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Let $A$ be the set of possible states of the world, or possible preferences a person could have. Let $G(A)$ be the set of "gambles" or "lotteries", i.e. the set of probability distributions over $A$. Then each person would have a preferred ordering of the states in $A$, as well as a preferred ordering of the lotteries in $G(A)$. The von Neumann-Morgenstern theorem states that, assuming your preference ordering over $G(A)$ obeys certain rationality axioms, your preferences can be represented by a utility function $u: A → ℝ$. (This function is unique up to multiplication of scalars and addition of constants.) That means that for any two lotteries $L_1$ and $L_2$ in $G(A)$, you prefer $L_1$ to $L_2$ if and only if the expected value of $u$ under $L_1$ is greater than the expected value of $u$ under $L_2$. In other words, you maximize the expected value of the utility function.

Now just because you maximize the expected value of your utility function does not mean that you maximize the expected value of actual things like money. After all, people are often risk averse; they say "a bird in the hand is worth two in the bush". Risk aversion means that you value a gamble less than expected value of the money you'll gain. If we express this notion in terms of the von Neumann-Morgenstern utility function, we get the following result through Jensen's inequality: a person is risk averse if and only if their utility function is a concave function of your money, i.e. the extent to which you're risk averse is the same as the extent to which you have a diminishing marginal utility of money. (See page 13 of this PDF.)

My question is, which direction does the causation run? Do the values of the von Neumann-Morgenstern utility function reflect the intensity of your preferences, and is risk aversion due to discounting the preferences of future selves who are well-off compared to the preferences of future versions of yourself who are poorer and thus value money more (as Brad Delong suggests here)? Or does the causation run the other way: does your tolerance for risk determine the shape of your utility function, so that the von Neumann-Morgenstern utility function tells you nothing about the relative intensity of your preferences?

luchonacho
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Keshav Srinivasan
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3 Answers3

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I think I've found an answer to my question, in this excerpt from Nobel laureate John C. Harsanyi's 1994 paper "Normative validity and meaning of von neumann-morgenstern utilities", presented at the Ninth International Congress of Logic, Methodology and Philosophy of Science. Harsanyi starts by proving the same lemma that Alecos proved in his answer, namely that if $u$ is a vNM utility function of an individual, then $u(10) - u(5) < u(5) - u(0)$ if and only if they would prefer a guaranteed 5 dollars compared to a 50% of 10 dollars and a 50% chance of 0 dollars. In the comments section I said that was insufficient to demonstrate that the vNM utility function represented intensity of preferences, because what if the individual's actual pleasure and pain was accurately described by some other utility function $v$, which is a monotonic transformation but not an affine transformation of $u$? In that case couldn't $v$ fail to satisfy the expected value property, and couldn't $v(10) - v(5) = v(5) - v(0)$?

Harsanyi has a clever argument dealing with this issue. Let $L_1$ be the lottery where you get 5 dollars guaranteed, let $L_2$ be the lottery where you have a 50% chance of 10 dollars and a 50% chance of 0 dollars, and let $L_3$ be the lottery where you have a 50% chance of 10 dollars and a 50% chance of 5 dollars. Then obviously the person prefers $L_3$ to both $L_1$ and $L_2$. And Harsanyi argues that $L_3$ is preferred to $L_1$ less strongly than $L_3$ is preferred to $L_2$ if and only if $v(10) - v(5) < v(5) - v(0)$. That's because in the choice between, $L_3$ vs $L_1$, 50% of the time they get 5 dollars, and 50% of the time they have to make a choice between 10 and 5. Similarly in the choice between $L_3$ and $L_2$, 50% of the time they get 10 dollars, and 50% of the time they have to make a choice between 5 and 0.

Now here comes the master stroke: $L_1$ is preferred to $L_2$ if and only if $L_3$ is preferred to $L_1$ less strongly than $L_3$ is preferred to $L_2$. Therefore, $L_1$ is preferred to $L_2$ if and only if $v(10)-v(5) < v(5) -v(0)$. And thus we reach the grand conclusion that $u(10) - u(5) < u(5) - u(0)$ if and only if $v(10)-v(5) < v(5) -v(0)$.

