Super sympathetic to the spirit of these examples, but it's my nature to pick nits.
On IVT, don't you think a lot of people are intuitively modeling these systems as discontinuous? I think cases you identify as failures to appreciate the IVT are really cases where you think people are wrongly modeling continuous systems as if they were discontinuous.
And on the taxation point, i'm not sure how rare it is for effective marginal tax rates to be greater than a hundred percent. When it happens, the main culprit is benefits that sharply cut off once you're above a given income threshold. I think it's generally recognized that that's a bad way to design benefit programs, but that doesn't mean it doesn't happen.
Re your first points, I think it's confusing because I think the real answer is that discontinuous functions are continuous enough. Eg "votes" are a discrete quantity, you can't have half a vote. But I think it basically still works even though the math isn't perfect.
I'm not sure what level of precision is correct, but the core insight is something like "if a system transitions from state A to state B as you add inputs one at a time, some specific input must be the one that causes the transition." Which sounds even more banal when put that way! But ppl don't model reality in this way for some reason.
Re taxation, I believe this is true for benefits but not for taxes alone, at least in the American system. And certainly people have been confused about both! I dunno if it's worth digging into the empirics here.
I definitely agree there's a deep mistake involved in thinking one vote/burger can't make a difference, but I've thought of it more along the lines of the informal gloss you just gave than the IVT. Basically, I've thought the right model is that in both cases you have a small chance of making a big difference, such that EV really depends on the details. (I think the voting case is interestingly different from the meat case because in the meat case all burgers are essentially on a par as far as likelihood of being a tipping point--at least given the info of a typical consumer--whereas with votes, at least in US, you have swing states vs non swing states.)
"but I've thought of it more along the lines of the informal gloss you just gave than the IVT"
I think this is related to the general concept of unknown knowns!
I think it's very uncommon for math-y beliefs that become part of your worldview to still stay in your head as a math-y belief; unless maybe you're a professional mathematician or something. It's much more common that they just become naturally integrated (same thing with the idea of differentiable functions being locally linear, most people's actual experience with probability and Bayes theorem, the pigeonhole principle, etc)
Yeah the actual probabilities are very different between swing states and non-swing states, at least for presidential elections. Local elections matter too of course, but not nearly as much, for most of the things I care about.
I actually don’t see how IVT applies in these examples. It’s not a matter of the domain being “only approximately” continuous; that’s not really a problem, in that one could go from the integers (discrete voters, burgers, etc) to the reals (arbitrary fractions of them) via interpolation. The issue is that a “tipping point” in this context is conventionally and literally a discontinuity. What we have here are step functions: for every n burgers demanded, an extra pallet of chickens (corresponding to K number of chickens) is produced; ie, (0,n] burgers maps to 1 pallet, (n,2n] maps to 2 pallets, (2n,3n] maps to 3 pallets, etc. Similarly w/malaria nets. I submit that 1) it’s highly unnatural for ppl to think that they may individually contribute to such a jump; and 2) they don’t actually think that way — they recognize they make a tiny difference but simply don’t care.
The solution would seem to be to do away w/IVT altogether, eliding the existence of these large jumps and instead emphasizing the small ones (bc in fact none of this is truly continuous!): “Hey, you individual rational agent, however infinitesimal you may think it is, you make a whole-ass increment of change w/your choice, so choose wisely!” Individuals know that however many chicken tenders they consume corresponds to an entire chicken life, and that a penny corresponds to however many square inches of a malaria net; it’s just a question of how much those things matter to them when they know the actual numbers. Likewise w/voting: in a setting where there are 2 candidates and 2N-1 voters, the N-th vote for a candidate will flip the switch for them (again a step function, 0 loss to 1 win); I think people just have a hard time wrapping their heads around being 1/N fraction responsible for a flip. Not to go all evopsych theorizing but I’d assume we mostly evolved to think of cause and effect of our own efforts as incremental — so indeed continuous or approximately so; and this sometimes falls apart in systems (society) where decisions are made in big jumps (and someone technically triggers the jump)…and sometimes it doesn’t fall apart but we just are indifferent in ways that others find objectionable.
