If a person is indifferent between [x+$10] and [55% chance of x+$20, 45% chance of x], for any x, then this attitude cannot reasonably be explained by expected utility maximization. The required utility function for money would curve so sharply as to be nonsensical (for example, U($2000)-U($1000) would have to be less than U($1000)-U($950)). This result is shown in a specific case as a classroom demonstration in Section 5 of a paper of mine in the American Statistician in 1998 and, more generally, as a mathematical theorem in a paper by my old economics classmate Matthew Rabin in Econometrica in 2000.
I was thinking about this stuff recently because of a discussion I had with Deb Frisch on her blog. I like Matt’s 2000 paper a lot, but Deb seems to be really irritated by it. Her main source of irritation seems to be that Matt writes, “The theorem is entirely ‘nonparametric,’ assuming nothing about the utility function except concavity.” But actually he assumes fairly strong assumptions about preferences (basically, a more general version of my [x, x+$10, x+$20] gamble above), and under expected utility, this has strong implications about the utility function.
Matt’s key assumption could be called “translation invariance”–the point is that the small-stakes risk aversion holds at a wide range of wealth levels. That’s the key assumption–the exact functional form isn’t the issue. Deb compares to a power-law utility function, but expected-utility preferences under this power law would not show substantial small-scale risk aversion across a wide range of initial wealth levels.
Deb did notice one mistake in Matt’s paper (and in mine too). Matt attributes the risk-averse attitude at small scales to “loss aversion.” As Deb points out, this can’t be the explanation, since if the attitude is set up as “being indifferent between [x+$10] and [55% chance of x+$20, 45% chance of x]”, then no losses are involved. I attributed the attitude to “uncertainty aversion,” which has the virtue of being logically possible in this example, but which, thinking about it now, I don’t really believe.
Right now, I’m inclined to attribute small-stakes risk aversion to some sort of rule-following. For example, it makes sense to be risk averse for large stakes, and a natural generalization is to continue that risk aversion for payoffs in the $10, $20, $30 range. Basically, a “heuristic” or a simple rule giving us the ability to answer this sort of preference question.
Attitudes, not preference or actions
By the way, I’ve used the term “attitude” above, rather than “preference.” I think “preference” is too much of a loaded word. For example, suppose I ask someone, “Do you prefer $20 or [55% chance of $30, 45% chance of $10]?” If he or she says, “I prefer the $20,” I don’t actually consider this any sort of underlying preference. It’s a response to a question. Even if it’s set up as a real choice, where they really get to pick, it’s just a preference in a particular setting. But for most of these studies, we’re really talking about attitudes.