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University of California, Berkeley *

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Economics

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Jan 9, 2024

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12/15/23, 10:00 PM View Submission | Gradescope https://www.gradescope.com/courses/572452/assignments/3582873/submissions/219835206 1/24

12/15/23, 10:00 PM View Submission | Gradescope https://www.gradescope.com/courses/572452/assignments/3582873/submissions/219835206 2/24 Q1 Decisions 21 Points You've been job hunting, and you've narrowed your options to two companies: Acme and Google. You already have an offer from Acme, but it expires today, and you are still waiting for a response from Google. You are faced with the dilemma of whether or not to accept the offer from Acme, which is modeled by the following decision network: The prior probability distribution for whether Google will hire you and the utilities over possible outcomes are as follows:

12/15/23, 10:00 PM View Submission | Gradescope https://www.gradescope.com/courses/572452/assignments/3582873/submissions/219835206 3/24 Q1.1 5 Points What is the expected utility of each action? (Note: throughout this problem answers will be evaluated to whole-number precision, so your answer should differ by no more than 1 from the exact answer.) Action: accept Acme offer 6500 Action: reject Acme offer 2500 Which action should you take? reject accept

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12/15/23, 10:00 PM View Submission | Gradescope https://www.gradescope.com/courses/572452/assignments/3582873/submissions/219835206 4/24 Q1.2 16 Points Suddenly, the phone rings. It's your uncle, who works at Google. Your uncle tells you he has some inside information about the status of your application. Your uncle won't tell you what the information is yet, but he might be willing to divulge it for the right price. You model the new situation by adding a new node to your decision network: You create a CPT to model the relationship between the inside information and Google's future hiring decision: We'll help grind through the probabilistic inference. The resulting distributions are:

12/15/23, 10:00 PM View Submission | Gradescope https://www.gradescope.com/courses/572452/assignments/3582873/submissions/219835206 5/24 (That these are identical to P(Google outcome) and P(Info | Google outcome) is just a numerical coincidence.) Fill in the expected utilities for each action, for each possible type of information we could be given: EU(accept Acme offer | good news) 3800 EU(reject Acme offer | good news) 7000 EU(accept Acme offer | bad news) 7400 EU(reject Acme offer | bad news) 1000 What is the maximum expected utility for each type of information we could be given? MEU(good news) 7000 MEU(bad news) 7400 If we are given the inside information, what is the expected value of MEU? 7300 What is the value of perfect information of the random variable Inside Info?

12/15/23, 10:00 PM View Submission | Gradescope https://www.gradescope.com/courses/572452/assignments/3582873/submissions/219835206 6/24 800 Q2 Value of Perfect Information 10 Points Consider the expected value of perfect information (VPI) of observing some node in an arbitrary decision network. Which of the following are true statements? VPI is guaranteed to be positive ( ). > 0 VPI is guaranteed to be nonnegative ( ). ≥ 0 VPI is guaranteed to be nonzero. The MEU after observing a node could potentially be less than the MEU before observing that node. For any two nodes and , . That is, the sum of individual VPI's for two nodes is always greater than or equal to the VPI of observing both nodes. X Y VPI( X ) + VPI( Y ) ≥ VPI( X , Y ) VPI is guaranteed to be exactly zero for any node that is conditionally independent (given the evidence so far) of all parents of the utility node.

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12/15/23, 10:00 PM View Submission | Gradescope https://www.gradescope.com/courses/572452/assignments/3582873/submissions/219835206 7/24 Q3 Rationality of Utilities 9 Points Q3.1 3 Points Consider a lottery = , where the utility values of each of the outcomes are , , , . What is the utility of this lottery, ? 3.3 Q3.2 3 Points Consider a lottery = , where , and = is a lottery, and , . What is the utility of the the first lottery, ? 5 L [0.2, A ; 0.3, B ; 0.4, C ; 0.1, D ] U ( A ) = 1 U ( B ) = 3 U ( C ) = 5 U ( D ) = 2 U ( L ) L 1 [0.5, A ; 0.5, L 2] U ( A ) = 4 L 2 [0.5, X ; 0.5, Y ] U ( X ) = 4 U ( Y ) = 8 U ( L 1)

12/15/23, 10:00 PM View Submission | Gradescope https://www.gradescope.com/courses/572452/assignments/3582873/submissions/219835206 8/24 Q3.3 3 Points Assume , , where = , and . Assuming rational preferences, which of the following statements are guaranteed to be true? A ≻ B B ≻ L L [0.5, C ; 0.5, D ] D ≻ A A ≻ L A ≻ C A ≻ D B ≻ C B ≻ D