“Example 2: People often have the intuition that altruists should be more careful with their money and more risk-sensitive than selfish people, even though the opposite is true. Altruistic people care about global welfare, so zoomed out, almost any individual altruist’s donation budget is linearly good for the world at large.”
There’s a pretty good post on the EA forum arguing that altruists should still be risk averse investors (https://forum.effectivealtruism.org/posts/cf8Dth9vpxX9ptgma/against-much-financial-risk-tolerance) . It says a lot of things, but one of the main arguments I retained from it was that the success of your investments are not fully independent of the success of other donors in the market, meaning that if the stock market performs poorly, all donors who keep their investments in stocks will be able to donate less. If this were to happen, there would be a greater marginal benefit to your smaller dollar amount being donated, meaning that risk-averse savings can be higher in expected utility.
I’m not an expert in the math here, but I think the optimal way to invest is to be exactly as risk-averse as you should be if you were the sole donor for global charities, assuming that every other donor follows that same pattern. If people are skewed toward personal EV maxing or risk aversion, you should take your risks in a way such that it moves the total market risk level towards the optimal level.
Right, the actual empirics are nuanced due to correlations with other altruists.
"I’m not an expert in the math here, but I think the optimal way to invest is to be exactly as risk-averse as you should be if you were the sole donor for global charities, assuming that every other donor follows that same pattern. If people are skewed toward personal EV maxing or risk aversion, you should take your risks in a way such that it moves the total market risk level towards the optimal level."
You can do better, in theory! Specifically if you know that other altruists are biased towards a specific risk, you can take actions that are uncorrelated with or negatively correlated with that risk. Concretely, you can, buy stocks that are less correlated with Anthropic and Meta's expected performances, or anticorrelated. (harder to do in practice than theory of course, for various reasons including but not limited to inside views on Anthropic)
> Example 3: People worry about “being pushed into a higher bracket” as if earning one more dollar could make them worse off overall. But tax liability is a continuous (piecewise linear) function of income. No additional dollar in income can result in greater than one dollar of tax liability, other than very narrow pathological cases.
i think it’s because the take-home $/hour rate for marginal hours might be less than they feel their time is worth (when it’s marginal hours that could tip the bracket, rather than increased pay per hour)
The point about these 'unknown knowns' really resonated, it feels like a natural continuation from your piece on implicit biases. What sparked your own realizaton that these mathy concepts aren't universal?
"What sparked your own realizaton that these mathy concepts aren't universal?" I think I've always known this at some level, but many of these specific concepts on from debates with people both online and offline.
The reasoning you are using is something I would mostly expect to work.
Calculus pedant time. I don't think "locally linear" has a formal definition. "Differentiable" and "smooth" are usually used to mean different things. C^1 vs C^infinity. https://en.wikipedia.org/wiki/Smoothness
There exists functions like x^2*sin(1/x)
This function has a gradient of 0 at 0. But in the vicinity of 0, the gradient ranges between +1 and -1. Alternating infinitely often.
abs(x)^1.1*sin(1/x) is even worse. Differentiable everywhere. And yet the gradients are unbounded in the vicinity of 0.
But, in reality rather than a real analysis class, you usually don't have to worry about functions like this.
Thanks! I knew about pathological continuous functions like the Weierstrass function but didn't know/remember differentiable functions can be really weird in this way too! Is this standard material covered in real analyses classes? (I took them but it's been over a decade!)
Sure there are other assumptions too, like growth of food prices continuing to be at or below inflation (or at least investment returns), population staying constant, falling, or at least not increasing very quickly, the people who need food aid as a percentage of population not increasing much, etc.
But the point I was trying to make is that this is not even *conceptually* very difficult as a problem, never mind impossible. (Also it's not at all clear that the non-donation methods are more sustainable in the long run, either).
