This sounds like regular risk aversion, which is normally easy to model by transforming utility by some concave function.
Assuming you privilege some reference point as your x-axis origin, sure. But there’s no good reason to do that. It turns out that people are risk averse no matter what origin point you select (this is one of the major findings of prospect theory), and thus the concave function you apply will be different (will be in different places along the x-axis) depending on which reference point you present to a person. This is clearly irrational; this kind of “regular risk aversion” is what Dawes refers to when he talks about independence axiom violation due to framing effects, or “pseudocertainty”.
Note that the axioms require that there exist a utility function u :: outcome → real such that you maximise expected utility, not that some particular function (such as the two graphs you’ve drawn) actually represents your utility.
The graphs are not graphs of utility functions. See the first paragraph of my post here.
How do you show that there’s an actual violation of the independence axiom from this example? … the two distributions don’t have the form ApC, BpC with A≅B.
Indeed they do; because, as one of the other axioms states, each outcome in an alternative may itself be an alternative; i.e, alternatives may be constructed as probability mixtures of other alternatives, which may themselves be… etc. If it’s the apparent continuity of the graphs that bothers you, then you have but to zoom in on the image, and you may pretend that the pixelation you see represents discreteness of the distribution. The point stands unchanged.
Simply postulating that the expected utilities are the same only shows that that particular utility function is not correct, not that no valid utility function exists.
The point Dawes is making, I think, is that any utility function (or at least, any utility function where the calculated utilities more or less track an intuitive notion of “personal value”) will lead to this sort of preference between two such distributions. In fact, it is difficult to imagine any utility function that both corresponds to a person’s preferences and doesn’t lead to preferences for less negatively skewed probably distributions over outcomes, over more negatively skewed ones.
It turns out that people are risk averse no matter what origin point you select (this is one of the major findings of prospect theory), and thus the concave function you apply will be different (will be in different places along the x-axis) depending on which reference point you present to a person.
Couldn’t this still be rational in general if the fact that a particular reference point is presented provides information under normal circumstances (though perhaps not rational in a laboratory setting)?
Assuming you privilege some reference point as your x-axis origin, sure.
What? This has nothing to do with “privileged reference points”. If I am [VNM-]rational, with utility function U, and you consider an alternative function $ = exp(U) (or an affine transformation thereof), I will appear to be risk averse with respect to $. This doesn’t mean I am irrational, it means you don’t have the correct utility function. And in this case, you can turn the wrong utility function into the right one by taking log($).
That is what I mean by “regular risk aversion”.
The graphs are not graphs of utility functions.
I know, they are graphs of P(U). Which is implicitly a graph of the composition of a probability function over outcomes with (the inverse of) a utility function.
Indeed they do; because, as one of the other axioms states, each outcome in an alternative may itself be an alternative;
Okay, which parts, specifically, are A, B and C, and how is it established that the agent is indifferent between A and B?
The point Dawes is making, I think, is that any utility function (or at least, any utility function where the calculated utilities more or less track an intuitive notion of “personal value”) will lead to this sort of preference between two such distributions.
And I say that is assuming the conclusion. And, if only established for some set of utility functions that “more or less track an intuitive notion of “personal value”″, fails to imply the conclusion that the independence axiom is violated for a rational human.
What? This has nothing to do with “privileged reference points”. If I am [VNM-]rational, with utility function U, and you consider an alternative function $ = exp(U) (or an affine transformation thereof), I will appear to be risk averse with respect to $. This doesn’t mean I am irrational, it means you don’t have the correct utility function. And in this case, you can turn the wrong utility function into the right one by taking log($).
That is what I mean by “regular risk aversion”.
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.
I know, they are graphs of P(U). Which is implicitly a graph of the composition of a probability function over outcomes with (the inverse of) a utility function.
True enough. I rounded your objection to the nearest misunderstanding, I think.
And I say that is assuming the conclusion.
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!
The core of Dawes’ argument is not a mathematical one, to be sure (and it would be difficult to make it into a mathematical argument, without some sort of rigorous account of what sorts of outcome distribution shapes humans prefer, which in turn would presumably require substantial field data, at the very least). It’s an argument from intuition: Dawes is saying, “Look, I prefer this sort of distribution of outcomes. [Implied: ‘And so do other people.’] However, such a preference is irrational, according to the VNM axioms...” Your objection seems to be: “No, in fact, you have no such preference. You only think you do, because your are envisioning your utility function incorrectly.” Is that a fair characterization?
