General comments:
1. Traditional utility-based approach to decision making.
1a. What is utility? Give a precise definition.
1b. How do you select the very bad alternative A_{−} and the very good alternative A_{+} that are needed to define utility?
1c. 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.
1d. Which of the two actions is better: B or C? For action B, there are two possible outcomes: a_{1} with probability 0.3 and utility 100, and a_{2} with probability 0.7 and utility −20. For action C, there are also two possible outcomes: b_{1} with probability 0.4 and utility 40, and b_{2} with probability 0.6 and utility −10.
2a. By using the fact that utility is proportional to the square root of money, explain why most people prefer $70 cash to a lottery in which they get $100 with probability 0.7.
2b. Give an example explaining why the simplified assumption -- that utility is proportional to money amount -- does not lead to a reasonable behavior.
2c. What properties were used to justify the power-law dependence of utility on money?
4a. Which of these three alternatives will be selected:
4b. 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 change the starting point (shift-invariance) or if we change the measuring unit (scale-invariance).
5a. Suppose that a decision maker follows McFadden's formula with β = ln(5). If we have three alternatives, with utilities 0, 1, and 2, with what probability will the expert select each of these three alternatives?
5b. What property was used to justify McFadden's formula?
5c. If we have three alternatives, with utilities 0, 1, and 2, 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 third powers?
5d. What property was used to justify the power-function modification of McFadden's formula?
6a. What will be the combined estimate if we use the standard least squares methods (i.e., l^{2}).
6b. What will be the combined estimate if we use the l^{1} method? Explain in what sense this method is more robust.
6c. 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?
7a. 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?
7b. What property was used to justify Nash's bargaining solution?
7c. 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).
8a. Which of the two investments leads to better end results?
8b. Which of the two investments will the investor prefer if he/she follows the peak-end rule?
8c. What is the justification for the peak-end rule?
9a. Let us assume that the trade volume between the two countries is described by the gravity model, with α = 3.
9b. What properties were used to justify the gravity model?
12a. 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 1 of the random variable located on the interval [0,3].
12b. What is the justification for using the generalized maximum entropy method, of maximizing the integral of ρ^{p}?
12c. 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 0.