Chapter 16, Making Simple Decisions Video Solutions, Artificial Intelligence. A Modern Approach [Global Edition]

Problem 1

(Adapted from David Heckerman.) This exercise concerns the Almanac Game, which is used by decision analysts to calibrate numeric estimation. For each of the questions that follow, give your best guess of the answer, that is, a number that you think is as likely to be too high as it is to be too low. Also give your guess at a 25th percentile estimate, that is, a number that you think has a $25 \%$ chance of being too high, and a $75 \%$ chance of being too low. Do the same for the 75th percentile. (Thus, you should give three estimates in all-low, median, and high-for each question.)
a. Number of passengers who flew between New York and Los Angeles in 1989.
b. Population of Warsaw in 1992.
c. Year in which Coronado discovered the Mississippi River.
d. Number of votes received by Jimmy Carter in the 1976 presidential election.
e. Age of the oldest living tree, as of 2002.
f. Height of the Hoover Dam in feet.
g. Number of eggs produced in Oregon in 1985.
h. Number of Buddhists in the world in 1992.
i. Number of deaths due to AIDS in the United States in 1981.
j. Number of U.S. patents granted in 1901 .

The correct answers appear after the last exercise of this chapter. From the point of view of decision analysis, the interesting thing is not how close your median guesses came to the real answers, but rather how often the real answer came within your $25 \%$ and $75 \%$ bounds. If it was about half the time, then your bounds are accurate. But if you're like most people, you will be more sure of yourself than you should be, and fewer than half the answers will fall within the bounds. With practice, you can calibrate yourself to give realistic bounds, and thus be more useful in supplying information for decision making. Try this second set of questions and see if there is any improvement:
a. Year of birth of Zsa Zsa Gabor.
b. Maximum distance from Mars to the sun in miles.
c. Value in dollars of exports of wheat from the United States in 1992.
d. Tons handled by the port of Honolulu in 1991.
e. Annual salary in dollars of the governor of California in 1993 .
f. Population of San Diego in 1990.
g. Year in which Roger Williams founded Providence, Rhode Island.
h. Height of Mt. Kilimanjaro in feet.
i. Length of the Brooklyn Bridge in feet.
j. Number of deaths due to automobile accidents in the United States in 1992.

Charles Carter

Numerade Educator

04:04

Problem 2

Chris considers five used cars before buying the one with maximum expected utility. Pat considers eleven cars and does the same. All other things being equal, which one is more likely to have the better car? Which is more likely to be disappointed with their car's quality? By how much (in terms of standard deviations of expected quality)?

Lucas Finney

Numerade Educator

06:36

Problem 3

In 1713, Nicolas Bemoulli stated a puzzle, now called the St. Petersburg paradox, which works as follows. You have the opportunity to play a game in which a fair coin is tossed repeatedly until it comes up heads. If the first heads appears on the $n$th toss, you win $2^n$ dollars.
a. Show that the expected monetary value of this game is infinite.
b. How much would you, personally, pay to play the game?
c. Nicolas's cousin Daniel Bernoulli resolved the apparent paradox in 1738 by suggesting that the utility of money is measured on a logarithmic scale (i.e., $U\left(S_n\right)=a \log _2 n+b$, where $S_n$ is the state of having $$\$ n$$ ). What is the expected utility of the game under this assumption?
d. What is the maximum amount that it would be rational to pay to play the game, assuming that one's initial wealth is $$\$ k$$ ?

Oluwadamilola Ameobi

Numerade Educator

02:55

Problem 4

Write a computer program to automate the process in Exercise 16.8. Try your program out on several people of different net worth and political outlook. Comment on the consistency of your results, both for an individual and across individuals.

Patina Herring

Numerade Educator

02:19

Problem 5

The Surprise Candy Company makes candy in two flavors: $75 \%$ are strawberry flavor and $25 \%$ are anchovy flavor. Each new piece of candy starts out with a round shape; as it moves along the production line, a machine randomly selects a certain percentage to be trimmed into a square; then, each piece is wrapped in a wrapper whose color is chosen randomly to be red or brown. $70 \%$ of the strawberry candies are round and $70 \%$ have a red wrapper, while $90 \%$ of the anchovy candies are square and $90 \%$ have a brown wrapper. All candies are sold individually in sealed, identical, black boxes.

