I would posit that humans behave in a much more optimal manner in terms of long-run quality of life than are given credit for, excluding gambling addicts.
A lot of people who are willing to bet everything (ie. follow a linear utility function) are lower income. It is more that just that, however. Lower income people just by necessity have less savings relative to income, so losing all their savings isn’t a big deal compared to work-derived income. Losing a couple months of pay sucks, but eh.
People who like to think they’re being more rational by not betting the farm usually just have more to lose. If you’re a professional who accrued a few million over a few decades of work, you can’t make it back; you will invest prudently, with diversification across asset classes and markets; you might not even keep everything in one country.
What would the optimal utility function look like for someone who has a steady income? I would expect it to smoothly transition from a linear to a log regime, as the total winnings exceed the income per turn. Fun textbook exercise for stats undergrads, maybe I’ll use it sometime.
I’m not sure I’ve ever seen a treatment of utility functions that deals with this problem? (The problem being “what if your utility function is such that maximizing expected utility at time t1 doesn’t maximize expected utility at time t2?”) It’s no more a problem for Linda than for Logan, it’s just less obvious for Logan given this setup.
In economics, this is considered via the time-value of money. It is considered at the market level, however, not at the level of individuals so plausibly it could have individual variances.
What would the optimal utility function look like for someone who has a steady income?
I think I know what you mean, but personally I’d try to avoid talking about utility functions here. A utility function is the thing one optimizes with respect to, trying to choose an “optimal utility function” suggests you have something outside the utility function that you value, and in that case it’s not really a utility function.
That said I’m not sure how I would ask the question myself. Maybe something about optimal levels of risk aversion?
In economics, this is considered via the time-value of money.
So this doesn’t really deal with the problem I’m thinking of.
I think what you’re thinking is: instead of having a utility function that’s (say) linear in money, you’d have it be linear in money and negative-exponential in time. So instead of U(m)=m, you’d have something isomorphic to U(m,t)=m⋅2−t. And so U(1,0)=U(2,1)=1.
But does that mean such an agent is indifferent between receiving £0.01 now and £0.02 in a second? That’s not obvious to me, because if they’re making the decision at time 0 they need to choose between “Utility U(1,0)=1 now and utility U(1,1)=1/2 in one second”; and “utility U(0,0)=0 now and utility U(2,1)=1 in one second”. Which do they choose? The fact that the “now” in one choice equals the “later” in another doesn’t answer that question for me.
(We can postulate that they might be able to use £0.01 at time 0 to have more than £0.01 at time 1. But that makes things more complicated, not less. I feel like if we want to claim we can answer questions about expected utility maximizers, we should be able to answer them in simple situations.)
And then there are even weirder cases, like what if we have an agent whose utility is U(m,t)=msin(t)?
I can imagine answers like “you integrate the utility function over all of time” or “you take the max value” or “the limit as t→∞”, but then it seems to me that that is the actual utility function? And also all of those possibilities will diverge in a lot possible situations. Now that I bring this up I have a vague feeling I’ve seen this sort of thing discussed? (I admittedly haven’t gone looking for explanations of VNM utility.)
The sort of direction I’d try to explore myself is: so the idea behind a utility function is that if your behavior regarding certain bets satisfies the VNM axioms, then we can model you as having a utility function that you’re maximizing. Okay, so let’s take an agent who chooses £0.01 now; what utility function do we derive for them? And now the same for the agent who chooses £0.02 later.
My weak guess is that the derivation of U from the bets an agent will take, assumes that for any given bet the agent will be maximizing their instantaneous U. And so if we postulate an agent who takes future money into account, then the utility function we derive for them will itself have a term for future money.
And then we could actually still have U(m,t)=m⋅2−t. It would just be a different interpretation than I gave it above: instead of “at time t, my utility given that I have m money is...”, it would be “right now, my utility from receiving m money at future time t is...”.
This doesn’t seem to me like it obviously causes big problems. Maybe this is all just standard when people study this more formally than I have. But I’m not sure.
