The key assumption that leads to problems in trying to descriptively model people’s decisions is just that people have a single consistent utility function, which is defined in terms of the amount of money that they have.
If someone starts with $18,000 and then gets $40, the assumption is that the benefit can be expressed as U(18,040)-U(18,000). Or, in words, the person thinks: getting $40 brought me from a world where I have $18,000 to a world where I have $18,040. I value a world where I have $18,000 at this amount, and I value a world where I have $18,040 at that amount, and the benefit of getting the $40 is just the difference between those two.
In this model, there is a single curve that you can plot of how much you value a world where you had $X. Think about what that curve would look like, with X ranging from 10,000 to 100,000. In nearly every plausible case, that curve will be close to linear on a small scale (in the range of 10s or 100s) almost everywhere. There may be some curvature to it (perhaps your curve resembles f(x)=log(x)), but if you zoom in then that will mostly go away and a straight line will give you a good fit (e.g., if you are looking at changes in x that are 2 orders of magnitude smaller than x, then a log function will look pretty much linear). In a few special cases, there may be a large sudden jump in the curve, if there is some specific thing that you really want to buy and there is a large benefit to suddenly being able to afford it, but those cases are rare. For the most part, U(x) will be relatively smooth, and it will be growing perceptibly over the whole range (even if your utility function is bounded, it’s not like it will be almost at that bound before you even have $100,000).
And if your curve is approximately linear over small scales, then expected utility theory basically reduces to expected value theory when the stakes are small (e.g., a 50% of gaining $40 has an EV of $20). If U(x) is close to linear from x=18,000 to x=18,040, then U(18,020) must be about halfway in between U(18,000) and U(18,040). If you have $18,000, and are basing your decisions on a consistent utility function U(x), then for pretty much any plausible U(x) you’ll prefer a 51% chance of gaining $40 to a 100% chance of gaining $20 (unless you just happen to have one of those rare big jumps in U(x) between $18,000 and $18,020 - perhaps you really really want something that costs $18,010?). The expected value is 2% higher ($20.40 vs. $20), and it’s not plausible that your U(x) would be so sharply curved that you’d be willing to give up 2% EV over such a narrow range of x (it’s just a 0.2% increase in x from $18,000 to $18,040).
Probably the most important feature of prospect theory is that it does away with this assumption of a single consistent utility function, and says that people value gambles based on the change from the status quo (or occasionally some other reference point, but we’ll ignore that wrinkle here). So people think about the value of gaining $40 as U(+40) - it’s whatever I have now plus forty dollars. The gamble in the previous paragraph now involves comparing U(+0), U(+20), and U(+40), rather than U(18,000), U(18,020), and U(18,040). It is no longer true that the scale of the change is small relative to the total amount, because the scale of the change sets the scale. So if there is any nonlinear curvature in your utility function, we can’t get rid of it by zooming in to the point where we can use linear approximations, because no matter what we’ll be looking at the function from U(+0) to U(+x). The utility function is at its curviest (least linear) near zero (think about log(x), or even sqrt(x)), and every change is defined relative to the status quo U(+0), so the curviest part of the curve is influencing every decision.
An assumption here that needs to be abandoned in order to have an accurate descriptive model of human decision making is that people have a single consistent utility function, which is defined in terms of the amount of money that they have.
That wasn’t an assumption to be abandoned, that was the beginning of a proof by contradiction.
No disagreement; that was just sloppy wording on my part. Edited. My comment is basically just repeating the argument in the original post, with less math and different emphasis.
The key assumption that leads to problems in trying to descriptively model people’s decisions is just that people have a single consistent utility function, which is defined in terms of the amount of money that they have.
If someone starts with $18,000 and then gets $40, the assumption is that the benefit can be expressed as U(18,040)-U(18,000). Or, in words, the person thinks: getting $40 brought me from a world where I have $18,000 to a world where I have $18,040. I value a world where I have $18,000 at this amount, and I value a world where I have $18,040 at that amount, and the benefit of getting the $40 is just the difference between those two.
In this model, there is a single curve that you can plot of how much you value a world where you had $X. Think about what that curve would look like, with X ranging from 10,000 to 100,000. In nearly every plausible case, that curve will be close to linear on a small scale (in the range of 10s or 100s) almost everywhere. There may be some curvature to it (perhaps your curve resembles f(x)=log(x)), but if you zoom in then that will mostly go away and a straight line will give you a good fit (e.g., if you are looking at changes in x that are 2 orders of magnitude smaller than x, then a log function will look pretty much linear). In a few special cases, there may be a large sudden jump in the curve, if there is some specific thing that you really want to buy and there is a large benefit to suddenly being able to afford it, but those cases are rare. For the most part, U(x) will be relatively smooth, and it will be growing perceptibly over the whole range (even if your utility function is bounded, it’s not like it will be almost at that bound before you even have $100,000).
And if your curve is approximately linear over small scales, then expected utility theory basically reduces to expected value theory when the stakes are small (e.g., a 50% of gaining $40 has an EV of $20). If U(x) is close to linear from x=18,000 to x=18,040, then U(18,020) must be about halfway in between U(18,000) and U(18,040). If you have $18,000, and are basing your decisions on a consistent utility function U(x), then for pretty much any plausible U(x) you’ll prefer a 51% chance of gaining $40 to a 100% chance of gaining $20 (unless you just happen to have one of those rare big jumps in U(x) between $18,000 and $18,020 - perhaps you really really want something that costs $18,010?). The expected value is 2% higher ($20.40 vs. $20), and it’s not plausible that your U(x) would be so sharply curved that you’d be willing to give up 2% EV over such a narrow range of x (it’s just a 0.2% increase in x from $18,000 to $18,040).
Probably the most important feature of prospect theory is that it does away with this assumption of a single consistent utility function, and says that people value gambles based on the change from the status quo (or occasionally some other reference point, but we’ll ignore that wrinkle here). So people think about the value of gaining $40 as U(+40) - it’s whatever I have now plus forty dollars. The gamble in the previous paragraph now involves comparing U(+0), U(+20), and U(+40), rather than U(18,000), U(18,020), and U(18,040). It is no longer true that the scale of the change is small relative to the total amount, because the scale of the change sets the scale. So if there is any nonlinear curvature in your utility function, we can’t get rid of it by zooming in to the point where we can use linear approximations, because no matter what we’ll be looking at the function from U(+0) to U(+x). The utility function is at its curviest (least linear) near zero (think about log(x), or even sqrt(x)), and every change is defined relative to the status quo U(+0), so the curviest part of the curve is influencing every decision.
That wasn’t an assumption to be abandoned, that was the beginning of a proof by contradiction.
No disagreement; that was just sloppy wording on my part. Edited. My comment is basically just repeating the argument in the original post, with less math and different emphasis.