# Does risk aversion cause diminishing marginal utility, or vice versa?

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?

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.)

• @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. 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: economics.stackexchange.com/q/5304/4447 Apr 25 '15 at 14:55

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.

• "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. Apr 15 '15 at 7:44
• 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. Apr 15 '15 at 7:47
• @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. Apr 15 '15 at 13:51
• @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. Apr 15 '15 at 18:02
• 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. Apr 15 '15 at 22:53

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.

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?

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".

• 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? Apr 16 '15 at 19:35
• @KeshavSrinivasan It ranks intensity, but it does not measure intensity. 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? 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 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. Apr 16 '15 at 20:40