So the reason why the time value of money works, and it makes sense to say that we can say that the utility of $1000 today and $1050 in a year are about the same, is because of the existence of the wider financial system. In other words, this isn’t necessarily true in a vacuum; however if I wanted $1050 in a year, I can invest the $1000 I have right now into 1 year treasuries. The converse is more complex; if I am guaranteed $1050 in a year I may not be able to get a loan for $1000 right now from a bank because I’m not the fed and loans to me have a higher interest rate, but perhaps I can play some tricks on the options market to do this? At any rate, I can get pretty close if I were getting an asset-backed loan, such as a mortgage.
Note that I’m not saying that actors are indifferent to which option they get, but that it is viewed with equal utility (when discounted by your cost of financing, basically).
This is a bit of a cop-out, but I would say modelling the utility of money without considering the wider world is a bit silly anyway, because money only has value due to its use as a medium of exchange and as a store of value, both of which depend on the existence of the rest of the world. The utility of money thus cannot be truly divorced from the influence of eg. finance.
Note that I’m not saying that actors are indifferent to which option they get, but that it is viewed with equal utility
I think this is precisely what “equal utility” means in context.
To be clear, in this post I’m trying to talk about expected utility maximizers, the simple mathematical abstraction of “agent who has a utility function (which satisfies certain technical conditions) and attempts to maximize its expected value”. And the reason I’m trying to talk about that type of agent is because I think the things I’m replying to are also trying to talk about that type of agent.
Possibly it would be clearer to simply leave “money” out of it, but reusing examples from prior art seems useful. Also I think it makes the post less dry. Perhaps I should have started with a big disclaimer that I’m trying to talk about expected utility maximizers and references to a thing labeled “money” are not intended to invoke concepts like “global economy” or “purchasing goods and services”. I’m talking about a hypothetical agent who values this thing labeled “money” for its own sake.
The reason I’m talking about that type of agent is because I think understanding that type of agent can be useful when we try to think about more-realistic agents, who more-realistically value a real-world thing called “money” that exists in a global economy and can be used to purchase goods and services. But those more-realistic agents, and that more-realistic money, are not what I’m talking about currently. And I’m not here trying to justify why I think that can be useful.
(Or maybe the things I’m replying to aren’t trying to talk about that type of agent, they just use words like “expected utility” without intending to point at their technical definition and that confuses things. But if that’s the case, then I’m probably not the only person who thinks that’s what they’re trying to talk about; and so it still seems good for me to clarify what happens with that type of agent, in the sort of situation in question.)
(Probably it would be good for me to find some examples of the things I’m responding to, but that would be a different type of effort than I feel like putting in currently.)
I would posit that humans behave in a much more optimal manner in terms of long-run quality of life than are given credit for, excluding gambling addicts.
A lot of people who are willing to bet everything (ie. follow a linear utility function) are lower income. It is more that just that, however. Lower income people just by necessity have less savings relative to income, so losing all their savings isn’t a big deal compared to work-derived income. Losing a couple months of pay sucks, but eh.
People who like to think they’re being more rational by not betting the farm usually just have more to lose. If you’re a professional who accrued a few million over a few decades of work, you can’t make it back; you will invest prudently, with diversification across asset classes and markets; you might not even keep everything in one country.
What would the optimal utility function look like for someone who has a steady income? I would expect it to smoothly transition from a linear to a log regime, as the total winnings exceed the income per turn. Fun textbook exercise for stats undergrads, maybe I’ll use it sometime.
In economics, this is considered via the time-value of money. It is considered at the market level, however, not at the level of individuals so plausibly it could have individual variances.
I think I know what you mean, but personally I’d try to avoid talking about utility functions here. A utility function is the thing one optimizes with respect to, trying to choose an “optimal utility function” suggests you have something outside the utility function that you value, and in that case it’s not really a utility function.
That said I’m not sure how I would ask the question myself. Maybe something about optimal levels of risk aversion?
So this doesn’t really deal with the problem I’m thinking of.
I think what you’re thinking is: instead of having a utility function that’s (say) linear in money, you’d have it be linear in money and negative-exponential in time. So instead of U(m)=m, you’d have something isomorphic to U(m,t)=m⋅2−t. And so U(1,0)=U(2,1)=1.
But does that mean such an agent is indifferent between receiving £0.01 now and £0.02 in a second? That’s not obvious to me, because if they’re making the decision at time 0 they need to choose between “Utility U(1,0)=1 now and utility U(1,1)=1/2 in one second”; and “utility U(0,0)=0 now and utility U(2,1)=1 in one second”. Which do they choose? The fact that the “now” in one choice equals the “later” in another doesn’t answer that question for me.
