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108 Decision making under uncertainty
0.15 probability of −$10 000. Similarly, she will have a 0.4 probability of
a profit, which is equivalent to a lottery ticket offering a 0.6 probability
of $60 000 and a 0.4 chance of −$10 000. Therefore the Luxuria Hotel
offers her the equivalent of a 0.6 × 0.85 + 0.4 × 0.6 = 0.75 probability of
the best outcome (and a 0.25 probability of the worst outcome). Note
that 0.75 is the expected utility of choosing the Luxuria Hotel.
Obviously, choosing the Maxima Center offers her the equivalent
of only a 0.5 probability of the best outcome on the tree (and a 0.5
probability of the worst outcome). Thus, as shown in Figure 5.4(b), utility
allows us to express the returns of all the courses of action in terms of
simple lotteries all offering the same prizes, namely the best and worst
outcomes, but with different probabilities. This makes the alternatives
easy to compare. The probability of winning the best outcome in these
lotteries is the expected utility. It therefore seems reasonable that we
should select the option offering the highest expected utility.
Note that the use here of the term ‘expected’ utility is therefore
somewhat misleading. It is used because the procedure for calculating
expected utilities is arithmetically the same as that for calculating
expected values in statistics. It does not, however, necessarily refer to an
average result which would be obtained from a large number of repetitions
of a course of action, nor does it mean a result or consequence
which should be ‘expected’. In decision theory, an ‘expected utility’
is only a ‘certainty equivalent’, that is, a single ‘certain’ figure that is
equivalent in preference to the uncertain situations.
Interpreting utility functions
The business woman’s utility function has been plotted on a graph in
Figure 5.5. If we selected any two points on this curve and drew a
straight line between them then it can be seen that the curve would
always be above the line. Utility functions having this concave shape
provide evidence of risk aversion (which is consistent with the business
woman’s avoidance of the riskiest option).
This is easily demonstrated. Consider Figure 5.6, which shows a utility
function with a similar shape, and suppose that the decision maker, from
whom this function has been elicited, has assets of $1000. He is then
offered a gamble which will give him a 50% chance of doubling his
money to $2000 and a 50% chance of losing it all, so that he finishes
with $0. The expected monetary value of the gamble is $1000 (i.e.