The reason is that, for me, changing my wealth by a relatively small amount won’t radically change my risk preference, and that implies an exponential curve
Is that because there have been points in time when you have made 200K and 400K respectively and found that your preferences didn’t change much. Or is that simply expected utility?
For the specific quote: I know that, for a small enough change in wealth, I don’t need to re-evaluate all the deals I own. They all remain pretty much the same. For example, if you told me a had $100 more in my bank account, I would be happy, but it wouldn’t significantly change any of my decisions involving risk. For a utility curve over money, you can prove that that implies an exponential curve. Intuitively, some range of my utility curve can be approximated by an exponential curve.
Now that I know it is exponential over some range, I needed to figure out which exponential and over what range does it apply. I assessed for myself that I am indifferent between having and not having a deal with a 50-50 chance of winning $400K and losing $200K. The way I thought about that was how I thought about decisions around job hunting and whether I should take or not take job offers that had different salaries.
If that is true, you can combine it with the above and show that the exponential curve should look like u(x) = 1 - exp(-x/400K). Testing it against my intuitions, I find it an an okay approximation between $400K and minus $200K. Outside that range, I need better approximations (e.g. if you try it out on a 50-50 shot of $10M, it gives ridiculous answers).
It makes sense however you mention that you test it against your intuitions. My first reaction would be to say that this is introducing a biased variable which is not based on a reasonable calculation.
That may not be the case as you may have done so many complicated calculations such that your unconscious “intuitions” may give your conscious the right answer. However from the millionaires biographies I have read and rich people I have talked to a better representation of money and utility according to them is logarithmic rather than exponential. This would indicate to me that the relationship between utility and money would be counter-intuitive for those who have not experienced those levels which are being compared.
I have not had the fortune to experience anything more than a 5 figure income so I cannot reasonably say how my preferences would be modeled. I can reasonably believe that I would be better off at 500K than 50K through simple comparison of lifestyle between myself and a millionaire. I cannot make an accurate enough estimation of my utility and as a result I would not be prepared to make a estimation of what model would best represent it because the probability of that being accurate is likely the same as coin flipping.
Ed: I had a much better written post but an errant click lost the whole thing—time didn’t allow the repetition of the better post.
As I said in my original post, for larger ranges, I like logarithmic-type u-curves better than exponential, esp. for gains. The problem with e.g. u(x)=ln(x) where x is your total wealth is that you must be indifferent between your current wealth and a 50-50 shot of doubling vs. halving your wealth. I don’t like that deal, so I must not have that curve.
Note that a logarithmic curve can be approximated by a straight line for some small range around your current wealth. It can also be approximated by an exponential for a larger range. So even if I were purely logarithmic, I would still act risk neutral for small deals and would act exponential for somewhat larger deals. Only for very large deals indeed would you be able to identify that I was really logarithmic.
Further to this, it’s also worth pointing out that, to the extent that Andew’s biographies and rich acquaintances are talking about a logarithmic experienced utility function that maps wealth into a mind state something like “satisfaction”, this doesn’t directly imply anything about the shape of the decision utility function they should use to represent their preferences over gambles.
It’s only if they’re also risk neutral with respect to experienced utility that the implied decision utility function needs to be log(x). If they’re risk averse with respect to experienced utility then their decision utility function will be a concave function of log(x), while if they’re risk loving it will be a convex function of it.
P.S. For more on the distinction between experienced and decision utility (which I seem constantly to be harping on about) see: Kahneman, Wakker and Sarin (1997) “Back to Bentham? Explorations of Experienced Utility”
It’s only if they’re also risk neutral with respect to experienced utility
I am curious how this would look in terms of decisions under experience. Does this imply that they are expecting to change their risk assessment once they are experienced?
I’m afraid I have no idea what you mean, perhaps because I failed to adequately explain the distinction between experienced utility and decision utility, and you’ve taken it to mean something else entirely. Roughly: experienced utility is something you experience or feel (e.g. positive emotions); decision utility is an abstract function that describes the decisions you make, without necessarily corresponding to anything you actually experience.
Follow the link I gave, or see my earlier comment here (experienced utility is 1., decision utility is 2.)