(We can postulate that they might be able to use £0.01 at time 0 to have more than £0.01 at time 1. But that makes things more complicated, not less. I feel like if we want to claim we can answer questions about expected utility maximizers, we should be able to answer them in simple situations.)
And then there are even weirder cases, like what if we have an agent whose utility is U(m,t)=msin(t)?
I can imagine answers like “you integrate the utility function over all of time” or “you take the max value” or “the limit as t→∞”, but then it seems to me that that is the actual utility function? And also all of those possibilities will diverge in a lot possible situations. Now that I bring this up I have a vague feeling I’ve seen this sort of thing discussed? (I admittedly haven’t gone looking for explanations of VNM utility.)
The sort of direction I’d try to explore myself is: so the idea behind a utility function is that if your behavior regarding certain bets satisfies the VNM axioms, then we can model you as having a utility function that you’re maximizing. Okay, so let’s take an agent who chooses £0.01 now; what utility function do we derive for them? And now the same for the agent who chooses £0.02 later.
My weak guess is that the derivation of U from the bets an agent will take, assumes that for any given bet the agent will be maximizing their instantaneous U. And so if we postulate an agent who takes future money into account, then the utility function we derive for them will itself have a term for future money.
And then we could actually still have U(m,t)=m⋅2−t. It would just be a different interpretation than I gave it above: instead of “at time t, my utility given that I have m money is...”, it would be “right now, my utility from receiving m money at future time t is...”.
This doesn’t seem to me like it obviously causes big problems. Maybe this is all just standard when people study this more formally than I have. But I’m not sure.
So the reason why the time value of money works, and it makes sense to say that we can say that the utility of $1000 today and $1050 in a year are about the same, is because of the existence of the wider financial system. In other words, this isn’t necessarily true in a vacuum; however if I wanted $1050 in a year, I can invest the $1000 I have right now into 1 year treasuries. The converse is more complex; if I am guaranteed $1050 in a year I may not be able to get a loan for $1000 right now from a bank because I’m not the fed and loans to me have a higher interest rate, but perhaps I can play some tricks on the options market to do this? At any rate, I can get pretty close if I were getting an asset-backed loan, such as a mortgage.
Note that I’m not saying that actors are indifferent to which option they get, but that it is viewed with equal utility (when discounted by your cost of financing, basically).
This is a bit of a cop-out, but I would say modelling the utility of money without considering the wider world is a bit silly anyway, because money only has value due to its use as a medium of exchange and as a store of value, both of which depend on the existence of the rest of the world. The utility of money thus cannot be truly divorced from the influence of eg. finance.
I think this is precisely what “equal utility” means in context.
To be clear, in this post I’m trying to talk about expected utility maximizers, the simple mathematical abstraction of “agent who has a utility function (which satisfies certain technical conditions) and attempts to maximize its expected value”. And the reason I’m trying to talk about that type of agent is because I think the things I’m replying to are also trying to talk about that type of agent.
Possibly it would be clearer to simply leave “money” out of it, but reusing examples from prior art seems useful. Also I think it makes the post less dry. Perhaps I should have started with a big disclaimer that I’m trying to talk about expected utility maximizers and references to a thing labeled “money” are not intended to invoke concepts like “global economy” or “purchasing goods and services”. I’m talking about a hypothetical agent who values this thing labeled “money” for its own sake.
The reason I’m talking about that type of agent is because I think understanding that type of agent can be useful when we try to think about more-realistic agents, who more-realistically value a real-world thing called “money” that exists in a global economy and can be used to purchase goods and services. But those more-realistic agents, and that more-realistic money, are not what I’m talking about currently. And I’m not here trying to justify why I think that can be useful.
(Or maybe the things I’m replying to aren’t trying to talk about that type of agent, they just use words like “expected utility” without intending to point at their technical definition and that confuses things. But if that’s the case, then I’m probably not the only person who thinks that’s what they’re trying to talk about; and so it still seems good for me to clarify what happens with that type of agent, in the sort of situation in question.)
(Probably it would be good for me to find some examples of the things I’m responding to, but that would be a different type of effort than I feel like putting in currently.)