Normative vs. descriptive

Following a link from Rajiv Sethi’s blog, I encountered this blog by Eilon Solan, who writes:

One of the assumptions of von-Neumann and Morgenstern’s utility theory is continuity: if the decision maker prefers outcome A to outcome B to outcome C, then there is a number p in the unit interval such that the decision maker is indifferent between obtaining B for sure and a lottery that yields A with probability p and C with probability 1-p.

When I [Solan] teach von-Neumann and Morgenstern’s utility theory I always provide criticism to their axioms. The criticism to the continuity axiom that I use is when the utility of C is minus infinity: C is death. In that case, one cannot find any p that would make the decision maker indifferent between the above two lotteries.

The funny thing is, this is an example I’ve used (see section 6 of this article from 1998) to demonstrate that you can, completely reasonably, put dollars and lives on the same scale. As I wrote:

We begin this demonstration by asking the students what is the dollar value of their lives—how much money would they accept in exchange for being killed? They generally answer that they would not be killed for any amount of money. Now flip it around: suppose you have the choice of (a) your current situation, or (b) a probability p$of dying and a probability (1-p) of gaining $1. For what value of p are you indifferent between (a) and (b)? Many students will answer that there is no value of p; they always prefer (a). What about p=10^{-12}? If they still prefer (a), let them consider the following example.

To get a more precise value for p, it may be useful to consider a gain of $1000 instead of $1 in the above decision. To see that $1000 is worth a nonnegligible fraction of a life, consider that people will not necessarily spend that much for air bags for their cars. Suppose a car will last for 10 years; the probability of dying in a car crash in that time is of the order of 10*40,000/280,000,000 (the number of car crash deaths in ten years divided by the U.S. population), and if an air bag has a 50% chance of saving your life in such a crash, this gives a probability of about 7*10^{-4} that the bag will save your life. Once you have modified this calculation to your satisfaction (for example, if you do not drive drunk, the probability of a crash should be adjusted downward) and determined how much you would pay for an air bag, you can put money and your life on a common utility scale. At this point, you can work your way down to the value of $1 (as illustrated in a different demonstration). This can all be done with a student volunteer working at the blackboard and the other students making comments and checking for coherence.

The student discussions can be enlightening. For example, one student, Julie, was highly risk averse: when given the choice between (a) the current situation, and (b) a 0.000 01 probability of dying and a 0.999 99 of gaining $10,000, she preferred (a).
Another student in the class pointed out that 0.000 01 is approximately the probability of dying in a car crash in any given three-week period. After correcting for the fact that Julie does not drive drunk, and that she drives less than the average American, perhaps this is her probability of dying in a car crash, with herself as a driver, in the next six months. By driving, she is accepting this risk; is the convenience of being able to drive
for six months worth $10,000 to her?

This demonstration is especially interesting to students because it shows that they really do put money and lives on a common scale, whether they like it or not.

So . . . is this a violation of the continuity axiom, or not? In a way, it is, because people’s stated preferences in these lotteries do not satisfy the axiom. In a way, it’s not, because people can be acting in a way consistent with the axiom without realizing it. From this perspective, the axiom (and the associated mathematics) are valuable because they give us an opportunity to confront our inconsistencies.

In that sense, the opposition isn’t really normative vs. descriptive, but rather descriptive in two different senses.

(Regular readers of this blog will know that I have big problems with the general use of utility theory in either the normative or the descriptive sense, but that’s another story. Here I’m talking about a circumscribed problem where I find utility theory to be helpful.)