I was referring to the line toward the end about "2000 years after a Jewish dude was nailed to a tree." Though on reading it again maybe that wasn't intended as a connection to Christmas after all, so sorry!
Thanks, yeah it was a Hitchhiker's Guide reference that I couldn't quite exactly work in perfectly. I agree making the line about Jesus's birth instead of death would be more fitting but I can't figure out how to do that while still going for everything else that line wanted to serve.
Your post is good too, and answers related and different questions! "Your decisions matter, at least a little" is a pretty important concept/ideology to hold.
I'm not sure how to best express this idea in a way that's novel, interesting, and appealing. Will have a think!
Also, you talked about ideas similar to mine in very different language, so maybe it could help others "click" with those ideas more.
I easily get most of the things on your list, but I’m really not sure why the fact that any curve if you zoom enough, will look like a straight line implies that you should not buy insurance on Small products. Could you explain how the one follows from the other? Especially because I’m really not sure why utility function would behave so differently around large losses of utility like life, compare to small losses of utility like a phone.
For a lot of people, the money-utility curve isn't a straight line. What that means is that for a lot of people, gaining 100,000 dollars offers less utility than the lost utility of losing 100,000 dollars. That's because after a certain point, gaining more money adds less and less utility.
That's why home insurance can make sense, even if it's negative EV in a strictly monetary sense. Paying $1000 today is a very different part of the utility curve than the 500,000 loss of a house.
However, take the example of a coin flip where you win $1 for heads and lose $1 for tails. At this super small scale, the value of winning and losing a dollar is essentially equal. That's what is meant by this curve being locally linear; if you're dealing with dollar values that are very close together on the curve, the difference in utility is basically linear (i.e. each dollar, or 20 dollars gained is equal to each dollar/20 dollars lost).
So, even if your money-utility curve is super risk averse, you should be risk neutral at super small scales.
I can't understand the author's reasoning well enough to disagree with it, but there's an entirely different reason which is more straightforward: you and yours aren't in a position to net the actual value of many lives versus life insurance payouts, whereas you can net smaller insurance choices (so actual value =~ EV, and you shouldn't buy them).
These are good applications. I have to admit I kind of got the theory of mind thing by reading about Myers-Briggs and all the other personality types as a teenager (yeah, I knew about the Big Five but it wasn't as much fun)--see, other people aren't like you, so you can't expect them to behave like you.
NPV: I learned about this one in econ class. Makes sense. I have, in fact, invested extensively in index funds and am therefore down with NPV. (Yeah, you know me.)
I would, however, add that Grice's maxims are mostly for purely educational purposes; most of them are commonly ignored in politics! Quantity...well, it is true you want to cut it short enough they can't meme it into something else. If you read Al Franken's book, he says the donors wouldn't take him seriously until he learned to 'pivot', i.e. ignore the maximum of relevance. Manner...well, would you describe most of our politicians (and of course I have one particular one in mind) as clear, brief, and orderly? And as for quality... "Only say what you believe to be true and can be supported." I think we all know how that one turns out.
Is it really that people don't understand that continuous utility functions are linear locally, or is it just that people want to change their behavior at some local maxima, but in real life there's often no easy way to determine where those maxima are, or if there even are local maxima? Your examples just sound like situations where people did the math wrong or didn't do it at all, but they are still looking for a maxima
Super sympathetic to the spirit of these examples, but it's my nature to pick nits.
On IVT, don't you think a lot of people are intuitively modeling these systems as discontinuous? I think cases you identify as failures to appreciate the IVT are really cases where you think people are wrongly modeling continuous systems as if they were discontinuous.
And on the taxation point, i'm not sure how rare it is for effective marginal tax rates to be greater than a hundred percent. When it happens, the main culprit is benefits that sharply cut off once you're above a given income threshold. I think it's generally recognized that that's a bad way to design benefit programs, but that doesn't mean it doesn't happen.
