It actually doesn’t matter what the values are, because we know from prospect theory that people’s preferences about risks can be reversed merely by framing gains as losses, or vice versa. No matter what shape the function has, it has to have some shape — it can’t have one shape if you frame alternatives as gains but a different, opposite shape if you frame them as losses.
Yes, framing effects are irrational, I agree. I’m saying that the mere existence of risk aversion with respect to something does not demonstrate the presence of framing effects or any other kind of irrationality (departure from the VNM axioms).
“No, in fact, you have no such preference. You only think you do, because your are envisioning your utility function incorrectly.”
That would be one way of describing my objection. The argument Dawes is making is simply not valid. He says “Suppose my utility function is X. Then my intuition says that I prefer certain distributions over X that have the same expected value. Therefore my utility function is not X, and in fact I have no utility function.” There are two complementary ways this argument may break:
If you take as a premise that the function X is actually your utility function (ie. “assuming I have a utility function, let X be that function”) then you have no license to apply your intuition to derive preferences over various distributions over the values of X. Your intuition has no facilities for judging meaningless numbers that have only abstract mathematical reasoning tying them to your actual preferences. If you try to shoehorn the abstract constructed utility function X into your intuition by imagining that X represents “money” or “lives saved” or “amount of something nice” you are making a logical error.
On the other hand, if you start by applying your intuition to something it understands (such as “money” or “amount of nice things”) you can certainly say “I am risk averse with respect to X”, but you have not shown that X is your utility function, so there’s no license to conclude “I (it is rational for me to) violate the VNM axioms”.
Are you able to conceive of a utility function, or even a preference ordering, that does not give rise to this sort of preference over distributions? Even in rough terms? If so, I would like to hear it!
No, but that doesn’t mean such a thing does not exist!
Yes, framing effects are irrational, I agree. I’m saying that the mere existence of risk aversion with respect to something does not demonstrate the presence of framing effects or any other kind of irrationality (departure from the VNM axioms).
Well, now, hold on. Dawes is not actually saying that (and neither am I)! The claim is not “risk aversion demonstrates that there’s a framing effect going on (which is clearly irrational, and not just in the ‘violates VNM axioms’ sense)”. The point is that risk aversion (at least, risk aversion construed as “preferring less negatively skewed distributions”) constitutes departure from the VNM axioms. The independence axiom strictly precludes such risk aversion.
Whether risk aversion is actually irrational upon consideration — rather than merely irrational by technical definition, i.e. irrational by virtue of VNM axiom violation — is what Dawes is questioning.
The argument Dawes is making is simply not valid. He says …
That is not a good way to characterize Dawes’ argument.
I don’t know if you’ve read Rational Choice in an Uncertain World. Earlier in the same chapter, Dawes, introducing von Neumann and Morgenstern’s work, comments that utilities are intended to represent personal values. This makes sense, as utilities by definition have to track personal values, at least insofar as something with more utility is going to be preferred (by a VNM-satisfying agent) to something with less utility. Given that our notion of personal value is so vague, there’s little else we can expect from a measure that purports to represent personal value (it’s not like we’ve got some intuitive notion of what mathematical operations are appropriate to perform on estimates of personal value, which utilities then might or might not satisfy...). So any VNM utility values, it would seem, will necessarily match up to our intuitive notions of personal value.
So the only real assumption behind those graphs is that this agent’s utility function tracks, in some vague sense, an intuitive notion of personal value — meaning what? Nothing more than that this person places greater value on things he prefers, than on things he doesn’t prefer (relatively speaking). And that (by definition!) will be true of the utility function derived from his preferences.
It seems impossible that we can have a utility function that doesn’t give rise to such preferences over distributions. Whatever your utility function is, we can construct a pair of graphs exactly like the ones pictured (the x-axis is not numerically labeled, after all). But such a preference constitutes independence axiom violation, as mentioned...
The point is that risk aversion (at least, risk aversion construed as “preferring less negatively skewed distributions”) constitutes departure from the VNM axioms.
