Questions related to the course CS 5354/CS 4365 Advanced Computational Methods in Economics and Finance, Fall 2018

1. What is the purpose of utility? What it utility?

A brief answer: The purpose of utility is to provide numerical values to the user's preference of different alternatives. Once utility of different alternatives is determined, the most preferable alternative is the one with the largest utility value.

To define utility, we select two alternatives: a very bad one A and a very good one A+. For each value p from the interval [0,1], we can then define a lottery L(p) in which we get the very good alternative with probability p and the very bad alternative with the remaining probability 1 − p. For each alternative A, there exists a probability p for which the corresponding lottery L(p) is equivalent to A. This probability is called the utility of the alternative A and is denoted by u(A).

2. Who invented modern decision making theory, with its notion of utility? Where was he from?

Answer: John von Neumann, in collaboration with the economist Oscar Morgenstern, wrote a pioneer book on this topic in 1944. John von Neumann was born in Hungary, then emigrated to the US.

3. How do we select its utility a very bad alternative A and a very good alternative A+?

Answer: The main requirement is that A should be worse than anything that we will actually encounter, and that A+ should be better than anything that we will encounter.

4. Once we select A and A+, how do we actually determine the utility u(A) of a given alternative A?

Answer: We do it by bisection. In the beginning, all we know is that the utility is somewhere in the interval [0,1], i.e., in the interval [u, u+], in which u = 0 and u+ = 1. Once we know such an interval containing u(A), we compute the midpoint m and ask the user to compare the alternative A with the lottery L(m).

• if A is better than the lottery L(m), this means that the utility u(A) belongs to the interval [m, u+];
• if the lottery L(m) is better than A, this means that utility u(A) belongs to the interval [u, m].
In both cases, we get an interval of half-width. We continue this procedure with the new interval until the resulting interval becomes sufficiently narrow.

5. With what accuracy we can determine u(A) after asking k questions?

Answer: We start with an interval of width 1. After each question, the width of the interval halves. So, after k questions, we get an interval of width 2-k. As the desired estimate, we can take the midpoint of this interval. This difference between this point and any value from this interval does not exceed half-width 2-(k+1). Thus, after k questions, we can determine the utility with accuracy 2-(k+1).

6. How many questions do we have to ask to get u(A) with a given accuracy 1/N? For example, 1% = 1/100? 0.1% = 1/1000?

Answer: As we discussed in our answer to Question 4, after k questions, we get utility with accuracy 2-(k+1). Thus, to answer this question, we need to find the smallest k for which 2-(k+1) is smaller than or equal to 1/N, i.e., equivalently, for which 2k+1 is larger than or equal to N.

• For N = 100, the smallest power of 2 exceeding 100 is 27 = 128, so here k + 1 = 7 and k = 6.
• For N = 1000, the smallest power of 2 exceeding 1000 is 210 = 1024, so here k + 1 = 10 and k = 9.

7. If we know the utility with respect to one pair of a very bad and a very good alternatives, and we now select a new pair, how does utility change?

Answer: look at the corresponding paper; it explains that the relation between the old utility u(A) and the new utility u'(A) is always linear: u'(A) = k * u(A) + b, for some k > 0 and b, and it explains how to compute the values k and b based on the utilities of the old very bad and very good alternatives with respect to the new pair.

9. Why do we need to consider decision making under interval uncertainty?

Answer: The above algorithm assumes that the user can always select between two alternatives. In reality, when alternatives are close, user often cannot decide which one is better. They may meaningfully compare the alternative A with lotteries in which A+ appears with probability 50% or 60%, but few folks can meaningfully compare with a lottery where the probability is, say, 50.3%. So, at some point, we cannot narrow down the utility interval -- and instead of the exact utility value, get an interval.

10. Who came up with the main idea of decision making under interval uncertainty? Where was he from?

Answer: a Nobelist Leo Hurwicz. He was born in Poland, which at that time was part of the Russian Empire, later emigrated to the US.

11. What is the main idea behind Hurwicz's solution to the problem of decision making under interval uncertainty?

Answer: We need to assign a single utility value u to each utility interval [x, y]. Since utility is defined modulo linear re-scaling u → k * u + b, it is reasonable to require that such re-scaling does not change the assignment.

12. What formula can be derived from this idea?

Answer: we should select u = αH * y + (1 − αH) * x, for some coefficient αH from the interval [0,1]. So, between several alternatives, we select the one for which this value is the largest.

13. Why is Hurwicz approach known as optimism-pessimism criterion?

Answer: If αH = 1, this means that the decision-maker completely ignores all but the best-case scenario; this is would expect from a perfect optimist. If αH = 0, this means that the decision-maker completely ignores all but the worst-case scenario; this is would expect from a perfect pessimist. A realist should use values of αH between 0 and 1. See, e.g., Section 4 from the following paper.