Re your first points, I think it's confusing because I think the real answer is that discontinuous functions are continuous enough. Eg "votes" are a discrete quantity, you can't have half a vote. But I think it basically still works even though the math isn't perfect.
I'm not sure what level of precision is correct, but the core insight is something like "if a system transitions from state A to state B as you add inputs one at a time, some specific input must be the one that causes the transition." Which sounds even more banal when put that way! But ppl don't model reality in this way for some reason.
Re taxation, I believe this is true for benefits but not for taxes alone, at least in the American system. And certainly people have been confused about both! I dunno if it's worth digging into the empirics here.
I definitely agree there's a deep mistake involved in thinking one vote/burger can't make a difference, but I've thought of it more along the lines of the informal gloss you just gave than the IVT. Basically, I've thought the right model is that in both cases you have a small chance of making a big difference, such that EV really depends on the details. (I think the voting case is interestingly different from the meat case because in the meat case all burgers are essentially on a par as far as likelihood of being a tipping point--at least given the info of a typical consumer--whereas with votes, at least in US, you have swing states vs non swing states.)
"but I've thought of it more along the lines of the informal gloss you just gave than the IVT"
I think this is related to the general concept of unknown knowns!
I think it's very uncommon for math-y beliefs that become part of your worldview to still stay in your head as a math-y belief; unless maybe you're a professional mathematician or something. It's much more common that they just become naturally integrated (same thing with the idea of differentiable functions being locally linear, most people's actual experience with probability and Bayes theorem, the pigeonhole principle, etc)
Yeah the actual probabilities are very different between swing states and non-swing states, at least for presidential elections. Local elections matter too of course, but not nearly as much, for most of the things I care about.
I actually don’t see how IVT applies in these examples. It’s not a matter of the domain being “only approximately” continuous; that’s not really a problem, in that one could go from the integers (discrete voters, burgers, etc) to the reals (arbitrary fractions of them) via interpolation. The issue is that a “tipping point” in this context is conventionally and literally a discontinuity. What we have here are step functions: for every n burgers demanded, an extra pallet of chickens (corresponding to K number of chickens) is produced; ie, (0,n] burgers maps to 1 pallet, (n,2n] maps to 2 pallets, (2n,3n] maps to 3 pallets, etc. Similarly w/malaria nets. I submit that 1) it’s highly unnatural for ppl to think that they may individually contribute to such a jump; and 2) they don’t actually think that way — they recognize they make a tiny difference but simply don’t care.
The solution would seem to be to do away w/IVT altogether, eliding the existence of these large jumps and instead emphasizing the small ones (bc in fact none of this is truly continuous!): “Hey, you individual rational agent, however infinitesimal you may think it is, you make a whole-ass increment of change w/your choice, so choose wisely!” Individuals know that however many chicken tenders they consume corresponds to an entire chicken life, and that a penny corresponds to however many square inches of a malaria net; it’s just a question of how much those things matter to them when they know the actual numbers. Likewise w/voting: in a setting where there are 2 candidates and 2N-1 voters, the N-th vote for a candidate will flip the switch for them (again a step function, 0 loss to 1 win); I think people just have a hard time wrapping their heads around being 1/N fraction responsible for a flip. Not to go all evopsych theorizing but I’d assume we mostly evolved to think of cause and effect of our own efforts as incremental — so indeed continuous or approximately so; and this sometimes falls apart in systems (society) where decisions are made in big jumps (and someone technically triggers the jump)…and sometimes it doesn’t fall apart but we just are indifferent in ways that others find objectionable.
“Example 2: People often have the intuition that altruists should be more careful with their money and more risk-sensitive than selfish people, even though the opposite is true. Altruistic people care about global welfare, so zoomed out, almost any individual altruist’s donation budget is linearly good for the world at large.”