Your talk of the utility function possibly being wrong makes me vaguely suspect a misunderstanding. It’s likely I’m just misunderstanding you, however, so if you already know this, I apologize, but just in case:
If you have some set of preferences, then (assuming your preferences satisfy the axioms), we can construct a utility function (up to positive affine transformation). But having constructed this function — which is the only function you could possibly construct from that set of preferences (up to positive affine transformation) — you are not then free to say “oh, well, maybe this is the wrong utility function; maybe the right function is something else”.
Of course you might instead be saying “well, we haven’t actually constructed any actual utility function from any actual set of preferences; we’re only imagining some vague, hypothetical utility function, and a vague hypothetical utility function certainly can be the wrong function”. Fair enough, if so. However, I once again invite you to exhibit a utility function — or even a preference ordering — which does not give rise to a preference for less-negatively-skewed distributions.
Okay, which parts, specifically, are A, B and C, and how is it established that the agent is indifferent between A and B?
I’m afraid an answer to this part will have to wait until I have some free time to do some math.
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.
Assuming you privilege some reference point as your x-axis origin, sure. But there’s no good reason to do that. It turns out that people are risk averse no matter what origin point you select (this is one of the major findings of prospect theory), and thus the concave function you apply will be different (will be in different places along the x-axis) depending on which reference point you present to a person. This is clearly irrational; this kind of “regular risk aversion” is what Dawes refers to when he talks about independence axiom violation due to framing effects, or “pseudocertainty”.
The graphs are not graphs of utility functions. See the first paragraph of my post here.
Indeed they do; because, as one of the other axioms states, each outcome in an alternative may itself be an alternative; i.e, alternatives may be constructed as probability mixtures of other alternatives, which may themselves be… etc. If it’s the apparent continuity of the graphs that bothers you, then you have but to zoom in on the image, and you may pretend that the pixelation you see represents discreteness of the distribution. The point stands unchanged.
The point Dawes is making, I think, is that any utility function (or at least, any utility function where the calculated utilities more or less track an intuitive notion of “personal value”) will lead to this sort of preference between two such distributions. In fact, it is difficult to imagine any utility function that both corresponds to a person’s preferences and doesn’t lead to preferences for less negatively skewed probably distributions over outcomes, over more negatively skewed ones.
Couldn’t this still be rational in general if the fact that a particular reference point is presented provides information under normal circumstances (though perhaps not rational in a laboratory setting)?
I think you’ll have to give an example of such a scenario before I could comment on whether it’s plausible.
What? This has nothing to do with “privileged reference points”. If I am [VNM-]rational, with utility function
U, and you consider an alternative function$ = exp(U)(or an affine transformation thereof), I will appear to be risk averse with respect to$. This doesn’t mean I am irrational, it means you don’t have the correct utility function. And in this case, you can turn the wrong utility function into the right one by takinglog($).That is what I mean by “regular risk aversion”.
I know, they are graphs of
P(U). Which is implicitly a graph of the composition of a probability function over outcomes with (the inverse of) a utility function.Okay, which parts, specifically, are A, B and C, and how is it established that the agent is indifferent between A and B?
And I say that is assuming the conclusion. And, if only established for some set of utility functions that “more or less track an intuitive notion of “personal value”″, fails to imply the conclusion that the independence axiom is violated for a rational human.
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.
True enough. I rounded your objection to the nearest misunderstanding, I think.
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!
The core of Dawes’ argument is not a mathematical one, to be sure (and it would be difficult to make it into a mathematical argument, without some sort of rigorous account of what sorts of outcome distribution shapes humans prefer, which in turn would presumably require substantial field data, at the very least). It’s an argument from intuition: Dawes is saying, “Look, I prefer this sort of distribution of outcomes. [Implied: ‘And so do other people.’] However, such a preference is irrational, according to the VNM axioms...” Your objection seems to be: “No, in fact, you have no such preference. You only think you do, because your are envisioning your utility function incorrectly.” Is that a fair characterization?
Your talk of the utility function possibly being wrong makes me vaguely suspect a misunderstanding. It’s likely I’m just misunderstanding you, however, so if you already know this, I apologize, but just in case:
If you have some set of preferences, then (assuming your preferences satisfy the axioms), we can construct a utility function (up to positive affine transformation). But having constructed this function — which is the only function you could possibly construct from that set of preferences (up to positive affine transformation) — you are not then free to say “oh, well, maybe this is the wrong utility function; maybe the right function is something else”.
Of course you might instead be saying “well, we haven’t actually constructed any actual utility function from any actual set of preferences; we’re only imagining some vague, hypothetical utility function, and a vague hypothetical utility function certainly can be the wrong function”. Fair enough, if so. However, I once again invite you to exhibit a utility function — or even a preference ordering — which does not give rise to a preference for less-negatively-skewed distributions.
I’m afraid an answer to this part will have to wait until I have some free time to do some math.
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.