No, it doesn’t. Not unless it’s literally risk aversion with respect to utility.
So any VNM utility values, it would seem, will necessarily match up to our intuitive notions of personal value.
That seems to me a completely unfounded assumption.
Whatever your utility function is, we can construct a pair of graphs exactly like the ones pictured (the x-axis is not numerically labeled, after all).
The fact that the x-axis is not labeled is exactly why it’s unreasonable to think that just asking your intuition which graph “looks better” is a good way of determining whether you have an actual preference between the graphs. The shape of the graph is meaningless.
Yes, framing effects are irrational, I agree. I’m saying that the mere existence of risk aversion with respect to something does not demonstrate the presence of framing effects or any other kind of irrationality (departure from the VNM axioms).
That would be one way of describing my objection. The argument Dawes is making is simply not valid. He says “Suppose my utility function is X. Then my intuition says that I prefer certain distributions over X that have the same expected value. Therefore my utility function is not X, and in fact I have no utility function.” There are two complementary ways this argument may break:
If you take as a premise that the function X is actually your utility function (ie. “assuming I have a utility function, let X be that function”) then you have no license to apply your intuition to derive preferences over various distributions over the values of X. Your intuition has no facilities for judging meaningless numbers that have only abstract mathematical reasoning tying them to your actual preferences. If you try to shoehorn the abstract constructed utility function X into your intuition by imagining that X represents “money” or “lives saved” or “amount of something nice” you are making a logical error.
On the other hand, if you start by applying your intuition to something it understands (such as “money” or “amount of nice things”) you can certainly say “I am risk averse with respect to X”, but you have not shown that X is your utility function, so there’s no license to conclude “I (it is rational for me to) violate the VNM axioms”.
No, but that doesn’t mean such a thing does not exist!
Well, now, hold on. Dawes is not actually saying that (and neither am I)! The claim is not “risk aversion demonstrates that there’s a framing effect going on (which is clearly irrational, and not just in the ‘violates VNM axioms’ sense)”. The point is that risk aversion (at least, risk aversion construed as “preferring less negatively skewed distributions”) constitutes departure from the VNM axioms. The independence axiom strictly precludes such risk aversion.
Whether risk aversion is actually irrational upon consideration — rather than merely irrational by technical definition, i.e. irrational by virtue of VNM axiom violation — is what Dawes is questioning.
That is not a good way to characterize Dawes’ argument.
I don’t know if you’ve read Rational Choice in an Uncertain World. Earlier in the same chapter, Dawes, introducing von Neumann and Morgenstern’s work, comments that utilities are intended to represent personal values. This makes sense, as utilities by definition have to track personal values, at least insofar as something with more utility is going to be preferred (by a VNM-satisfying agent) to something with less utility. Given that our notion of personal value is so vague, there’s little else we can expect from a measure that purports to represent personal value (it’s not like we’ve got some intuitive notion of what mathematical operations are appropriate to perform on estimates of personal value, which utilities then might or might not satisfy...). So any VNM utility values, it would seem, will necessarily match up to our intuitive notions of personal value.
So the only real assumption behind those graphs is that this agent’s utility function tracks, in some vague sense, an intuitive notion of personal value — meaning what? Nothing more than that this person places greater value on things he prefers, than on things he doesn’t prefer (relatively speaking). And that (by definition!) will be true of the utility function derived from his preferences.
It seems impossible that we can have a utility function that doesn’t give rise to such preferences over distributions. Whatever your utility function is, we can construct a pair of graphs exactly like the ones pictured (the x-axis is not numerically labeled, after all). But such a preference constitutes independence axiom violation, as mentioned...
No, it doesn’t. Not unless it’s literally risk aversion with respect to utility.
That seems to me a completely unfounded assumption.
The fact that the x-axis is not labeled is exactly why it’s unreasonable to think that just asking your intuition which graph “looks better” is a good way of determining whether you have an actual preference between the graphs. The shape of the graph is meaningless.