The cholera example was definitely a bit silly—after all, “cholera” and “apple vs orange” are usually really independent in the real world, you’ve to make very far-fetched circumstances for them to be dependent. But an axiom is supposed to be valid everywhere—even in far-fetched circumstances ;)
But overall, I understand the thing much better now: in fact, the independence principle doesn’t strictly hold in the real world, like there are no strictly right angle in the real world. But yet, like we do use the Pythagoras theorem in the real world, assuming an angle to be right when it’s “close enough” to be right, we apply the VNM axioms and the related expected utility theory when we consider the independence principle to have enough validity?
But do we have any way to measure the degree of error introduced by this approximation? Do we have ways to recognize the cases where we shouldn’t apply the expected utility theory, because we are too far from the ideal model?
My point never was to fully reject VNM and expected utility theory—I know they are useful, they work in many cases, … My point was to draw attention on a potential problem (making it an approximation, making it not always valid) that I don’t usually see being addressed (actually, I don’t remember ever having seen it that explicitly).
I think we have almost reached agreement, just a few more nitpicks I seem to have with your current post.
the independence principle doesn’t strictly hold in the real world, like there are no strictly right angle in the real world
Its pedantic, but these two statements aren’t analogous. A better analogy would be
“the independence principle doesn’t strictly hold in the real world, like the axiom that all right angles are equal doesn’t hold in the real world”
“there are no strictly identical outcomes in the real world, like there are no strictly right angle in the real world”
Personally I prefer the second phrasing. The independence principle and the right angle principle do hold in the real world, or at least they would if the objects they talked about ever actually appeared, which they don’t.
I’m in general uncomfortable with talk of the empirical status of mathematical statements, maybe this makes me a Platonist or something. I’m much happier with talk of whether idealised mathematical objects exist in the real world, or whether things similar to them do.
What this means is we don’t apply VNM when we think independence is relatively true, we apply them when we think the outcomes we are facing are relatively similar to each other, enough that any difference can be assumed away.
But do we have any way to measure the degree of error introduced by this approximation?
This is an interesting problem. As far as I can tell, its a special case of the interesting problem of “how do we know/decide our utility function?”.
Do we have ways to recognize the cases where we shouldn’t apply the expected utility theory
I’ve suggested one heuristic that I think is quite good. Any ideas for others?
(Once again, I want to nitpick the language. “Do we have ways to recognize the cases where two outcomes look equal but aren’t” is the correct phrasing.
The cholera example was definitely a bit silly—after all, “cholera” and “apple vs orange” are usually really independent in the real world, you’ve to make very far-fetched circumstances for them to be dependent. But an axiom is supposed to be valid everywhere—even in far-fetched circumstances ;)
But overall, I understand the thing much better now: in fact, the independence principle doesn’t strictly hold in the real world, like there are no strictly right angle in the real world. But yet, like we do use the Pythagoras theorem in the real world, assuming an angle to be right when it’s “close enough” to be right, we apply the VNM axioms and the related expected utility theory when we consider the independence principle to have enough validity?
But do we have any way to measure the degree of error introduced by this approximation? Do we have ways to recognize the cases where we shouldn’t apply the expected utility theory, because we are too far from the ideal model?
My point never was to fully reject VNM and expected utility theory—I know they are useful, they work in many cases, … My point was to draw attention on a potential problem (making it an approximation, making it not always valid) that I don’t usually see being addressed (actually, I don’t remember ever having seen it that explicitly).
I think we have almost reached agreement, just a few more nitpicks I seem to have with your current post.
Its pedantic, but these two statements aren’t analogous. A better analogy would be
“the independence principle doesn’t strictly hold in the real world, like the axiom that all right angles are equal doesn’t hold in the real world”
“there are no strictly identical outcomes in the real world, like there are no strictly right angle in the real world”
Personally I prefer the second phrasing. The independence principle and the right angle principle do hold in the real world, or at least they would if the objects they talked about ever actually appeared, which they don’t.
I’m in general uncomfortable with talk of the empirical status of mathematical statements, maybe this makes me a Platonist or something. I’m much happier with talk of whether idealised mathematical objects exist in the real world, or whether things similar to them do.
What this means is we don’t apply VNM when we think independence is relatively true, we apply them when we think the outcomes we are facing are relatively similar to each other, enough that any difference can be assumed away.
This is an interesting problem. As far as I can tell, its a special case of the interesting problem of “how do we know/decide our utility function?”.
I’ve suggested one heuristic that I think is quite good. Any ideas for others?
(Once again, I want to nitpick the language. “Do we have ways to recognize the cases where two outcomes look equal but aren’t” is the correct phrasing.