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.
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.