Thus Harasanyi reaches the conclusion that the vNM utility function represents preferences intensities. So the answer to my question seems to be that diminishing marginal utility in the vNM utility function reflects genuine diminishing marginal utility when it comes to intensity of preferences, and thus (assuming the vNM axioms are true) diminishing marginal utility really is the cause of risk aversion.

By the way, on a side note I wonder whether we could identify the set of all functions $v$ that satisfy the constraint that $u(x) - u(y) < u(z) - u(w)$ if and only if $v(x)-v(y) < v(z) -v(w)$ (and similarly for greater than and equal to). (EDIT: I asked about this on Mathematics.SE here.)

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Keshav Srinivasan
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  • @AlecosPapadopoulos Thanks! But this proof isn't really a case of "working the axioms"; the function $v$ doesn't have to satisfy the expected value property at all. – Keshav Srinivasan Apr 17 '15 at 14:31
  • @AlecosPapadopoulos By the way, I just posted another question relating to expected-utility theory that you may be interested in: http://economics.stackexchange.com/q/5304/4447 – Keshav Srinivasan Apr 25 '15 at 14:55
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The utility function is a representation of preferences, which are traditionally inferred from choices. Preferences come before utility. I would not call the connection between utility and preferences causality, just a mathematical relationship.

Risk aversion (risk preference) is not connected to discounting, which measures time preference. It does not make sense to say that risk aversion is due to discounting the preferences of future selves.

Sander Heinsalu
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  • "I would not call the connection between utility and preferences causality, just a mathematical relationship." Well, the core of my question isn't about the whether preferences lead to utility functions. Here's what I'm fundamentally asking: do the values of the von Neumann-Morgenstern utility function reflect intensity of preferences, or do they merely reflect attitudes toward risk that have nothing to do with intensity of preferences? And by the way, by discounting I don't mean time discounting. I mean valuing versions of yourself in some possible futures more than versions in other futures. – Keshav Srinivasan Apr 15 '15 at 07:44
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    The expected utility representation of preferences is unique up to strictly increasing affine transformations. Utility values have no meaning, only their ranking has meaning. You can multiply the utility function by 2 for example with preferences unchanged. – Sander Heinsalu Apr 15 '15 at 07:47
  • @SanderHeinslau Yes, I'm aware of that, and my question is actually related to that very issue. Here's the heart of the matter: vNM utility functions convey more information than ordinary utility functions, because they're unique upto affine transformations as opposed to only being unique upto monotonic transformations. So what does that extra information represent? Does it indicate intensity of preferences, or does it just indicate risk preferences? – Keshav Srinivasan Apr 15 '15 at 07:53
  • To be specific, the quantity (u(x) - u(y)) / (u(y) - u(z)) is invariant under affine transformations, i.e. it's independent of which vNM utility function we choose. So this quantity represents real information about you, not just some arbitrary mathematical construct. So does this quantity reflect how much you prefer x to y relative to how much you prefer y to z? Or does it just reflect your tolerance for risk, and tells nothing about preference intensities? – Keshav Srinivasan Apr 15 '15 at 07:57
  • I now think I understand the question. vNM utility treats the same good (call it consumption) in different states of the world as different goods, but with the restriction that the marginal rate of substitution between goods equals the ratio of the probabilities of the states. This is the extra information, captured precisely in one of the vNM axioms (if I remember correctly, "Sure thing principle"). – Sander Heinsalu Apr 15 '15 at 12:07
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    @KeshavSrinivasan Perhaps both of want to update question/answer with the additional information you put into the comments here. Perhaps the question is also asked too formally (and as that, too length). I feel that I learned something just reading these comments here. – FooBar Apr 15 '15 at 13:51
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    @SanderHeinsalu Let's distinguish between two things. There's the extra information conveyed by the existence of a vNM utility function, namely the information that the person satisfies the vNM axioms. But I'm talking about the information conveyed by the vNM function itself. That is to say, if x, y, and z are three fixed elements of A, then the quantity (u(x) - u(y))/(u(y) - u(z)) varies from person to person (among people who satisfy the vNM axioms), but it does not vary among different vNM utility functions for the same person. So this quantity conveys something specific to a person. – Keshav Srinivasan Apr 15 '15 at 18:02
  • @SanderHeinsalu So the question, is what information about the person does this quantity convey? Does it convey something about intensity of preferences, like "this person prefers x to y much less strongly than he prefers y to z"? Or does it merely indicate inherent attitudes toward risk that have nothing to do with intensity of preferences? – Keshav Srinivasan Apr 15 '15 at 18:06
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    Attitude to risk is part of preferences. So it conveys both attitude to risk and intensity of preference in some sense. But there is also the state-independent utility in vNM, which is relaxed in some later decision theory. This can be interpreted as same intensity of preference for consumption in different states, with the whole difference in utility from consumption in different states ascribed to the probabilities of the states. – Sander Heinsalu Apr 15 '15 at 22:53
  • @SanderHeinsalu By the way, I just posted another question about expected-utility theory that you may be interested in: http://economics.stackexchange.com/q/5304/4447 – Keshav Srinivasan Apr 25 '15 at 14:58
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The Expected Utility property is not a property that depends on the functional form of the utility function. Its existence depends on satisfying certain "axioms" (which would be more acurately be described as "conditions"), that have to do with preferences/behavior of human beings. They may be given a strict mathematical expression (which is good), but they have to do with preferences, i.e. before any functional form for the utility function is specified. Let's see what that means. In a comment the OP wrote

