## CS 5354/CS 4365 Advanced Computational Methods in Economics and Finance Fall 2018, Test 1

Name: ____________________________

• you are allowed up to 5 pages of handwritten notes;
• if you need extra pages, place your name on each extra page;
• the main goal of most questions is to show that you know the corresponding algorithms; so, if you are running of time, just follow the few first steps of the corresponding algorithm;
Good luck!

1. What is utility? Give a precise definition.

2. How do you select the very bad alternative A and the very good alternative A+ that are needed to define utility?

3. Once you have selected the very bad and the very good alternatives, how do we find the utility of a given alternative A? Explain, in detail.

4. With what accuracy can we determine the utility if we only ask 3 questions? Explain your answer.

5. How many questions do you need to ask a person to determine his/her utility with accuracy 10%? Explain your answer.

6. Suppose that an alternative A has utility u(A) = 0.7 with respect to the original pair (A, A+), and that with respect to a new pair (A', A'+), we have u'(A) = 0.1 and u'(A+) = 0.9. What is the utility u'(A) of the alternative A with respect to the new pair? Explain your answer.

7-9. Suppose that:
• for alternative A, the utility is from 0.5 to 0.7,
• for alternative B, it is from 0.4 and 0.8, and
• for alternative C, it is from 0.1 to 0.9.
Which of these three alternatives will be selected:
• by a perfect optimist, for whom the Hurwicz optimism-pessimism parameter is αH = 1?
• by a perfect pessimist, for whom the Hurwicz optimism-pessimism parameter is αH = 0?
• by a realist, for whom the Hurwicz optimism-pessimism parameter is αH = 0.6?

10. Explain, in detail, how we can deduce the Hurwicz optimism-pessimism formula from the requirement that decision making under interval uncertainty should not change if we re-scale the utility values.