Apologies if I’m failing to understand you for some other reason, such as not having slept. ;)
Unfortunately the better parts of my post were lost—or rather more of the main point.
I posit that the utility valuation is an impossibility currently. I was not really challenging whether your function was exponential or logarithmic—but questioning how you came to the conclusion; how you decide, for instance where exactly the function changes especially having not experienced the second state. The “logarithmic” point I was making was designed to demonstrate that true utility may differ significantly from expected utility once you are actually at point 2 and thus may not be truly representative.
Mainly I am curious as to what value you place on “intuition” and why.
If you wanted to, we could assess at least a part of your u-curve. That might show you why it isn’t an impossibility, and show what it means to test it by intuitions.
Would you, right now, accept a deal with a 50-50 chance of winning $100 versus losing $50?
If you answer yes, then we know something about your u-curve. For example, over a range at least as large as (100, −50), it can be approximated by an exponential curve with a risk tolerance parameter of greater than 100 (if it were less that 100, then you wouldn’t accept the above deal).
Here, I have assessed something about your u-curve by asking you a question that it seems fairly easy to answer. That’s all I mean by “testing against intuitions.” By asking a series of similar questions I can assess your u-curve over whatever range you would like.
You also might want to do calculations: for example, $10K per year forever is worth around $300K or so. Thinking about losing or gaining $10K per year for the rest of your life might be easier than thinking about gaining or losing $200-300K.
I think this greatly oversimplifies the issue. Whatever my response to the query is, it is only an estimation as to my preferences. It also assumes that my predicted risk will, upon the enactment of an actual deal, stay the same; if only for the life of the deal.
A model like this, even if correct for right now, could be significantly different tomorrow or the next day. It could be argued that some risk measurements do not change at intervals so fast as would technically prohibit recalculation. Giving a fixed metric puts absolutes on behaviors which are not fixed, or which unpredictably change. Today, because I have lots of money in my account, I might agree to your deal. Tomorrow I may not. This is what I mean by intuitions—I may think I want the deal but I may in reality be significantly underestimating the chance of −50 or any other number of factors which may skew my perception.
I know of quite a few examples of people getting stuck in high load mutual funds or other investments because their risk preferences significantly changed over a much shorter time period than they expected because they really didn’t want to take that much risk in their portfolio but could not cognitively comprehend the probability as most people cannot.
This in no way advocates going further to correcting for these mistakes after the fact—however the tendencies for economists and policy makers is to suggest modeling such as this. In fact most consequentialists make the case that modeling this way is accurate however I have yet to see a true epistemic study of a model which reliably demonstrates accurate “utility” or valuation. The closest to accurate models I have seen take stated and reveled preferences together and work towards a micro estimation which still has moderate error variability where not observed (http://ideas.repec.org/a/wly/hlthec/v13y2004i6p563-573.html). Even with observed behavior applied it is still terribly difficult and unreliable to apply broadly—even to an individual.
Just to be clear, you know that an exponential utility function (somewhat misleadingly ) doesn’t actually imply that utility is exponential in wealth, right? Bill’s claimed utility function doesn’t exhibit increasing marginal utility, if that’s what you’re intuitively objecting to. It’s 1-exp(-x), not exp(x).
Many people do find the constant absolute risk aversion implied by exponential utility functions unappealing, and prefer isoelastic utility functions that exhibit constant relative risk aversion, but it does have the advantage of tractability, and may be reasonable over some ranges.
Example of the “unappealingness” of constant absolute risk aversion. Say my u-curve were u(x) = 1-exp(-x/400K) over all ranges. What is my value for a 50-50 shot at 10M?
Answer: around $277K. (Note that it is the same for a 50-50 shot at $100M)
Given the choice, I would certainly choose a 50-50 shot at $10M over $277K. This is why over larger ranges, I don’t use an exponential u-curve.
However, it is a good approximation over a range that contains almost all the decisions I have to make. Only for huge decisions to I need to drag out a more complicated u-curve, and they are rare.
Just to be clear, you know that he means negative exponential, right? His claimed utility function doesn’t exhibit increasing marginal utility, if that’s what you’re intuitively objecting to.