Now you, the customer, have just bought a Surprise candy at the store but have not yet opened the box. Consider the three Bayes nets in Figure 16.11.
c. Does network (i) assert that $\mathbf{P}($ Wrapper $\mid$ Shape $)=\mathbf{P}($ Wrapper $)$ ?
d. What is the probability that your candy has a red wrapper?
e. In the box is a round candy with a red wrapper. What is the probability that its flavor is strawberry?
f. A unwrapped strawberry candy is worth $s$ on the open market and an unwrapped anchovy candy is worth $a$. Write an expression for the value of an unopened candy box.
g. A new law prohibits trading of unwrapped candies, but it is still legal to trade wrapped candies (out of the box). Is an unopened candy box now worth more than less than, or the same as before?

Dominador Tan

Numerade Educator

Problem 6

Prove that the judgments $B \succ A$ and $C \succ D$ in the Allais paradox (page 620) violate the axiom of substitutability.

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Problem 7

Consider the Allais paradox described on page 620: an agent who prefers $B$ over $A$ (taking the sure thing), and $C$ over $D$ (taking the higher EMV) is not acting rationally, according to utility theory. Do you think this indicates a problem for the agent, a problem for the theory, or no problem at all? Explain.

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Problem 8

Assess your own utility for different incremental amounts of money by running a series of preference tests between some definite amount $M_1$ and a lottery $\left[p, M_2 ;(1-p), 0\right]$. Choose different values of $M_1$ and $M_2$, and vary $p$ until you are indifferent between the two choices. Plot the resulting utility function.

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09:39

Problem 9

How much is a micromort worth to you? Devise a protocol to determine this. Ask questions based both on paying to avoid risk and being paid to accept risk.

Arulmozhi T

Numerade Educator

02:42

Problem 10

Let continuous variables $X_1, \ldots, X_k$ be independently distributed according to the same probability density function $f(x)$. Prove that the density function for $\max \left\{X_1, \ldots, X_k\right\}$ is given by $k f(x)(F(x))^{k-1}$, where $F$ is the cumulative distribution for $f$.

Ahmad Reda

Numerade Educator

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Problem 11

Economists often make use of an exponential utility function for money: $U(x)=$ $-e^{x / R}$, where $R$ is a positive constant representing an individual's risk tolerance. Risk tolerance reflects how likely an individual is to accept a lottery with a particular expected monetary value (EMV) versus some certain payoff. As $R$ (which is measured in the same units as $x$ ) becomes larger, the individual becomes less risk-averse.
a. Assume Mary has an exponential utility function with $$R=\$ 400$$. Mary is given the choice between receiving $$\$ 400$$ with certainty (probability 1 ) or participating in a lottery which has a $60 \%$ probability of winning $$\$ 5000$$ and a $40 \%$ probability of winning nothing. Assuming Marry acts rationally, which option would she choose? Show how you derived your answer.
b. Consider the choice between receiving $$\$ 100$$ with certainty (probability 1 ) or participating in a lottery which has a $50 \%$ probability of winning $$\$ 500$$ and a $50 \%$ probability of winning nothing. Approximate the value of $\mathrm{R}$ (to 3 significant digits) in an exponential utility function that would cause an individual to be indifferent to these two alternatives. (You might find it helpful to write a short program to help you solve this problem.)

Lainey Roebuck

Numerade Educator

02:12

Problem 12

Alex is given the choice between two games. In Game 1, a fair coin is flipped and if it comes up heads, Alex receives $$\$ 100$$. If the coin comes up tails, Alex receives nothing. In Game 2, a fair coin is flipped twice. Each time the coin comes up heads, Alex receives $$\$ 50$$, and Alex receives nothing for each coin flip that comes up tails. Assuming that Alex has a monotonically increasing utility function for money in the range $$[\$ 0, \$ 100]$$, show mathematically that if Alex prefers Game 2 to Game 1, then Alex is risk averse (at least with respect to this range of monetary amounts).

A M

Numerade Educator

01:47

Problem 13

Show that if $X_1$ and $X_2$ are preferentially independent of $X_3$, and $X_2$ and $X_3$ are preferentially independent of $X_1$, then $X_3$ and $X_1$ are preferentially independent of $X_2$.

Jacob Fry

Numerade Educator

02:00

Problem 14

Repeat Exercise 16.18, using the action-utility representation shown in Figure 16.7.

Khoobchandra Agrawal

Numerade Educator

03:54

Problem 15

For either of the airport-siting diagrams from Exercises 16.18 and 16.14 , to which conditional probability table entry is the utility most sensitive, given the available evidence?

Barsha Rana

Numerade Educator

Problem 16

Modify and extend the Bayesian network code in the code repository to provide for creation and evaluation of decision networks and the calculation of information value.

