Comment author:Academian
16 July 2011 03:25:37PM
3 points
[-]

This post mis-uses the term "utility". Expected utility theory does not treat utility as linear in money, as you suggest.

... trying to decide whether or not to buy an hurricane insurance policy costing $5000/year. Prospero owns assets worth $10,000, and estimates a 50%/year chance of a hurricane destroying his assets; to make things simple, he will be moving in one year and so need not consider the future. Under expected utility theory, he should feel neutral about the policy.

The main descriptive difference between prospect theory and EU theory is that for monetary decisions, EU theory uses one curve (utility function), whereas prospect theory uses two curves (a value function and weight function) as well as a framing variable... it's about three times as suspect for overfitting, so I think I'll wait until it pays a little more rent :)

Comment author:Unnamed
16 July 2011 07:40:21PM
3 points
[-]

The other big difference is that the prospect theory value function is defined relative to a reference point (which typically represents the status quo) while the EU theory utility function is defined based on total wealth. So (as jimmy said) the nonlinearity of the prospect theory curve has a big effect on pretty much any decision (since any change from the current state is taking you through the curviest part of the curve), but the nonlinearity of EU theory curve is relatively minor unless the stakes are large relative to your total wealth. Under those conditions, EU theory (based on the utility of total wealth) is essentially equivalent to expected value.

Let's say that you have $30,000 in total wealth and you're given a choice of getting $10 for sure or getting $21 with p=.5. On the EU curve, the relationship between U($30,000), U($30,010), and U($30,021) should be nearly linear, so with any reasonable curve EU theory predicts that you prefer the 50% chance at $21 (indeed, you'd even prefer a 50% chance at $20.01 to $10 for sure as long as your curve is something like the square root function or even the natural log function). But on the prospect theory curve, V($0), V($10), and V($21) are very nonlinear, so even if we just treat probabilities as probabilities (rather than using the probability weighting function) prospect theory predicts that you'll prefer the certain $10 (at least, it will if the V(x) curve is the square root function, or x^.88 as is commonly used).

When people are actually given choices like $10 for sure vs. $21 w. p=.5, they tend to choose $10 for sure just as prospect theory predicts (and EU theory does not). That's paying rent in anticipated experiences. Prospect theory was developed by asking people a bunch of questions like that one, seeing what they did, and fitting curves to the data so that predictions about hundreds of similar decisions could be made based on a model with only a few parameters. That research produced a lot of data which was inconsistent with expected value (which, for these types of gambles, implies that it was also inconsistent with EU theory based on utility-of-wealth) and so Kahneman & Tversky developed a relatively simple model that did fit the empirical data, prospect theory.

Comment author:Academian
23 July 2011 09:22:46AM
*
1 point
[-]

The main descriptive difference between prospect theory and EU theory is that for monetary decisions, EU theory uses one curve (utility function), whereas prospect theory uses two curves (a value function and weight function) as well as a framing variable

The other big difference is that the prospect theory value function is defined relative to a reference point

That's what Yvain and I are calling framing.

When people are actually given choices like $10 for sure vs. $21 w. p=.5, they tend to choose $10 for sure just as prospect theory predicts (and EU theory does not).

What you're calling EU theory is a very restricted version of EU theory, where you require utility to be a function of total monetary wealth, or total material wealth. You might call it "Expected Utility of Wealth" theory. EU theory is actually much more general, and assigns utility to outcomes rather than amounts of money or even lists of possessions. This is all discussed in

But for predictive purposes, EU theory is so ridiculously general (there are so many situational parameters) that, as far as anyone knows, it has almost no predictive power. So for the purposes of prediction, I think you're justified in talking about "EUW" theory, because without a highly restrictive assumption like utility being a function of wealth, EU theory has no chance of making predictions.

Nonetheless, I want to encourage you, and anyone else, to make explicit the assumption "utility is a function of wealth" when you're making it. My reason is that, in toy decision-theory problems, EU theory is usually part of the framework, and it's a reasonable framework provided we don't impose the restrictions that make it predictively meaningful and false.

Comment author:DanielLC
15 February 2012 06:23:05PM
1 point
[-]

Utility is generally accepted to be differentiable in money, which means that it's approximately linear in amounts that are insignificant over your lifetime earnings. If you use a non-linear utility to explain risk aversion for a small amount of money, and extend this until you get large amounts of money, it results in absurdly huge utility falloff. I remember someone posted an article on this. I can't seem to find it at the moment.