(If that’s not what you’re intuitively objecting to, then is there a specific aspect of the negative exponential that you find unappealing?)
How have you come to these conclusions?
For example:
Is that because there have been points in time when you have made 200K and 400K respectively and found that your preferences didn’t change much. Or is that simply expected utility?
For the specific quote: I know that, for a small enough change in wealth, I don’t need to re-evaluate all the deals I own. They all remain pretty much the same. For example, if you told me a had $100 more in my bank account, I would be happy, but it wouldn’t significantly change any of my decisions involving risk. For a utility curve over money, you can prove that that implies an exponential curve. Intuitively, some range of my utility curve can be approximated by an exponential curve.
Now that I know it is exponential over some range, I needed to figure out which exponential and over what range does it apply. I assessed for myself that I am indifferent between having and not having a deal with a 50-50 chance of winning $400K and losing $200K. The way I thought about that was how I thought about decisions around job hunting and whether I should take or not take job offers that had different salaries.
If that is true, you can combine it with the above and show that the exponential curve should look like u(x) = 1 - exp(-x/400K). Testing it against my intuitions, I find it an an okay approximation between $400K and minus $200K. Outside that range, I need better approximations (e.g. if you try it out on a 50-50 shot of $10M, it gives ridiculous answers).
Does this make sense?
It makes sense however you mention that you test it against your intuitions. My first reaction would be to say that this is introducing a biased variable which is not based on a reasonable calculation.
That may not be the case as you may have done so many complicated calculations such that your unconscious “intuitions” may give your conscious the right answer. However from the millionaires biographies I have read and rich people I have talked to a better representation of money and utility according to them is logarithmic rather than exponential. This would indicate to me that the relationship between utility and money would be counter-intuitive for those who have not experienced those levels which are being compared.
I have not had the fortune to experience anything more than a 5 figure income so I cannot reasonably say how my preferences would be modeled. I can reasonably believe that I would be better off at 500K than 50K through simple comparison of lifestyle between myself and a millionaire. I cannot make an accurate enough estimation of my utility and as a result I would not be prepared to make a estimation of what model would best represent it because the probability of that being accurate is likely the same as coin flipping.
Ed: I had a much better written post but an errant click lost the whole thing—time didn’t allow the repetition of the better post.
As I said in my original post, for larger ranges, I like logarithmic-type u-curves better than exponential, esp. for gains. The problem with e.g. u(x)=ln(x) where x is your total wealth is that you must be indifferent between your current wealth and a 50-50 shot of doubling vs. halving your wealth. I don’t like that deal, so I must not have that curve.
Note that a logarithmic curve can be approximated by a straight line for some small range around your current wealth. It can also be approximated by an exponential for a larger range. So even if I were purely logarithmic, I would still act risk neutral for small deals and would act exponential for somewhat larger deals. Only for very large deals indeed would you be able to identify that I was really logarithmic.
Further to this, it’s also worth pointing out that, to the extent that Andew’s biographies and rich acquaintances are talking about a logarithmic experienced utility function that maps wealth into a mind state something like “satisfaction”, this doesn’t directly imply anything about the shape of the decision utility function they should use to represent their preferences over gambles.
It’s only if they’re also risk neutral with respect to experienced utility that the implied decision utility function needs to be log(x). If they’re risk averse with respect to experienced utility then their decision utility function will be a concave function of log(x), while if they’re risk loving it will be a convex function of it.
P.S. For more on the distinction between experienced and decision utility (which I seem constantly to be harping on about) see: Kahneman, Wakker and Sarin (1997) “Back to Bentham? Explorations of Experienced Utility”
I am curious how this would look in terms of decisions under experience. Does this imply that they are expecting to change their risk assessment once they are experienced?
I’m afraid I have no idea what you mean, perhaps because I failed to adequately explain the distinction between experienced utility and decision utility, and you’ve taken it to mean something else entirely. Roughly: experienced utility is something you experience or feel (e.g. positive emotions); decision utility is an abstract function that describes the decisions you make, without necessarily corresponding to anything you actually experience.
Follow the link I gave, or see my earlier comment here (experienced utility is 1., decision utility is 2.)
Apologies if I’m failing to understand you for some other reason, such as not having slept. ;)
Unfortunately the better parts of my post were lost—or rather more of the main point.