"...if x, y, and z are three fixed elements of A, then the quantity $[u(x) - u(y)]/[u(y) - u(z)]$ varies from person to person (among people who satisfy the vNM axioms), but it does not vary among different vNM utility functions for the same person. So this quantity conveys something specific to a person."

It does.

Quoting from Jehle & Renyi (2011) "Advanced Microeconomic Theory" (3d ed), ch. 2 p. 108

"We conclude that the ratio of utility differences has inherent meaning regarding the individual’s preferences and they must take on the same value for every VNM utility representation of (the weak preference relation). Therefore, VNM utility representations provide distinctly more than ordinal information about the decision maker’s preferences, for otherwise, through suitable monotone transformations, such ratios could assume many different values."

In their example just before the quote they show that

$$\frac {[u(x) - u(y)]}{[u(y) - u(z)]} = \frac {1-\alpha}{\alpha}$$

where $\alpha$ is a probability that reflects the preferences that we are modelling. Quote again (p. 107)

"Note well that the probability number $\alpha$ is determined by, and is a reflection of, the decision maker’s preferences. It is a meaningful number. One cannot double it, add a constant to it, or transform it in any way without also changing the preferences with which it is associated."

And $(1-\alpha)/\alpha$ is an odds (not "odds ratio").

So here you are: a vNM utility function is associated with the odds that can characterize a person's preferences.

ADDENDUM
After an interesting but too lengthy exchange of opinions and thoughts in the comments with the OP, I decided to enhance this answer with an example, in order to show that in the context of the specific theory of preferences we are discussing, "preference intensity" (as is informally discussed here) cannot be dissociated from "attitude towards risk" -they are inextricably linked.

Assume that an individual declares (as he has every right to): "My preferences are monotonic and I prefer more to less. Moreover, the next five euros will give me exactly the same utility as the five after them". Note that this is the individual speaking -we cannot question him by whether utility can be cardinal or not etc. Starting from zero for convenience, we symbolize his statement as

$$u(10) - u(5) = u(5) - u(0) \implies u(5) = \frac 12 u(0) + \frac 12 u(10) \tag {1}$$

In the context of the discussion with the OP, this is a statement about "preference intensity".

Next we present to this individual the following choice: he can either get $5$ euros, or he can participate to a gamble $G$ where he will get $0$ euros with probability $1/2$ or $10$ euros with probability $1/2$. The individual then declares that he strictly prefers to get the $5$ euros with certainty. This is a statement revealing "attitude towards risk".

