ELICITATION OF UTILITY FUNCTIONS
Utility or preference elicitation is a process of assessing preferences and the attitude to risk of a
decision maker or, more precisely, utility functions.
The classical technique for evaluating utility functions involves an interaction between the decision
analyst and the decision maker, and include several steps. The central stage is specification of
quantitative restrictions and selection of a utility function. During this stage, the decision analyst
asks various queries in an attempt to model a utility function, specific for the decision maker's
preferences.
In order to derive a utility function to represent the decision maker's attitude to risk, first we need to
rank all the outcomes from the best to the worst and assign a utility of 1 to the best and 0 to the
worst. According to von Neumann and Morgestern theory, the main idea is to have the decision
maker stating preferences over lotteries, or gambles; then utilities are calculated reasoning
backwards. In other words, if we find two gambles that are equally attractive, we can equalize the
expected utilities. In this way, if two utilities are known, also the third will be known.
Specifically:
Let's suppose we have three different outcomes A, B, C, and we can rank them from the best to the
worst, so that for example A ≽ B ≽ C. As I said before, we can assign a utility 1 to A (i.e. the best
outcome) and 0 to C (i.e. the worst outcome). At this point, we can elicit the utility of B asking the
decision maker for what probability p he is indifferent between these two gambles:
–
–
B for certain, and
A with probability p and C with probability 1-p.
Recalling what is written above, if the decision maker succeeds in finding a probability p in order to
assess the gambles as equally attractive, then their expected utilities must be equal. Therefore:
1.0u(B) = pu(A) + (1-p)u(C)
Since we know that u(A) = 1 and u(C) = 0, once we know how much is p we can easily calculate
u(B). Then, we will have elicited all the values needed to derive and represent the utility function.
I think that an example could provide a better understanding of the method described (for the sake
of simplicity, I will just consider the part of the process we need to understand the elicitation
method).
Suppose that a decision maker has these three possible monetary outcomes:
(I chose monetary outcomes because I think it’s easier to understand how the rank of utilities works)
A = €4800
B = €3500
C = 1130
It is obvious that we can rank them in a way that we have A ≽ B ≽ C and therefore we can assign 1
to the best outcome A and 0 to the worst outcome C.
Outcome
A
B
C
Utility
1
?
0
In order to find the utility of outcome B, we have to offer the decision maker this kind of lotteries:
1.0
p
€3500
€4800
1–p
€1130
Imagine that the decision maker says the probability p that makes him indifferent between these two
gambles is 0.7 (i.e., 1 – p = 0.3), hence given this probability they are equally attractive.
At this point there is only one thing left: elicit the utility of B equalizing the expected utilities.
1.0u(B) = 0.7u(A) + 0.3u(C) ⟹ u(B) = 0.7*1 + 0.3*0 = 0.7
In the end, the final step of the process consists of using these values to represent the utility function
of the decision maker.
1.2
1
Utility
0.8
0.6
0.4
0.2
0
0
1000
2000
3000
4000
5000
6000
Monetary sum
References
Braziūnas D., Decision-Theoretic Elicitation of Generalized Additive Utilities, University of Toronto, 2012.
Goodwin P. & Wright G., Decision Analysis for Management Judgment, Third Edition, John Wiley & Sons, Ltd.