There’s a pretty good post on the EA forum arguing that altruists should still be risk averse investors (https://forum.effectivealtruism.org/posts/cf8Dth9vpxX9ptgma/against-much-financial-risk-tolerance) . It says a lot of things, but one of the main arguments I retained from it was that the success of your investments are not fully independent of the success of other donors in the market, meaning that if the stock market performs poorly, all donors who keep their investments in stocks will be able to donate less. If this were to happen, there would be a greater marginal benefit to your smaller dollar amount being donated, meaning that risk-averse savings can be higher in expected utility.
I’m not an expert in the math here, but I think the optimal way to invest is to be exactly as risk-averse as you should be if you were the sole donor for global charities, assuming that every other donor follows that same pattern. If people are skewed toward personal EV maxing or risk aversion, you should take your risks in a way such that it moves the total market risk level towards the optimal level.
Right, the actual empirics are nuanced due to correlations with other altruists.
"I’m not an expert in the math here, but I think the optimal way to invest is to be exactly as risk-averse as you should be if you were the sole donor for global charities, assuming that every other donor follows that same pattern. If people are skewed toward personal EV maxing or risk aversion, you should take your risks in a way such that it moves the total market risk level towards the optimal level."
You can do better, in theory! Specifically if you know that other altruists are biased towards a specific risk, you can take actions that are uncorrelated with or negatively correlated with that risk. Concretely, you can, buy stocks that are less correlated with Anthropic and Meta's expected performances, or anticorrelated. (harder to do in practice than theory of course, for various reasons including but not limited to inside views on Anthropic)
> Example 3: People worry about “being pushed into a higher bracket” as if earning one more dollar could make them worse off overall. But tax liability is a continuous (piecewise linear) function of income. No additional dollar in income can result in greater than one dollar of tax liability, other than very narrow pathological cases.
i think it’s because the take-home $/hour rate for marginal hours might be less than they feel their time is worth (when it’s marginal hours that could tip the bracket, rather than increased pay per hour)
Yeah I agree in that context it's very rational not to work more!
The point about these 'unknown knowns' really resonated, it feels like a natural continuation from your piece on implicit biases. What sparked your own realizaton that these mathy concepts aren't universal?
Which piece on "implicit biases?"
"What sparked your own realizaton that these mathy concepts aren't universal?" I think I've always known this at some level, but many of these specific concepts on from debates with people both online and offline.
> Differentiable functions are locally linear
The reasoning you are using is something I would mostly expect to work.
Calculus pedant time. I don't think "locally linear" has a formal definition. "Differentiable" and "smooth" are usually used to mean different things. C^1 vs C^infinity. https://en.wikipedia.org/wiki/Smoothness
There exists functions like x^2*sin(1/x)
This function has a gradient of 0 at 0. But in the vicinity of 0, the gradient ranges between +1 and -1. Alternating infinitely often.
abs(x)^1.1*sin(1/x) is even worse. Differentiable everywhere. And yet the gradients are unbounded in the vicinity of 0.
But, in reality rather than a real analysis class, you usually don't have to worry about functions like this.
Thanks! I knew about pathological continuous functions like the Weierstrass function but didn't know/remember differentiable functions can be really weird in this way too! Is this standard material covered in real analyses classes? (I took them but it's been over a decade!)
> Sometimes people say it’s impossible to fix a perpetual problem (e.g. SF homelessness, or world hunger)
This is tricky. Under the dubious assumption of a financial market willing to give interest, you can solve world hunger "forever".
Except it's not actually forever, it's just until someone figures out how to take the money and run, or there is some bank crisis or something.
Sure there are other assumptions too, like growth of food prices continuing to be at or below inflation (or at least investment returns), population staying constant, falling, or at least not increasing very quickly, the people who need food aid as a percentage of population not increasing much, etc.
But the point I was trying to make is that this is not even *conceptually* very difficult as a problem, never mind impossible. (Also it's not at all clear that the non-donation methods are more sustainable in the long run, either).
