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Single-attribute utility 105
(when the lottery ticket was preferred). Suppose that after trying several
probabilities we pose the following question.
Question: Which of the following would you prefer?
A $30 000 for certain; or
B A lottery ticket which will give you an 85% chance of $60 000 and a
15% chance of −$10 000?
Answer: I am now indifferent between the certain money and the lottery
ticket.
We are now in a position to calculate the utility of $30 000. Since the
business woman is indifferent between options A and B the utility of
$30 000 will be equal to the expected utility of the lottery. Thus:
u($30 000) = 0.85 u($60 000) + 0.15 u(−$10 000)
Since we have already allocated utilities of 1.0 and 0 to $60 000 and
−$10 000, respectively, we have:
u($30 000) = 0.85(1.0) + 0.15(0) = 0.85
Note that, once we have found the point of indifference, the utility of
the certain money is simply equal to the probability of the best outcome
in the lottery. Thus, if the decision maker had been indifferent between
the options which we offered in the first question, her utility for $30 000
would have been 0.7.
We now need to determine the utility of $11 000. Suppose that after
being asked a similar series of questions the business woman finally
indicates that she would be indifferent between receiving $11 000 for
certain and a lottery ticket offering a 60% chance of the best outcome
($60 000) and a 40% chance of the worst outcome (−$10 000). This implies
that u($11 000) = 0.6. We can now state the complete set of utilities and
these are shown below:
Monetary sum Utility
$60 000 1.0
$30 000 0.85
$11 000 0.60
−$10 000 0
These results are now applied to the decision tree by replacing the
monetary values with their utilities (see Figure 5.3). By treating these