Question: Can the preferences of this individual, as described by his two statements, be represented by a utility function that possesses the Expected Utility Property?

Answer: No.

Proof: By his second statement, the individual revealed that the Certainty Equivalent of the gamble $CE_G$ is strictly less than $5$ euros:

Therefore we have that

$$E[u(G)] = u(CE_G) < u(5) \tag{2}$$

Now for the Expected Utility property to hold, it must be the case that

$$u[G;p(G)] = E[u(G)] = \frac 12 u(0) + \frac 12 u(10) \tag{3}$$

Due to $(2)$ (which expresses the "attitude towards risk" of the individual) we have that

$$(2), (3) \implies \frac 12 u(0) + \frac 12 u(10) < u(5) \tag {4}$$

But this contradicts $(1)$, which expresses "preference intensity" of the individual.

So we conclude that an individual whose preferences are described by the above statements cannot be represented by a utility function that possesses the Expected Utility Property.

In other words, for the Expected Utility property to hold, "attitude towards risk" cannot be dissociated from "preference intensity". If the individual had declared that he was indifferent between the $5$ certain euros and the gamble $G$, then his preferences could be represented by a utility function that had the EU property. But in order to achieve that, we had to "align" the "attitude towards risk" with "preference intensity".

Alecos Papadopoulos
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  • OK, now that we've established that the quantity $\frac {u(x) - u(y)}{u(y) - u(z)}$ tells you something about a person, my question remains, does it tell you something about intensity of preferences? For example, if $ \frac {u(10) - u(5)}{u(5) - u(0)}= \frac{1}{3}$, does that necessarily mean that they prefer 10 dollars to 5 dollars less strongly than they prefer 5 dollars to 0 dollars? Or does it just indicate something about attitudes toward risk (i.e. preferences over $G(A)$) that says nothing about preference intensity? – Keshav Srinivasan Apr 16 '15 at 19:35
  • @KeshavSrinivasan It ranks intensity, but it does not measure intensity. – Alecos Papadopoulos Apr 16 '15 at 20:00
  • OK, but why does it even rank intensity? Why does the fact that $u(10) - u(5) < u(5) - u(10)$ necessary imply that the person's preference for 10 dollars over 5 dollars is less strong than the person's preference for 5 dollars over 0 dollars? – Keshav Srinivasan Apr 16 '15 at 20:11
  • if you look up the reference that I provided in my answer, you will see that your numerical examples says: "This person is indifferent between $5$ dollars with certainty, and a gamble where he gets $10$ dollars with probability $3/4$, and $0$ dollars with probability $1/4$. Ah, but that's "attitude towards risk" one could say, "not intensity preferences". And who said that "attitude towards risk" is something disassociated from "preference intensity"? CONTD – Alecos Papadopoulos Apr 16 '15 at 20:38
  • CONTD If I like "plus 5", less than I dislike "minus 5", isn't it logical to think that, when it comes to uncertainty I will err a bit more on the side of not losing 5, rather than winning 5 more? Remember, a utility function exhibiting risk aversion, exhibits also diminishing marginal utility of wealth. Attitude towards risk and "preference intensity" are very closely linked. – Alecos Papadopoulos Apr 16 '15 at 20:40
  • OK, so this is the core of my question: why must the reason that you're indifferent between 5 dollars and the gamble you mention necessarily be that you value going from 5 dollars to 10 dollars less than going from 0 dollars to 5 dollars. I agree that that's one possible reason why you might be indifferent, but I don't see why it's impossible that you value going from 5 to 10 just as much as you value going from 0 to 5, and that the reason you're indifferent between 5 dollars and the gamble is simply that you inherently don't like taking too much risk? – Keshav Srinivasan Apr 16 '15 at 20:52
  • "Remember, a utility function exhibiting risk aversion, exhibits also diminishing marginal utility of wealth." Yes, and my question is, why must the diminishing marginal utility of a vNM utility function necessarily indicate something about preference intensity? – Keshav Srinivasan Apr 16 '15 at 20:56
