ELICITING SUBJECTIVE UTILTES

 

Consider the following two decision makers. Anna is paying her way through college and is often a bit short on cash. Bob is a bachelor with no worries, mate, who is known for his fondness of poker. (The names have been changed, but the references are real.)

   [ut1]

Anna is offered to play the game with x = $50, the expected value of the lottery. She immediately picks Option C, assuring a $50 gain with no risk. So the analyst lowers x to $40. Anna is not pleased by the change but she still chooses C. The analyst now lowers x to $30. Anna takes her time to think things through (thus showing that, for her, both L and C are close in perceived value) and finally goes along with C. The analyst lowers x to $20. Anna quickly replies that she will now choose L.

Since Anna is willing to accept C with a lower payoff than the expected value of L, we know she is risk averse. Furthermore, we know that the limit to her risk aversion in this particular game lies somewhere in the interval 20 = x = 30. To ascertain the point of indifference, the analyst raises x to $25. Anna is now deep in thought. She has a hard time expressing her preference between L and C. We infer, therefore, that the worth of both L and C to Anna are the same or nearly so.

Subsequent questioning revealed that Anna was by and large indifferent to L and C in the range 24 = x = 26. Consequently, by Axiom 2, we have determined the certain equivalent of L, namely, the boundary quantity x* = $25. This is the equivalent value Anna places on the lottery L under conditions of no risk.

Now let’s see how Bob responded to the elicitation procedure. When offered to play the game with x = $50, he promptly chose the lottery. To Bob, L is more valuable than $50. Clearly, he is risk preferring. The analyst would now raise the value of x successively until the point of indifference is established. In Bob’s case, x*, his certain equivalent, turned out to be $65.

Certain equivalents provide a means of quantifying a person’s tolerance to risk. In Anna’s case, she was willing to forgo $25 in expected-value terms in order to eliminate the risk of a total loss. In effect, she “bought an insurance policy” to insure herself against the total-loss outcome and paid for it with $25 in expected-value dollars. This forgone amount is known as the risk premium RP:

 RP = EMV(L) – CE

where CE is the certain equivalent (x*).  In Bob’s case, RP is actually negative (–$15). This means that we would have to pay Bob a premium greater than $15 over the lottery’s expected value in order to dissuade him from choosing L. At $15 he would be indifferent between L and C.

In general, the following relations hold:

 RP > 0  ?  risk aversion
 RP = 0  ?  risk neutrality
 RP < 0  ?  risk preference

The greater the magnitude (absolute value) of the RP, the more risk averse/preferring is the person.

Coming up next: derivation of utility functions.