I was referring to the line toward the end about "2000 years after a Jewish dude was nailed to a tree." Though on reading it again maybe that wasn't intended as a connection to Christmas after all, so sorry!
Thanks, yeah it was a Hitchhiker's Guide reference that I couldn't quite exactly work in perfectly. I agree making the line about Jesus's birth instead of death would be more fitting but I can't figure out how to do that while still going for everything else that line wanted to serve.
Ah, makes sense. I missed the reference, lol.
Nice list! I have a different type of nitpick though - I think you confused Christmas and Good Friday!
How so?
IVT and ToM are very nice formalizations of thing I was grasping at in a post from earlier this year. Predictably outclassed by you! https://open.substack.com/pub/frommatter/p/the-world-is-not-fake-and-your-actions?utm_campaign=post-expanded-share&utm_medium=web
Your post is good too, and answers related and different questions! "Your decisions matter, at least a little" is a pretty important concept/ideology to hold.
I'm not sure how to best express this idea in a way that's novel, interesting, and appealing. Will have a think!
Also, you talked about ideas similar to mine in very different language, so maybe it could help others "click" with those ideas more.
I easily get most of the things on your list, but I’m really not sure why the fact that any curve if you zoom enough, will look like a straight line implies that you should not buy insurance on Small products. Could you explain how the one follows from the other? Especially because I’m really not sure why utility function would behave so differently around large losses of utility like life, compare to small losses of utility like a phone.
The y axis is utility, the x axis is money
Is this clear enough or would you find it helpful for I or somebody else to elaborate?
For a lot of people, the money-utility curve isn't a straight line. What that means is that for a lot of people, gaining 100,000 dollars offers less utility than the lost utility of losing 100,000 dollars. That's because after a certain point, gaining more money adds less and less utility.
That's why home insurance can make sense, even if it's negative EV in a strictly monetary sense. Paying $1000 today is a very different part of the utility curve than the 500,000 loss of a house.
However, take the example of a coin flip where you win $1 for heads and lose $1 for tails. At this super small scale, the value of winning and losing a dollar is essentially equal. That's what is meant by this curve being locally linear; if you're dealing with dollar values that are very close together on the curve, the difference in utility is basically linear (i.e. each dollar, or 20 dollars gained is equal to each dollar/20 dollars lost).
So, even if your money-utility curve is super risk averse, you should be risk neutral at super small scales.
Thank you, appreciate the longer explanation! I hope it helps people! :)
I can't understand the author's reasoning well enough to disagree with it, but there's an entirely different reason which is more straightforward: you and yours aren't in a position to net the actual value of many lives versus life insurance payouts, whereas you can net smaller insurance choices (so actual value =~ EV, and you shouldn't buy them).
These are good applications. I have to admit I kind of got the theory of mind thing by reading about Myers-Briggs and all the other personality types as a teenager (yeah, I knew about the Big Five but it wasn't as much fun)--see, other people aren't like you, so you can't expect them to behave like you.
NPV: I learned about this one in econ class. Makes sense. I have, in fact, invested extensively in index funds and am therefore down with NPV. (Yeah, you know me.)
I would, however, add that Grice's maxims are mostly for purely educational purposes; most of them are commonly ignored in politics! Quantity...well, it is true you want to cut it short enough they can't meme it into something else. If you read Al Franken's book, he says the donors wouldn't take him seriously until he learned to 'pivot', i.e. ignore the maximum of relevance. Manner...well, would you describe most of our politicians (and of course I have one particular one in mind) as clear, brief, and orderly? And as for quality... "Only say what you believe to be true and can be supported." I think we all know how that one turns out.
Is it really that people don't understand that continuous utility functions are linear locally, or is it just that people want to change their behavior at some local maxima, but in real life there's often no easy way to determine where those maxima are, or if there even are local maxima? Your examples just sound like situations where people did the math wrong or didn't do it at all, but they are still looking for a maxima