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116 Decision making under uncertainty
the course of action which has the highest expected utility. Of course,
it may be possible to demonstrate that a particular decision maker does
not act according to the axioms of utility theory. However, this does not
necessarily imply that the theory is inappropriate in his case. All that is
required is that he wishes to behave consistently according to the axioms.
Applying decision analysis helps a decision maker to formulate preferences,
assess uncertainty and make judgments in a coherent fashion.
Thus coherence is the result of decision analysis, not a prerequisite.
More on utility elicitation
So far, we have only considered utility assessment based on the
probability-equivalence approach. A disadvantage of this approach is
that the decision maker may have difficulty in thinking in terms of
probabilities like 0.90 or 0.95. Because of this, a number of alternative
approaches have been developed (for example, Farquahar 4 reviews 24
different methods). Perhaps the most widely used of these is the certaintyequivalence
approach, which, in its most common form, only requires the
decision maker to think in terms of 50:50 gambles.
To illustrate the approach, let us suppose that we wish to elicit
a decision maker’s utility function for monetary values in the range
$0–40 000 (so that u($0) = 0andu($40 000) = 1). An elicitation session
might proceed as follows:
Analyst: If I offered you a hypothetical lottery ticket which gave a
50% chance of $0 and a 50% chance of $40 000, how much would you
be prepared to pay for it? Obviously, its expected monetary value is
$20 000, but I want to know the minimum amount of money you would
just be willing to pay for the ticket.
Decision maker: (after some thought) $10 000.
Hence u($10 000) = 0.5 u($0) + 0.5 u($40 000)
= 0.5(0) + 0.5(1) = 0.5
The analyst would now use the $10 000 as the worst payoff in a new
hypothetical lottery.
Analyst: If I now offered you a hypothetical lottery ticket which gave you
a 50% chance of $40 000 and a 50% chance of $10 000 how much would
you be prepared to pay for it?