Utility, Risk Aversion, and all that

2010 January 1
by Daniel Lakeland

Last summer I took a course on the very basics of Decision Theory. It didn't get into much of the theory, but did have a fair number of practical examples. Unfortunately I was more interested in the theory, especially the Philosophy, Psychology, and failures of the theory. When a model breaks down, it's useful to know when and how.

Wikipedia has a surprisingly readable article on Von Neumann-Morgenstern Utility theory (vNMU). Apparently there are known situations in which vNMU fails to predict people's actual choices. In particular, people are not always consistent, so that if A > B and B > C they don't always prefer A>C for example. These cases usually occur where there are some significant uncertainties involved. The whole general genre of theories basically refer to situations in which outcomes have probabilities associated. So "A" in an idealized example situation might mean something like (25% of the time you lose $1, 50% of the time you gain $10 and 25% of the time you're given an electric shock) so making decisions about preferences between uncertain lotteries is something that requires non-trivial thought. It's my intuition that these inconsistencies can in part be explained by errors in estimation of the effects and desirability of both outcomes and the probabilities associated.

For example, which would you prefer, a lottery ticket with p = 10^-9 and payout x = $10^6 vs (1-p) payout of x = -$1, or another lottery ticket p = 10^-10 and payout x = $10^7 vs 1-p for payout of x = -$2? Many people might resort to something along the lines of the following thought process:

"These are both almost zero probabilities of paying out, and I can easily lose either $1 or $2 without much pain, so I'll take the one that has the higher maximum payout".

Another person might just as reasonably resort to the following line of thought:

"Neither one of these is going to pay out, so I'll take the one that costs only $1 since I could use that other dollar towards the Double Cappuccino that I'm about to buy."

In other words, neither of these people is really considering the specifics of the payout very much, doing so itself takes some effort, time, and therefore resource cost, and the unlikeliness of the positive event makes it not worth considering for very long. The high cost of analysis can in part be blamed for heuristic reasoning which might lead to inconsistent results in repeated trials.

In a discussion on Gelman's blog, another commenter referred to "Prospect Theory" which is more of a descriptive theory. vNMU is more of a normative theory, in that it tells you how to make decisions if you're Spock and want total logical consistency. Failure to conform to vNMU seems to be common enough that Psychologists have creating things like Prospect Theory to better explain what really is going on, and while Prospect Theory informs us a bit about what might be really happening, does it tell us something about how we should make decisions? I'm not so sure.

2 Responses leave one →
  1. January 1, 2010

    Behavioral economics has documented a bunch of anomalies that we'd like to understand better. Unfortunately, they had less success in producing a general enough alternative to expected utility.

    There's also a literature that contrasts the failures of these assumptions in the laboratory versus their performance "in the field"... people seem to learn to "be rational".

    List, John A. "Neoclassical Theory Versus Prospect Theory: Evidence from the Marketplace," Econometrica (2004), 72(2): 615-625.

    ^ that paper finds that new traders are best modeled with Prospect Theory while veterans looks a lot more neoclassical/rational.

    Rubinstein has a lot of good and free material on decision theory, rationality and models of bounded rationality: http://arielrubinstein.tau.ac.il/

    P.S. It's awesome to see a *good* interdisciplinary blog!

  2. Daniel Lakeland
    January 4, 2010

    Thanks for the link to the List article. i'll go grab that and take a look. It seems to me that neoclassical/rational decision making may be a better normative model than the more descriptive models, especially when it comes to something like decisions involving engineering policy. Von Neumann Morgenstern utility seems to me to be the kind of thing that just gets better when defined in terms of averages over the entire population. The population-averaged "apparent utility for earthquake resistance" is something I'd like to see the Engineering community do some research on for example.

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