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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 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.

Thank You in Advance.

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? – amdouglas – 2014-01-07T01:03:39.167This seems mostly like a economics question though it does also deal with well-being. Simply put, humans are not perfect optimizers. – virmaior – 2014-02-06T06:58:30.140