Chapter 2, The decision matrix Video Solutions, An Introduction to Decision Theory

Problem 1

If you play roulette in Las Vegas and bet on a single number, the probability of winning is $1 / 38$ : There are 38 equally probable outcomes of the game, viz. 1-36, 0 and 00 . If the ball lands on the number you have chosen the croupier will pay you 35 times the amount betted, and return the bet.
(a) Formalise and visualise the decision problem in a decision matrix.
(b) Formalise and visualise the decision problem in a decision tree.
(c) How much money can you expect to lose, on average, for every dollar you bet?

Ryan Finn

Numerade Educator

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Problem 2

Formalise the following decision problem, known as Pascal's wager:
God either exists or He doesn't. . ... It is abundantly fair to conceive, that there is at least $50 \%$ chance that the Christian Creator God does in fact exist. Therefore, since we stand to gain eternity, and thus infinity, the wise and safe choice is to live as though God does exist. If we are right, we gain everything, and lose nothing. If we are wrong, we gain nothing and lose nothing. Therefore, based on simple mathematics, only a fool would choose to live a Godless life. Since you must choose, let us see which interests you least. You have nothing to lose. Let us estimate these two chances. If you gain, you gain all; if you lose, you lose nothing. Wager, then, without hesitation that $\mathrm{He}$ is.

Rashmi Sinha

Numerade Educator

02:10

Problem 3

(a) Is Pascal's argument convincing?
(b) Is it really necessary for Pascal to assume that, "there is at least $50 \%$ chance that the Christian Creator God does in fact exist'? What if the probability is much lower?

Pankaj Jain

Numerade Educator

07:39

Problem 4

Congratulations! You have won a free holiday in a city of your choice: London, New Delhi or Tokyo. You have been to London before, and you know that the city is okay, but expensive. New Delhi would be very exciting, unless you get a stomach infection; then it would be terrible. Tokyo would be almost as exciting, given that it is not too cold; then the trip would be rather boring.
(a) Formalise and visualise the decision problem in a decision matrix.
(b) Formalise and visualise the decision problem in a decision tree.
(c) Represent the five possible outcomes in an ordinal scale.

Jonathan Tapiwa

Numerade Educator

01:39

Problem 5

A friend offers you to invest all your savings, $$\$ 100,000$$, in his dot com company. You find it very hard to understand the business plan he presents to you, but your friend tells you that your $$\$ 100,000$$ will 'certainly' be worth at least $$\$ 10 \mathrm{M}$$ within two years. Naturally, your friend may be right, but he may also be wrong - you feel that you cannot estimate the probabilities for this. Consider the decision matrix below. What is wrong with this formalisation?
$$
\begin{array}{lll}
\hline & \text { Friend is right } & \text { Friend is wrong } \\
\hline \text { Invest } & \$ 10 \mathrm{M} & \$ 0 \\
\text { Do not } & \$ 100,000 & \$ 100,000 \\
\hline
\end{array}
$$

Manisha Sarker

Numerade Educator

04:47

Problem 6

Visualise the following vector (which is written on a single line, to save space $)$ in a decision matrix: $\left[\left[a_1, a_2, a_3\right] ;\left[s_1, s_2\right] ;\left[\left(a_1, s_1\right)=p,\left(a_1, s_2\right)=q,\left(a_2\right.\right.\right.$, $\left.\left.\left.s_1\right)=r,\left(a_2, s_2\right)=s,\left(a_3, s_1\right)=t,\left(a_3, s_2\right)=u\right]\right]$.

Cory Glover

Numerade Educator

05:08

Problem 7

Explain the difference between (a) ordinal and cardinal scales, and (b) interval scales and ratio scales.

Jennifer Stoner

Numerade Educator

07:26

Problem 8

Your rich aunt died some time ago. In her will she stipulated that you shall receive a painting of your choice from her collection of impressionist art. The aesthetical values of her four paintings are, as measured on your personal interval scale, as follows: Manet 5,000; Monet 8,000; Pissarro 6,000; Renoir 2,000 . Which of the following scales can be obtained from the original scale by a positive linear transformation?
(a) Manet 8; Monet 11; Pissarro 9; Renoir 5
(b) Manet - 250; Monet 2,750; Pissarro 750; Renoir - 3,250
(c) Manet 1,000; Monet 3,000; Pissarro 2,950; Renoir 995

Sheryl Ezze

Numerade Educator

Problem 9

Show that scales (a)-(c) are equivalent ordinal scales.

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00:22

Problem 10

Suppose that scale (a) in Exercise 2.8 is a ratio scale. Show that neither (b) nor (c) is equivalent to that ratio scale.

Amrita Bhasin

Numerade Educator

05:08

Problem 11

Prove that every ratio scale is an interval scale, and that every interval scale is an ordinal scale.

Jennifer Stoner

Numerade Educator

01:43

Problem 12

Suppose $f$ is a ratio scale that can be transformed into $f$ by multiplying all values of $f$ by $k$. Show that $f(x)=\frac{1}{k} \cdot f^{\prime}(x)$.

Umar Sohail Qureshi

Numerade Educator