Question

In this chapter a decision tree was developed for John Thompson. After completing the analysis, John was not completely sure that he is indifferent to risk. After going through a number of standard gambles, John was able to assess his utility for money. Here are some of the utility assessments: U( - $190,000) = 0, U( - $180,000) = 0.15,
U(- $30,000) = 0.10, U(-$20,000) = 0.15,
U (-$10,000) = 0.2, U($0) = 0.3,
($90,000) = 0.5, U($100,000)=0.6 U=($190,000)=.95
and U($200,000) = 1.0. If John maximizes his expected utility, does his decision change?

Answer 1

$$\mathrm{U}(-\$190,000) = 0, \mathrm{U}(-\$180,000) = 0.15,$$ $$\mathrm{U}(-\$30,000) = 0.10, \mathrm{U}(-\$20,000) = 0.15,$$ $$\mathrm{U}(-\$10,000) = 0.2, \mathrm{U}(\$0) = 0.3,$$ $$(\$90,000) = 0.5, \mathrm{U}(\$100,000) = 0.6, \mathrm{U} = (\$190,000) = 0.95$$ $$\mathrm{U}(\$200,000) = 1.0.$$

From this schedule we can see the dependence between U and $x(\$)$. We also see that U increases when $x\epsilon(-\$190000; -\$180000) \cup (-\$30000; \$200000)$ and decreases when $x\epsilon(-\$180000; -\$30000)$.