Homeworks for the course CS 5354/CS 4365, Fall 2018

*General comment:* Unless explicitly specified, each homework
should be turned in on a sheet of paper. Place it on the
instructor's desk in the classroom by the beginning of the class
on the day when the homework is due.

1. (Due September 5) With what accuracy can we determine the utility if we only ask 4 questions? Explain your answer.

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

3. (Due September 5) Suppose that an alternative A has utility
u(A) = 0.6 with respect to the original pair (A_{−},
A_{+}), and that with respect to a new pair
(A'_{−}, A'_{+}), we have
u'(A_{−}) = 0.3 and u'(A_{+}) = 0.8. What is
the utility u'(A) of the alternative A with respect to the new
pair? Explain your answer.

4. (Due September 17) Suppose that:

- for alternative A, the utility is from 0.4 to 0.6,
- for alternative B, it is from 0.3 and 0.7, and
- for alternative C, it is from 0.0 to 0.8.

- 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}= 0.5?

5. (Due September 26) 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 5, and a_{2} with probability
0.6 and utility −2. For action B, there are also two
possible outcomes: b_{1} with probability 0.3 and utility
4, and b_{2} with probability 0.7 and utility −1.

6. (Due September 26) To determine the subjective probability of snow in Alaska tomorrow, we asked an expert whether she prefers $10 with probability 0.6 or $10 if it snows tomorrow. She selected $10 with probability 0.6. Based on this answer, what can we conclude about the expert's subjective probability of snowing tomorrow? To be more precise, based on this answer, what is the interval of possible values of the corresponding subjective probability?

7. (Due September 26) Suppose that a decision maker follows McFadden's formula with β = ln(2). If we have three alternatives, with utilities 0, 1, and 4, with what probability will the expert select each of these three alternatives?

8. (Due October 1) In situation as described in Problem 7, 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?

9. (Due October 1) Suppose that a group of two people needs to select between two alternatives. For the first alternative, their utilities are 5 and 10; for the second alternative, their utilities are 7 and 8. Which alternative should they select if they use Nash's bargaining solution?

10. (Due October 1) 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).

11. (Due October 1) By using the fact that utility is proportional to the square root of money, explain why most people prefer $500 cash to a lottery in which they get $1,000 with probability 0.5.

12. (Due October 3) 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.

13. (Due October 3) Let us assume that the trade volume between the two countries is described by the gravity model, with α = 1.

- How will the trade volume change if the GDP of the first country doubles?
- How will the trade volume change if the GDP of the second country increases by 20%?
- How will the trade volume be different if countries with the same GDP were located at a twice larger distance than the given pair?

14. (Due October 3) 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}= 0, y_{1}= 0; - x
_{2}= 1, y_{2}= 0; - x
_{3}= 2, y_{3}= 2.

15. (Due October 15) Write a program that use the Least Squares
method to find coefficients a and b of linear regression y = a * x
+ b based on several pairs (x_{i}, y_{i}). Please
turn in a printout of the program, and the printout of the test
case -- that you checked by hand -- showing that your program is
correct. *For extra credit:* make your program capable of
finding a linear dependence on several variables.

16. (Due October 24) Use Lagrange multiplier method to solve the
following constraint optimization problem: find the point of the
line x_{1} − x_{2} = 1 which is the closest
to 0, i.e., in precise terms, minimize the sum
x_{1}^{2} + x_{2}^{2} under the
above constraint.

16a. (Due November 5; *the grade for this h/w will replace the
grade for h/w 16 if that grade was not perfect; no need to do this
problem if you already got 10/10 on h/w 16*) Use Lagrange
multiplier method to solve the following constraint optimization
problem: minimize the sum 2x_{1}^{2} −
x_{2}^{2} + 2x_{2} under the constraint
x_{1} + x_{2} = 2.

17. (Due November 5) Suppose that we have two investments, one with expected return 1 and variance 2, another with expected return 2 and variance 1, and we want to have a return of 1.5. Assuming that these two investments are independent, use the general formulas that we had in class to find the optimal portfolio.

18. (Due November 5) Same as in Problem 17, but this time, the two investments are not independent: the covariance is 0.5. Describe the optimal portfolio for this case.

19. (Due November 7) Assume that we have ten estimates for the annual economic growth: four estimate of 2%, four estimates of 3%, and two outliers: an over-pessimistic estimate of −50%, and an over-optimistic estimate of 10%.

- What will be the
combined estimate if we use the standard least squares methods
(i.e., l
^{2}). - What will be the combined estimate if
we use the l
^{1}method? Explain in what sense this method is more robust. - What is the general class of robust
techniques that includes both l
^{2}and l^{1}as particular cases? What is the justification for using methods from this class?

20. (Due November 12) Use the algorithm that we learned in class
(and which is also described here) to write a
program that use the robust l^{p} method to find
coefficients a and b of linear regression y = a * x + b based on
several pairs (x_{i}, y_{i}). Allow the user to
input the value α describing degree of robustness and the
value ε > 0 describing the desired accuracy. Please turn
in a printout of the program, and the printout of the test case --
that you checked by hand -- showing that your program is correct.
*For extra credit:* make your program capable of finding a
linear dependence on several variables.

21. (Due November 12) An investor placed her money into two hedge funds. The first one led to annual returns of 10%, 1%, 1%, 3%, and 9%. The second one lead to annual returns of 8%, 0%, 8%, 8%, and 8%.

- Which of the two investments leads to better end results?
- Which of the two investments will the investor prefer if she follows the peak-end rule?
- What is the justification for the peak-end rule?

22. (Due November 14) Suppose that we have three estimates for the same quantity:

- x
_{1}= 1 with accuracy σ_{1}= 0.1, - x
_{2}= 2 with accuracy σ_{2}= 0.2, and - x
_{3}= 3 with accuracy σ_{3}= 0.3.

23. (Due November 14) Use the generalized maximum entropy method
-- where we maximize the integral of ρ^{2} -- to find
the probability density function in a situation when the only
thing we know is the mean value 2 of the random variable located
on the interval [0,3]. What is the justification for using the
generalized maximum entropy method, of maximizing the integral of
ρ^{p}? Explain how we derive the traditional maximum
entropy method -- of maximizing the integral of
−ρ*ln(ρ) -- as a limit case of the generalized
maximum entropy method when p tends to 1.

24. (Due November 26) Write a program that use the weighted Least
Squares method to find coefficients a and b of linear regression y
= a * x + b based on several triples (x_{i},
y_{i}, σ_{i}), where σ_{i} is
the accuracy with which we measure y_{i}. Reminder: the
algorithm is the same as the auxiliary algorithm for the
l^{p}-method (see Problem 20), but with w_{i} =
(σ_{i})^{−2}. Please turn in a
printout of the program, and the printout of the test case -- that
you checked by hand -- showing that your program is correct.
*For extra credit:* make your program capable of finding a
linear dependence on several variables.

25. (Due November 28) What if the usual scale- and/or shift-invariant approximations are not accurate enough, and we try multi-D approximations to get a better accuracy? Explain what family of functions we need to use:

- if we require shift-invariance,
- if we require scale-invariance, and
- if we require both shift- and scale-invariance.