12/15/23, 10:00 PM View Submission | Gradescope https://www.gradescope.com/courses/572452/assignments/3582873/submissions/219835206 9/24 Q4 Certainty Equivalent Values 6 Points Consider the utility function shown below. Under the above utility function, what is the certainty equivalent monetary value in dollars ($) of the lottery ? I.e., what is such that = ? Hint: Keep in mind that is not equal to . 10 [0.6, $0; 0.4, $100] X U ($ X ) U ([0.6, $0; 0.4, $100]) U ([ p , A ; 1 − p , B ]) U ( pA + (1 − p ) B )

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12/15/23, 10:00 PM View Submission | Gradescope https://www.gradescope.com/courses/572452/assignments/3582873/submissions/219835206 10/24 Q5 Preferences and Utilities 14 Points Our Pacman board now has food pellets of 3 different sizes - pellet of radius 1, of radius 2 and of radius 3. In different moods, Pacman has different preferences among these pellets. In each of the following questions, you are given Pacman's preference for the different pellets. From among the options pick the utility functions that are consistent with Pacman's preferences, where each utility function is given as a function of the pellet radius , and is defined over non-negative values of . Q5.1 2 Points P 1 P 2 P 3 U ( r ) r r P ∼ 1 P ∼ 2 P 3 U ( r ) = 0 U ( r ) = 3 U ( r ) = r U ( r ) = 2 r + 4 U ( r ) = − r U ( r ) = r 2 U ( r ) = − r 2 U ( r ) = r U ( r ) = − r Irrational preferences!

12/15/23, 10:00 PM View Submission | Gradescope https://www.gradescope.com/courses/572452/assignments/3582873/submissions/219835206 11/24 Q5.2 2 Points P ≺ 1 P ≺ 2 P 3 U ( r ) = 0 U ( r ) = 3 U ( r ) = r U ( r ) = 2 r + 4 U ( r ) = − r U ( r ) = r 2 U ( r ) = − r 2 U ( r ) = r U ( r ) = − r Irrational preferences!

12/15/23, 10:00 PM View Submission | Gradescope https://www.gradescope.com/courses/572452/assignments/3582873/submissions/219835206 12/24 Q5.3 2 Points P ≻ 1 P ≻ 2 P 3 U ( r ) = 0 U ( r ) = 3 U ( r ) = r U ( r ) = 2 r + 4 U ( r ) = − r U ( r ) = r 2 U ( r ) = − r 2 U ( r ) = r U ( r ) = − r Irrational preferences!

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12/15/23, 10:00 PM View Submission | Gradescope https://www.gradescope.com/courses/572452/assignments/3582873/submissions/219835206 13/24 Q5.4 2 Points P ≺ P ≺ P and P ≺ 50-50 lottery among P and P ( 1 2 3 ) ( 2 ( 1 3 )) U ( r ) = 0 U ( r ) = 3 U ( r ) = r U ( r ) = 2 r + 4 U ( r ) = − r U ( r ) = r 2 U ( r ) = − r 2 U ( r ) = r U ( r ) = − r Irrational preferences!

12/15/23, 10:00 PM View Submission | Gradescope https://www.gradescope.com/courses/572452/assignments/3582873/submissions/219835206 14/24 Q5.5 2 Points P ≻ P ≻ P and P ≻ 50-50 lottery among P and P ( 1 2 3 ) ( 2 ( 1 3 )) U ( r ) = 0 U ( r ) = 3 U ( r ) = r U ( r ) = 2 r + 4 U ( r ) = − r U ( r ) = r 2 U ( r ) = − r 2 U ( r ) = r U ( r ) = − r Irrational preferences!

12/15/23, 10:00 PM View Submission | Gradescope https://www.gradescope.com/courses/572452/assignments/3582873/submissions/219835206 15/24 Q5.6 2 Points P ≺ P and P ≺ P and 50-50 lottery among P and P ≺ 50 ( 1 2 ) ( 2 3 ) (( 2 3 ) ( U ( r ) = 0 U ( r ) = 3 U ( r ) = r U ( r ) = 2 r + 4 U ( r ) = − r U ( r ) = r 2 U ( r ) = − r 2 U ( r ) = r U ( r ) = − r Irrational preferences!

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12/15/23, 10:00 PM View Submission | Gradescope https://www.gradescope.com/courses/572452/assignments/3582873/submissions/219835206 16/24 Q5.7 2 Points Which of the following would be a utility function for a risk- seeking preference? That is, for which utility(s) would Pacman prefer entering a lottery for a random food pellet, with expected size , over receiving a pellet of size ? s s U ( r ) = 0 U ( r ) = 3 U ( r ) = r U ( r ) = 2 r + 4 U ( r ) = − r U ( r ) = r 2 U ( r ) = − r 2 U ( r ) = r U ( r ) = − r