Comment author:roystgnr
15 February 2012 07:15:59PM
1 point
[-]

Unless you have a good estimate of your future earnings and can borrow up to that at low interest rates, I think "amounts that are insignificant compared to your current liquidity" might be a slightly more rational metric. Note also that any explanation of human risk aversion (as opposed to rational risk aversion) is trying to explain behaviors that evolved during a time when "borrowing at low interest rates" wasn't really an option. If a failed risk means you starve to death next year, it doesn't matter how copious a quantity of food you otherwise would have acquired in subsequent years.

## Comments (46)

BestThis post mis-uses the term "utility". Expected utility theory does not treat utility as linear in money, as you suggest.

See http://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Morgenstern_utility_theorem, or perhaps also

http://lesswrong.com/lw/244/vnm_expected_utility_theory_uses_abuses_and/

The main descriptive difference between prospect theory and EU theory is that for monetary decisions, EU theory uses one curve (utility function), whereas prospect theory uses two curves (a value function and weight function) as well as a framing variable... it's about three times as suspect for overfitting, so I think I'll wait until it pays a little more rent :)

The other big difference is that the prospect theory value function is defined relative to a reference point (which typically represents the status quo) while the EU theory utility function is defined based on total wealth. So (as jimmy said) the nonlinearity of the prospect theory curve has a big effect on pretty much any decision (since any change from the current state is taking you through the curviest part of the curve), but the nonlinearity of EU theory curve is relatively minor unless the stakes are large relative to your total wealth. Under those conditions, EU theory (based on the utility of total wealth) is essentially equivalent to expected value.

Let's say that you have $30,000 in total wealth and you're given a choice of getting $10 for sure or getting $21 with p=.5. On the EU curve, the relationship between U($30,000), U($30,010), and U($30,021) should be nearly linear, so with any reasonable curve EU theory predicts that you prefer the 50% chance at $21 (indeed, you'd even prefer a 50% chance at $20.01 to $10 for sure as long as your curve is something like the square root function or even the natural log function). But on the prospect theory curve, V($0), V($10), and V($21) are very nonlinear, so even if we just treat probabilities as probabilities (rather than using the probability weighting function) prospect theory predicts that you'll prefer the certain $10 (at least, it will if the V(x) curve is the square root function, or x^.88 as is commonly used).

When people are actually given choices like $10 for sure vs. $21 w. p=.5, they tend to choose $10 for sure just as prospect theory predicts (and EU theory does not). That's paying rent in anticipated experiences. Prospect theory was developed by asking people a bunch of questions like that one, seeing what they did, and fitting curves to the data so that predictions about hundreds of similar decisions could be made based on a model with only a few parameters. That research produced a lot of data which was inconsistent with expected value (which, for these types of gambles, implies that it was also inconsistent with EU theory based on utility-of-wealth) and so Kahneman & Tversky developed a relatively simple model that did fit the empirical data, prospect theory.

*1 point [-]That's what Yvain and I are calling framing.

What you're calling EU theory is a very restricted version of EU theory, where you require utility to be a function of total monetary wealth, or total material wealth. You might call it "Expected Utility of Wealth" theory. EU theory is actually much more general, and assigns utility to

outcomesrather than amounts of money or even lists of possessions. This is all discussed inhttp://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Morgenstern_utility_theorem , and

http://lesswrong.com/lw/244/vnm_expected_utility_theory_uses_abuses_and/

But for predictive purposes, EU theory is so ridiculously general (there are so many situational parameters) that, as far as anyone knows, it has almost no predictive power. So for the purposes of prediction, I think you're justified in talking about "EUW" theory, because without a highly restrictive assumption like utility being a function of wealth, EU theory has no chance of making predictions.

Nonetheless, I want to encourage you, and anyone else, to make explicit the assumption "utility is a function of wealth" when you're making it. My reason is that, in toy decision-theory problems, EU theory is usually part of the framework, and it's a reasonable framework provided we don't impose the restrictions that make it predictively meaningful and false.

Utility is generally accepted to be differentiable in money, which means that it's approximately linear in amounts that are insignificant over your lifetime earnings. If you use a non-linear utility to explain risk aversion for a small amount of money, and extend this until you get large amounts of money, it results in absurdly huge utility falloff. I remember someone posted an article on this. I can't seem to find it at the moment.

Unless you have a good estimate of your future earnings and can borrow up to that at low interest rates, I think "amounts that are insignificant compared to your current liquidity" might be a slightly more rational metric. Note also that any explanation of human risk aversion (as opposed to rational risk aversion) is trying to explain behaviors that evolved during a time when "borrowing at low interest rates" wasn't really an option. If a failed risk means you starve to death next year, it doesn't matter how copious a quantity of food you otherwise would have acquired in subsequent years.

http://lesswrong.com/lw/9oe/risk_aversion_vs_concave_utility_function/5svv

Are you looking for this?