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Problem 17

Consider a student who has the choice to buy or not buy a textbook for a course. We'll model this as a decision problem with one Boolean decision node, $B$, indicating whether the agent chooses to buy the book, and two Boolean chance nodes, $M$, indicating whether the student has mastered the material in the book, and $P$, indicating whether the student passes the course. Of course, there is also a utility node, $U$. A certain student, Sam, has an additive utility function: 0 for not buying the book and $$\$ 100$$ for buying it; and $$\$ 2000$$ for passing the course and 0 for not passing. Sam's conditional probability estimates are as follows:
$$
\begin{array}{ll}
P(p \mid b, m)=0.9 & P(m \mid b)=0.9 \\
P(p \mid b, \neg m)=0.5 & P(m \mid \neg b)=0.7 \\
P(p \mid \neg b, m)=0.8 & \\
P(p \mid \neg b, \neg m)=0.3 &
\end{array}
$$

You might think that $P$ would be independent of $B$ given $M$, But this course has an openbook final—so having the book helps.
a. Draw the decision network for this problem.
b. Compute the expected utility of buying the book and of not buying it.
c. What should Sam do?

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02:46

Problem 18

This exercise completes the analysis of the airport-siting problem in Figure 16.6.
a. Provide reasonable variable domains, probabilities, and utilities for the network, assuming that there are three possible sites.
b. Solve the decision problem.
c. What happens if changes in technology mean that each aircraft generates half the noise?
d. What if noise avoidance becomes three times more important?
e. Calculate the VPI for AirTraffic, Litigation, and Construction in your model.

Wendi Zhao

Numerade Educator

02:01

Problem 19

16.19 (Adapted from Pearl (1988).) A used-car buyer can decide to carry out various tests with various costs (e.g., kick the tires, take the car to a qualified mechanic) and then, depending on the outcome of the tests, decide which car to buy. We will assume that the buyer is deciding whether to buy car $c_1$, that there is time to carry out at most one test, and that $t_1$ is the test of $c_1$ and costs $$\$ 50$$.

A car can be in good shape (quality $q^{+}$) or bad shape (quality $q^{-}$), and the tests might help indicate what shape the car is in. Car $c_1$ costs $$\$ 1,500$$, and its market value is $$\$ 2,000$$ if it is in good shape; if not, $$\$ 700$$ in repairs will be needed to make it in good shape. The buyer's estimate is that $c_1$ has a $70 \%$ chance of being in good shape.
a. Draw the decision network that represents this problem.
b. Calculate the expected net gain from buying $c_1$, given no test.
c. Tests can be described by the probability that the car will pass or fail the test given that the car is in good or bad shape. We have the following information:
$P\left(\right.$ pass $\left.\left(c_1, t_1\right) \mid q^{+}\left(c_1\right)\right)=0.8$
$P\left(\right.$ pass $\left.\left(c_1, t_1\right) \mid q^{-}\left(c_1\right)\right)=0.35$
Use Bayes' theorem to calculate the probability that the car will pass (or fail) its test and hence the probability that it is in good (or bad) shape given each possible test outcome.
d. Calculate the optimal decisions given either a pass or a fail, and their expected utilities.
e. Calculate the value of information of the test, and derive an optimal conditional plan for the buyer.

Amy Jiang

Numerade Educator

Problem 20

Recall the definition of value of information in Section 16.6.
a. Prove that the value of information is nonnegative and order independent.
b. Explain why it is that some people would prefer not to get some information-for example, not wanting to know the sex of their baby when an ultrasound is done.
c. A function $f$ on sets is submodular if, for any element $x$ and any sets $A$ and $B$ such that $A \subseteq B$, adding $x$ to $A$ gives a greater increase in $f$ than adding $x$ to $B$ :
$$
A \subseteq B \Rightarrow(f(A \cup\{x\})-f(A)) \geq(f(B \cup\{x\})-f(B)) .
$$

Submodularity captures the intuitive notion of diminishing returns. Is the value of information, viewed as a function $f$ on sets of possible observations, submodular? Prove this or find a counterexample.

The answers to Exercise 16.1 (where $\mathrm{M}$ stands for million): First set: $3 \mathrm{M}, 1.6 \mathrm{M}, 1541,41 \mathrm{M}$, 4768, 221, 649M, 295M, 132, 25,546. Second set: 1917, 155M, 4,500M, 11M, 120,000, $1.1 \mathrm{M}, 1636,19,340,1,595,41,710$.

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