I posit that the utility valuation is an impossibility currently. I was not really challenging whether your function was exponential or logarithmic—but questioning how you came to the conclusion; how you decide, for instance where exactly the function changes especially having not experienced the second state. The “logarithmic” point I was making was designed to demonstrate that true utility may differ significantly from expected utility once you are actually at point 2 and thus may not be truly representative.
Mainly I am curious as to what value you place on “intuition” and why.
If you wanted to, we could assess at least a part of your u-curve. That might show you why it isn’t an impossibility, and show what it means to test it by intuitions.
Would you, right now, accept a deal with a 50-50 chance of winning $100 versus losing $50?
If you answer yes, then we know something about your u-curve. For example, over a range at least as large as (100, −50), it can be approximated by an exponential curve with a risk tolerance parameter of greater than 100 (if it were less that 100, then you wouldn’t accept the above deal).
Here, I have assessed something about your u-curve by asking you a question that it seems fairly easy to answer. That’s all I mean by “testing against intuitions.” By asking a series of similar questions I can assess your u-curve over whatever range you would like.
You also might want to do calculations: for example, $10K per year forever is worth around $300K or so. Thinking about losing or gaining $10K per year for the rest of your life might be easier than thinking about gaining or losing $200-300K.
I think this greatly oversimplifies the issue. Whatever my response to the query is, it is only an estimation as to my preferences. It also assumes that my predicted risk will, upon the enactment of an actual deal, stay the same; if only for the life of the deal.
A model like this, even if correct for right now, could be significantly different tomorrow or the next day. It could be argued that some risk measurements do not change at intervals so fast as would technically prohibit recalculation. Giving a fixed metric puts absolutes on behaviors which are not fixed, or which unpredictably change. Today, because I have lots of money in my account, I might agree to your deal. Tomorrow I may not. This is what I mean by intuitions—I may think I want the deal but I may in reality be significantly underestimating the chance of −50 or any other number of factors which may skew my perception.
I know of quite a few examples of people getting stuck in high load mutual funds or other investments because their risk preferences significantly changed over a much shorter time period than they expected because they really didn’t want to take that much risk in their portfolio but could not cognitively comprehend the probability as most people cannot.
This in no way advocates going further to correcting for these mistakes after the fact—however the tendencies for economists and policy makers is to suggest modeling such as this. In fact most consequentialists make the case that modeling this way is accurate however I have yet to see a true epistemic study of a model which reliably demonstrates accurate “utility” or valuation. The closest to accurate models I have seen take stated and reveled preferences together and work towards a micro estimation which still has moderate error variability where not observed (http://ideas.repec.org/a/wly/hlthec/v13y2004i6p563-573.html). Even with observed behavior applied it is still terribly difficult and unreliable to apply broadly—even to an individual.
Just to be clear, you know that an exponential utility function (somewhat misleadingly ) doesn’t actually imply that utility is exponential in wealth, right? Bill’s claimed utility function doesn’t exhibit increasing marginal utility, if that’s what you’re intuitively objecting to. It’s 1-exp(-x), not exp(x).
Many people do find the constant absolute risk aversion implied by exponential utility functions unappealing, and prefer isoelastic utility functions that exhibit constant relative risk aversion, but it does have the advantage of tractability, and may be reasonable over some ranges.
Example of the “unappealingness” of constant absolute risk aversion. Say my u-curve were u(x) = 1-exp(-x/400K) over all ranges. What is my value for a 50-50 shot at 10M?
Answer: around $277K. (Note that it is the same for a 50-50 shot at $100M)
Given the choice, I would certainly choose a 50-50 shot at $10M over $277K. This is why over larger ranges, I don’t use an exponential u-curve.
However, it is a good approximation over a range that contains almost all the decisions I have to make. Only for huge decisions to I need to drag out a more complicated u-curve, and they are rare.
Just to be clear, you know that he means negative exponential, right? His claimed utility function doesn’t exhibit increasing marginal utility, if that’s what you’re intuitively objecting to.
(If that’s not what you’re intuitively objecting to, then is there a specific aspect of the negative exponential that you find unappealing?)