# 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 L1 and L2 in G(A) you prefer L1 to L2 if and only if the expected value of u under L1 is greater than the expected value of u under L2. 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: 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 possible future selves who are well-off compared to the preferences of possible 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?

Any help would be greatly appreciated.

• After a cursory glance, I am inclined to ask: why not both ways? And is there not a factor of how much you have to begin with? As you have more money, your tolerance for risk surely rises as you need not be all in; if you have fewer assets and are debating whether you invest a significant portion of what money you do have, are you not more inclined to think critically about risk and resist the temptation to gamble? Commented Jan 7, 2014 at 1:03
• This seems mostly like a economics question though it does also deal with well-being. Simply put, humans are not perfect optimizers. Commented Feb 6, 2014 at 6:58

Aristotle addresses roughly the same question (sans the contemporary mathematicization) in explaining why he does not think Utilitarian in right in the Nicomachean Ethics. (Of course I use the term Utilitarian here loosely).

The basic answer is that in maximizing our well-being (trying to achieve eudaimonea) there are parts to the puzzle. First, there is the goal that I am bothering to pursue. Second, there is my pursuit of it. According to Aristotle, we all pursue happiness (eudaimonia) a thesis that will go unchallenged until past the time of Kant. The genius of Aristotle's answer is that he maintains that we pursue what we think will make us happy. The problem is that we can be deeply deceived about this question.

In your question, the idea that we are mistaken about a risk-reward calculation would for Aristotle demonstrate a lack of phronesis (practical wisdom). In other words, it would mean that we are making a bad particular judgment. Despite how it may appear, the localized version falls under the second question which is my means of pursuing of happiness. My miscalculation refers to what I think will get me eudaimonia. This problem can then repeat itself on the larger scale if one's entire idea of what is good is mistaken.

On Aristotle's picture, the more you lack wisdom the poorer both your choices and your idea of what is good to have are. Thus, lack of practical wisdom can compound itself. Interestingly, you can encourage yourself to be wrong because you take pleasure in things that are not able to ultimately give you eudaimonia. For instance, the pursuit of raw pleasure might encourage you to pursue crack cocaine and then this experience might be such that you begin identifying it with the good you want in life thus distorting your idea of the good life and making your judgments increasingly bad.

Referencing his view to the language you are using, the utility function can be wrong (= having the wrong vision) OR one's risk aversion can cause one to err greatly ( = failure to pursue the right things vis-a-vis your vision).

There is no particular reason why it must be either way. The utility function just corresponds to what your choices would be; it doesn't constrain any particular reason why you might make those choices.

There is reason to suspect that various choices are made sub-optimally. For example, happiness research suggests that as long as you're making ends meet, experiences not stuff is what makes you happy. Of course, happiness may not be synonymous with utility, but people with money don't seem to do a very good job using it the best way they can to convert it into happiness.

So if you are wondering about any particular person's direction of causality with respect to their utility function, you probably need to examine that person (possibly even that decision) for evidence one way or the other.