tl;dr: If you have less than ~$13k saved and have only enough income to meet expenses, picking B might legitimately make you sad even if it’s correct.
I’d take B every time. But it depends on your financial situation. If the stakes are small relative to my reference wealth I maximize expected dollars, no regrets, and if you can’t without regrets then maybe try playing poker for a while until you can be happy with good decisions that result in bad outcomes because of randomness. You may not make that exact decision again, but you make decisions like it plenty often.
If the stakes are large relative to my reference wealth then the situation changes for two reasons. One, I probably won’t have the opportunity to take bets with stakes large relative to my wealth very often. Two, change in utility is no longer approximately proportional to change in dollars. Perhaps $240 is a non-trivial amount to you? For a hypothetical person living in an average American city with $50k saved and an annual income of $50k, an additional $240 is in no way life changing, so dU(+$240) ~= 0.24 dU(+$1000) and they should pick B. But with 1000x the stakes, it’s entirely possible that dU(+$240,000) >> 0.25 dU(+$1,000,000).
Another way of looking at this is investing with Kelly Criterion (spherical cow but still useful), which says if you start with $50k and no other annual income and have the opportunity to periodically pay $24x for a 25% chance at $100x, you should start by betting ~$657 a pop for maximum growth rate, which is within shouting distance of the proposed $240 - and KC is well known to be too risky for individuals without massive wealth, as people actually have to spend money during low periods. This is proportional to wealth, so the breakeven wealth before you’re sad that you have to bet so much at once ($240) is, under the too-risky KC, about $18k, which means actually it’s probably like $10k-$15k.
I have very little intuition how this translates if, for example, you have heaps of student load debt and are still trying to finish education in hopes of obtaining a promised-but-who-knows well paying job in a few years.
If you have a log utility function (which the KC maximizes), you can calculate the breakeven starting wealth with this formula.25%2Blog(x).75%3Dlog(x%2B240)).
Pretty much this; if we adjust the numbers to “A: 20 cents or B: 25% chance for 100 cents” then I’d take the option B, but scale it up to “A: $200,000 or B: 25% chance for $1,000,000″, and I’d take option A. Because $0 is 0 points, $1 million is something like 4 points, and $200,000 is about 2 points.
Human perception of scale for money is not linear (but not logarithmic either… not log 10, anyway, maybe log somethingelse). And since I’m running this flawed hardware...
Some of it was pointed out already as “prospect theory” but that seems to be more about perception of probability rather than the perception of the actual reward.
Log to base 10 and log to base anything_else differ only by a scale factor, so if you find log-to-base-10 unsatisfactory you won’t like any other sort of log much better.
It sounds like you’re suggesting that utility falls off slower than log(wealth) or log(income). I think there’s quite good evidence that it doesn’t—but if you’re looking at smallish changes in wealth or income then of course you can get that smaller falloff in change in utility. E.g., if you start off with $66,666 and gain $200k then your wealth has gone up by a factor of 4; if you gain $1M instead then your wealth has gone up by a factor of 16. If your logs are to base 2, you get exactly the numbers you described.
The right figure to use for “wealth” there may well not be exactly your total net wealth; it should probably include some figure for “effective wealth” arising from your social context—family and friends, government-funded safety net, etc. It seems like that should probably be at least a few tens of kilobucks in prosperous Western countries.
tl;dr: If you have less than ~$13k saved and have only enough income to meet expenses, picking B might legitimately make you sad even if it’s correct.
I’d take B every time. But it depends on your financial situation. If the stakes are small relative to my reference wealth I maximize expected dollars, no regrets, and if you can’t without regrets then maybe try playing poker for a while until you can be happy with good decisions that result in bad outcomes because of randomness. You may not make that exact decision again, but you make decisions like it plenty often.
If the stakes are large relative to my reference wealth then the situation changes for two reasons. One, I probably won’t have the opportunity to take bets with stakes large relative to my wealth very often. Two, change in utility is no longer approximately proportional to change in dollars. Perhaps $240 is a non-trivial amount to you? For a hypothetical person living in an average American city with $50k saved and an annual income of $50k, an additional $240 is in no way life changing, so dU(+$240) ~= 0.24 dU(+$1000) and they should pick B. But with 1000x the stakes, it’s entirely possible that dU(+$240,000) >> 0.25 dU(+$1,000,000).
Another way of looking at this is investing with Kelly Criterion (spherical cow but still useful), which says if you start with $50k and no other annual income and have the opportunity to periodically pay $24x for a 25% chance at $100x, you should start by betting ~$657 a pop for maximum growth rate, which is within shouting distance of the proposed $240 - and KC is well known to be too risky for individuals without massive wealth, as people actually have to spend money during low periods. This is proportional to wealth, so the breakeven wealth before you’re sad that you have to bet so much at once ($240) is, under the too-risky KC, about $18k, which means actually it’s probably like $10k-$15k.
I have very little intuition how this translates if, for example, you have heaps of student load debt and are still trying to finish education in hopes of obtaining a promised-but-who-knows well paying job in a few years.
If you have a log utility function (which the KC maximizes), you can calculate the breakeven starting wealth with this formula.25%2Blog(x).75%3Dlog(x%2B240)).
Pretty much this; if we adjust the numbers to “A: 20 cents or B: 25% chance for 100 cents” then I’d take the option B, but scale it up to “A: $200,000 or B: 25% chance for $1,000,000″, and I’d take option A. Because $0 is 0 points, $1 million is something like 4 points, and $200,000 is about 2 points.
Human perception of scale for money is not linear (but not logarithmic either… not log 10, anyway, maybe log somethingelse). And since I’m running this flawed hardware...
Some of it was pointed out already as “prospect theory” but that seems to be more about perception of probability rather than the perception of the actual reward.
Log to base 10 and log to base anything_else differ only by a scale factor, so if you find log-to-base-10 unsatisfactory you won’t like any other sort of log much better.
It sounds like you’re suggesting that utility falls off slower than log(wealth) or log(income). I think there’s quite good evidence that it doesn’t—but if you’re looking at smallish changes in wealth or income then of course you can get that smaller falloff in change in utility. E.g., if you start off with $66,666 and gain $200k then your wealth has gone up by a factor of 4; if you gain $1M instead then your wealth has gone up by a factor of 16. If your logs are to base 2, you get exactly the numbers you described.
The right figure to use for “wealth” there may well not be exactly your total net wealth; it should probably include some figure for “effective wealth” arising from your social context—family and friends, government-funded safety net, etc. It seems like that should probably be at least a few tens of kilobucks in prosperous Western countries.