law about them); I donât think we should mix edge cases like St. Petersburg (and problems with unbounded utility in general) with the optimizerâs curse â itâs already hard to analyze them separately.
Pace Buchak, I agree with that, but I wouldnât say it aloud without adding caveats: in the real world, our problems are often of dynamic choice (and so one may have to think about optimal stopping and strategies, information gathering, etc.), we donât observe utility-functions, we have limited cognitive resources, and we are evaluated and have to cooperate with others, etc. So I guess some âpureâ risk-aversion might be a workable satisficing heuristics to [signal you] try to avoid the worst outcomes when you canât account for all that. But thatâs not talking about utility itselfâand certainly not talking probability /â uncertainty itself.
In line with the spirit of your comment, I believe, I think that itâs useful to recognise that not all discussions related to pros and cons of probabilities or how to use them or that sort of thing can or should address all potential issues. And I think that itâs good to recognise/âacknowledge when a certain issue or edge case actually applies more broadly than just to the particular matter at hand (e.g., how St Petersburg is relevant even aside from the optimizerâs curse). An example of roughly the sort of reasoning I mean with that second sentence, from Tarsney writing on moral uncertainty:
But I certainly donât think you need to apologise for raising those issues! They are relevant and very worthy of discussionâI just donât know if theyâre in the top 7 issues Iâd discuss in this particular post, given its intended aims and my current knowledge base.
Oh, I only apologised because, well, if we start discussing about catchy paradoxes, weâll soon lose the track of our original point.
But if you enjoy it, and since it is a relevant subject, I think people use 3 broad âstrategiesâ to tackle St. Petersburg paradoxes and the like:
[epistemic status: low, but it kind makes sense]
a) âeconomistâ: âif you use a bounded version, or takes time into account, the paradox disappears: just apply a logarithmic function for diminishing returns...â
b) âphilosopherâ: âunbounded utility is weirdâ or âbeware, itâs Pascalâs Wager with objective probabilities!â
c) âstatisticianâ: âthe problem is this probability distribution, you canât apply central limit /â other theorem, or the indifference principle, or etc., and calculate its expectationâ