Fall 2018, Test 1

General comments:

- 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;

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.

- by a perfect
optimist, for whom the Hurwicz optimism-pessimism parameter is
α
_{H}= 1? - by a perfect pessimist, for whom the
Hurwicz optimism-pessimism parameter is α
_{H}= 0? - by a realist, for whom the Hurwicz optimism-pessimism
parameter is α
_{H}= 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.