Fall 2018, Test 2

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. Which of the two actions is better: A or B? For action A, there
are two possible outcomes: a_{1} with probability 0.4 and
utility 50, and a_{2} with probability 0.6 and utility
−20. For action B, there are also two possible outcomes:
b_{1} with probability 0.3 and utility 40, and
b_{2} with probability 0.7 and utility −10.

2. To determine the subjective probability that a student S will
get A on this test we asked his colleague C whether he prefers
$100 with probability 0.4 or $100 if S gets an A. The student C
selected $100 if S gets an A. Based on this answer, what can we
conclude about C's subjective probability that S will get an A in
this test? To be more precise, based on this answer, what is the
interval of possible values of the corresponding subjective
probability?

3a. Suppose that a decision maker follows McFadden's formula with
β = ln(3). If we have three alternatives, with utilities 1,
2, and 3, with what probability will the expert select each of
these three alternatives?

3b. What property was used to justify McFadden's formula?

4a. If we have three alternatives, with utilities 1, 2, and 3,
with what probability will the expert select each of the three
alternatives is this expert uses a power-function modification of
McFadden's formula: with the first powers? with the second
powers?

4b. What property was used to justify the power-function modification of McFadden's formula?

5a. Suppose that a group of two people needs to select between two
alternatives. For the first alternative, their utilities are 50
and 10; for the second alternative, their utilities are 70 and 8.
Which alternative should they select if they use Nash's bargaining
solution?

5b. What property was used to justify Nash's bargaining solution?

6. Give an example that Nash's bargaining solution provides a more
fair description of the country's economic situation than the
Gross Domestic Product (GDP).

7a. By using the fact that utility is proportional to the square
root of money, explain why most people prefer $64 cash to a
lottery in which they get $100 with probability 0.64.

7b. Give an example explaining why the simplified assumption -- that utility is proportional to money amount -- does not lead to a reasonable behavior.

7c. What properties were used to justify the power-law dependence of utility on money?

8. Provide the detailed proofs of the three main mathematical
results that we had:

- that every shift-invariant function F(x), i.e., a function for which F(x + c) = f(c) * F(x) for some f(c), has the form F(x) = const * exp(β * x);
- that every
scale-invariant function F(x), i.e., a function for which F(k * x)
= f(k) * F(x) for some f(k), has the form F(x) = const *
x
^{α}; - that every additive function, i.e., a function for which F(x + y) = F(x) + F(y), has the form F(x) = const * x.

9a. Let us assume that the trade volume between the two countries
is described by the gravity model, with α = 2.

- How will the trade volume change if the GDP of the first country increases by 50%?
- How will the trade volume change if the GDP of the second country increases by 10%?
- How will the trade volume be different if countries with the same GDP were located at a twice larger distance than the given pair?

9b. What properties were used to justify the gravity model?

10. Use the Least Squares method to find the values a and b for
which a * x_{i} + b ~ y_{i}, based on the
following observations:

- x
_{1}= −1, y_{1}= −1; - x
_{2}= 0, y_{2}= −1; - x
_{3}= 1, y_{3}= 1.