  • I can only repeat that you are disassociating "preferences towards risk" and "preference intensity", while the theory we are discussing doesn't. So it is my impression that perhaps you should look into other theoretical frameworks, that treat these specific aspects of preferences separately. – Alecos Papadopoulos Apr 16 '15 at 20:57
  • I'm not at all disputing the theory. I'm asking, even assuming that the von Neumann-Morgenstern axioms are true, why does that imply that risk preferences are related to preference intensity? Of to put it another way, If you assume that preferences over $G(A)$ have nothing to do with preference intensity, why does that imply that the von Neumann-Morgenstern axioms are false? – Keshav Srinivasan Apr 16 '15 at 20:59
  • Because we postulate a single unified concept of preferences. There is no such thing as "risk preferences" separately from "certainty preferences". If somebody offers to give you 5 and somebody else offers to take away 5, both with certainty, and you say "hmm, the pleasure in the one case would be the same as the pain in the other", then you cannot,if you obey the axioms, exhibit anything else than "risk neutrality" when considering a gamble of +/- 5. Perhaps you should try to formalize this into symbols and mathematics. – Alecos Papadopoulos Apr 16 '15 at 21:07
  • The von Neumann-Morgenstern theorem just states that you'll be risk-neutral with respect to your vNM utility function $u$. But how does that imply that you'll be risk neutral with respect to your actual pleasure and pain, unless you assume that $u$ measures actual pleasure and pain? What if the actual pleasure and pain experienced in your brain was accurately described by some other function $v$ (which is a monotonic but not affine transformation of $u$), and that you were risk-averse with respect to $v$ while being risk-neutral with respect to $u$? – Keshav Srinivasan Apr 16 '15 at 21:18
  • I am not following you. It appears that you are trying to extract from this theory more behavioral content than it possess. It was not constructed as mathematized psychology but as a semi-normative model of "logically consistent" behavior. So, under the axioms, there is no "other function" concurrently with the the vNM. – Alecos Papadopoulos Apr 16 '15 at 21:59
  • What I'm asking is, suppose someone's behavior completely follows the von Neumann-Morgernstern axioms. Then we don't need to worry about whether it's a normative model or not. So the question remains, just because someone is risk-neutral with respect to the von Neumann-Morgenstern utility function u, why must they be risk-neutral with respect to actual pleasure and pain? – Keshav Srinivasan Apr 16 '15 at 22:19
  • They mustn't. But if they are not, then they are not described well by this specific model of preferences. Work the axioms to find out why this is so. – Alecos Papadopoulos Apr 16 '15 at 22:22
  • "They mustn't. But if they are not, then they are not described well by this specific model of preferences." Sorry, I'm not sure what you mean. I'm saying suppose that someone's behavior and preferences are described perfectly by the von Neumann-Morgenstern axioms. Under that circumstance, is it or is it not necessarily true that they are risk-neutral with respect to actual pleasure and pain? "Work the axioms to find out why this is so." What consequence of the axioms are you referring to? The vNM theorem just states that they're risk-neutral with respect to the vNM utility function $u$. – Keshav Srinivasan Apr 16 '15 at 23:51
  • @KeshavSrinivasan I included an example in my answer. Deeper and more theoretical work is on you now. – Alecos Papadopoulos Apr 17 '15 at 01:07
  • Your addendum doesn't address my question at all. I'm saying suppose that someone satisfies the von Neumann-Morgenstern axioms, and that they have a vnM utility function $u$, but the function that describes their actual pleasure and pain, intensity of preferences, and so on is some other function $v$ which is a monotonic transformation of $u$. Then it's certainly true that $u$ satisfies the expected value property (by vNM theorem) and that $u(10) - u(5) < u(5) - u(0)$. But why can't $v$ fail to satisfy the expected value property, and why can't $v(10) - v(5) = v(5) - v(0)$? That's my question. – Keshav Srinivasan Apr 17 '15 at 01:36
  • I think I found an answer to my question in a paper by Harsanyi. See the answer I just posted. – Keshav Srinivasan Apr 17 '15 at 04:34
  • Maybe this conversation could be moved to a chat? – luchonacho Aug 21 '17 at 07:56