Sunday, December 16, 2012

Follow Up On Measuring Utility

Jonathan came back with this:

Hi, thanks for that Professor. IMO Utility does describe the feeling, or at least the “something” that kicks-in mentally, particularly with ""money"" based decisions - when you know (or don’t) you are detaching yourself from the wholly rational course of action (expected return) to take the money on the table offered, or a sum offered with certainty that is just enough of a push - for you to cash in your chips.
Where I’m stuck – is that there is a load of books and research about the abstract pro’s and cons – but I can’t find anything anywhere on the net that explains just how to go about discovering how to assess yourself (e.g. a clear example from the grass up).
Any ideas?
Thanks again,

My response:

If you took the standard expected utility theory at face value, then you would approximate the utility function locally with a quadratic.  The benefit of doing that is you get that for small gambles around the mean, the theory says the risk premium should be the Arrow-Pratt measure of absolute risk aversion at the mean, r, times the variance of the gamble.

For example suppose you face the gamble of $1,001 with probability .5 and  $999 with probability .5, so the mean is $1,000 and the variance of this gamble is $1, which is small relative to the mean.  You then try to elicit what amount of money for certain would make the person indifferent between having that or having the gamble.   Suppose you find the certainty equivalent determined experimentally is $999.60.  So in this case the risk premium is $.40 and hence the inferred Arrow-Pratt measure of absolute risk aversion is .4.

Now if you do this seriously, you would like to see whether the theory is really confirmed.  So you might try other small gambles with mean $1000.  For example you might consider the gamble (a) of $1001 with probability .8 and $996 with probability .2  as well as the gamble (b) of  $1004 with probability .2 and $999 with probability .8.  Each of these gambles has the same variance, $4.  So ahead of time you might guess based on what you discovered before that the measured risk premium would be $1.60, which must be the case if  the formula in the first paragraph held exactly and you measured the risk premium perfectly in the previous experiment.  You might get something close to that for the gamble (a) but you definitely won't for gamble (b) because that says the certainty equivalent is $998.40, which is lower than the $999, what is attained in the lower income state.

There are two possible source of error here: (1) measurement of the risk premium in the first experiment and (2) the formula that relates risk premium to the Arrow-Pratt measure and the variance of the gamble.   The second error becomes less as the variance gets smaller but the first error gets bigger that way.  So even if you take the theory as fully correct, you will have issues in measuring the utility function.

Let me make one more point on this.  The psychologist Daniel Kahneman, winner of the Nobel Prize in  Economics, has shown that the standard expected utility theory is wrong and that something else called Prospect Theory is closer to how we actually behave.  In that a reference point matters for evaluating gambles and then whether the outcome is a win with respect to the reference point (where the individual is then risk averse) or a loss with respect to the reference point (where the individual is then risk seeking).  Put another way, the utility function for Prospect Theory is convex-concave, with an inflection point at the reference point.   If you find this interesting you might read Kahneman's recent book, Thinking Fast and Slow.

Wednesday, December 12, 2012

Can (Expected) Utility Functions Be Measured Empirically?

Jonathan asked:

Hi Professor Arvan,
I just watched your ExpUty video on Youtube -
In reality, how would you go about capturing personal utility functions and preferences? Is there a defacto approach / way or template for doing this for Money, or other goods? I referring to the question construction, interpretation / ranking of the answers and then the maths behind plotting the curve? Or do you know of a spreadsheet / program solution? I take it ""Utils"" can only ever be ordinal, in reality?  I would appreciate any further advice on the subject - Thanks, Jon

My response:

There are lots of issues that question.  So it is a good one in bringing those to the surface.  Let's get to some of these:

(1)  Is the person rational a la the expected utility hypothesis or do "animal spirits" better serve as a guide to behavior?  And here instead of animal spirits think of Darwin and the decision to fight or flee. Moderate financial risk is qualitatively different, in my view, than the threat of somebody doing physical violence on your person, or the chance you may catch some serious disease.  For the latter two, I doubt expected utility theory is useful at all.  For the first, at least there is some hope it might be.  

(2)  How does the person assess the probability distribution in practice?  We understand how to do this in coin flipping, or casino games, but for real-world uncertainty do probability assessments at all conform with what the actuaries tell us we should believe?  There is psychological research on this and it confirms that people are bad at making probability assessments on their own and typically over estimate the chance that a threat will materialize.  The expression is "better safe than sorry" and the research supports that conclusion.  But it also means the individual is not being rational in the expected utility sense.  On the flip slide of this people of modest income are known to buy lottery tickets, even when the odds are quite bad for them.  They are fascinated with the prospect of a high payoff, irrespective of the odds.

(3) When there is more than just one good, money, but rather several commodities does it make sense to monetize them all and speak of a single dimension of risk preference or is it harder than that?  As far as I know there is no good theory of risk preference in a multi-dimensional commodity setting.  Since consumption bundles are themselves random - for example, if you buy a knock off computer instead of a name brand to save a few bucks how well does it function - the issues certainly appear there but whether there can be a coherent risk preference theoretically, I doubt it.  I do think that psychologically we tend to convert these sorts of risk into unto time units - as a measure of the possible inconvenience - and if necessary then try to monetize those, but we do it only in a very rough way.

(4)  Are a person's risk preferences stable over time or do they vary?  Let me give just one example here.  People may drink alcohol because it "loosens them up," which you might interpret as becoming less risk averse.  If the choice to drink alcohol in the first place is rational, and some might question that, then it is as if the risk aversion is a constraint that the person wants to shed.  (And this is why there is so much discussion about peer pressure and drinking, because it may be others who want the person to shed the risk aversion, not the person himself or herself.)  There are certain circumstances  where a normally mild person (one who will take flight most of the time) becomes extremely aggressive (opts to fight and then does so with a fiery intensity) so it's almost a Dr. Jekyll and Mr. Hyde thing.

Conclusion.  Given these various caveats, each which bring realism to the story, you might ask whether expected utility is at all useful as an approach.  I would say, yes it is useful especially if you restrict the domains where you apply it.  The first is that it provides a nice explanation of the demand for insurance.  The second is that in trading risks across individuals, it offers the reasonable intuition that with increasing wealth risk aversion should decline simply because there are better opportunities for diversifying the portfolio as one gets wealthier and hence suggests where there may be gains from trade from better sharing risks.