12/15/23, 10:00 PM View Submission | Gradescope https://www.gradescope.com/courses/572452/assignments/3582873/submissions/219835206 17/24 Q6 Rejection Sampling 12 Points We will work with a Bayes' net of the following structure. In this question, we will perform rejection sampling to estimate . Perform one round of rejection sampling, using the random samples given in the table below. Variables are sampled in the order , , , , . In the boxes below, choose the value (0 or 1) that each variable gets assigned to. Note that the sampling attempt should stop as soon as you discover that the sample will be rejected. In that case mark the assignment of that variable and write none for the rest of the variables. When generating random samples, use as many values as needed from the table below, which we generated independently and uniformly at random from . Use numbers from left to right. To sample a binary variable with probability and using a value from the table, choose if and if . P ( C = 1 ∣ B = 1, E = 1) A B C D E [0, 1) W P ( W = 0) = p P ( W = 1) = 1 − p a W = 0 a < p W = 1 a ≥ p

12/15/23, 10:00 PM View Submission | Gradescope https://www.gradescope.com/courses/572452/assignments/3582873/submissions/219835206 18/24 Enter either a 0 or 1 for each variable that you assign a value to. Upon rejecting a sample, enter its assigned value, and enter none for the remaining variables. For example, if C gets rejected, fill in none for D and E. A: 1 B: 0 C: none D: none E: none Which variable will get rejected?

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12/15/23, 10:00 PM View Submission | Gradescope https://www.gradescope.com/courses/572452/assignments/3582873/submissions/219835206 19/24 Q7 Estimating Probabilities from Samples 6 Points Below are a set of samples obtained by running rejection sampling for the Bayes' net from the previous question. Use them to estimate . The estimation cannot be made whenever all samples were rejected. In this case, input -1 into the box below. Estimation: 1/2 A B C D E None of the variables will get rejected P ( C = 1 ∣ B = 1, E = 1)

12/15/23, 10:00 PM View Submission | Gradescope https://www.gradescope.com/courses/572452/assignments/3582873/submissions/219835206 20/24 Q8 Likeliehood Weighting 12 Points We will work with a Bayes' net of the following structure. In this question, we will perform likelihood weighting to estimate . Generate a sample and its weight, using the random samples given in the table below. Variables are sampled in the order A, B, C, D, E. In the table below, select the assignments to the variables you sampled. When generating random samples, use as many values as needed from the table below, which we generated independently and uniformly at random from . Use numbers from left to right. To sample a binary variable with probability and using a value from the table, choose if and if . P ( C = 1 ∣ B = 1, E = 1) [0, 1) W P ( W = 0) = p P ( W = 1) = 1 − p a W = 0 a < p W = 1 a ≥ p

12/15/23, 10:00 PM View Submission | Gradescope https://www.gradescope.com/courses/572452/assignments/3582873/submissions/219835206 21/24 Input Answers Here A: 1 B: 1 C: 0 D: 0 E: 1 What is the weight for the sample you obtained above? .64

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12/15/23, 10:00 PM View Submission | Gradescope https://www.gradescope.com/courses/572452/assignments/3582873/submissions/219835206 22/24 Q9 Estimating Probabilities from Weighted Samples 6 Points Below are a set of weighted samples obtained by running likelihood weighting for the Bayes' net from the previous question. Use them to estimate . Input -1 in the box below if the estimation cannot be made. Estimation: 0.78 Q10 Feedback 0 Points Optionally, you can provide any feedback for the course that you have here. Anything you write here will not affect your grade and will stay anonymous. P ( C = 1 ∣ B = 1, E = 1)

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12/15/23, 10:00 PM View Submission | Gradescope https://www.gradescope.com/courses/572452/assignments/3582873/submissions/219835206 23/24 Graded Homework 7 (Part 1) 41 Days, 19 Hours Late Select each question to review feedback and grading details. Student Lucy Yang Total Points 96 / 96 pts Question 1 Decisions 21 / 21 pts 1.1 (no title) 5 / 5 pts 1.2 (no title) 16 / 16 pts Question 2 Value of Perfect Information 10 / 10 pts Question 3 Rationality of Utilities 9 / 9 pts 3.1 (no title) 3 / 3 pts 3.2 (no title) 3 / 3 pts 3.3 (no title) 3 / 3 pts Question 4 Certainty Equivalent Values 6 / 6 pts 

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12/15/23, 10:00 PM View Submission | Gradescope https://www.gradescope.com/courses/572452/assignments/3582873/submissions/219835206 24/24 Question 5 Preferences and Utilities 14 / 14 pts 5.1 (no title) 2 / 2 pts 5.2 (no title) 2 / 2 pts 5.3 (no title) 2 / 2 pts 5.4 (no title) 2 / 2 pts 5.5 (no title) 2 / 2 pts 5.6 (no title) 2 / 2 pts 5.7 (no title) 2 / 2 pts Question 6 Rejection Sampling 12 / 12 pts Question 7 Estimating Probabilities from Samples 6 / 6 pts Question 8 Likeliehood Weighting 12 / 12 pts Question 9 Estimating Probabilities from Weighted Samples 6 / 6 pts Question 10 Feedback 0 / 0 pts

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