Chapter 4, Decisions under risk Video Solutions, An Introduction to Decision Theory

Problem 1

Consider the decision problem illustrated below.
$$
\begin{array}{llll}
\hline & 1 / 2 & 1 / 4 & 1 / 4 \\
\hline a_1 & \$ 49 & \$ 25 & \$ 25 \\
a_2 & \$ 36 & \$ 100 & \$ 0 \\
a_3 & \$ 81 & \$ 0 & \$ 0 \\
\hline
\end{array}
$$
(a) The decision maker's utility $u$ of money is linear. Which act should be chosen according to the principle of maximising expected monetary value?
(b) The decision maker's utility $u$ of money $x$ is given by the formula $u(x)=\sqrt{x}$. Which act should be chosen according to the principle of maximising expected utility?

Zachary Watson

Numerade Educator

06:11

Problem 2

I am in my office in Cambridge, but I have to catch a flight from Heathrow this afternoon. I must decide whether to go to Heathrow by coach, which comes relatively cheap at $£ 40$, or buy a train ticket for $£ 70$. If I take the coach I might get stuck in an intense traffic jam and miss my flight. I would then have to buy a new ticket for $£ 100$. According to the latest statistics, the traffic jam on the M25 to Heathrow is intense one day in three. (a) Should I travel by train or coach? (b) This description of my decision problem overlooks a number of features that might be relevant. Which?
(a) You are in Las Vegas. The probability of winning a jackpot of $$\$ 350,000$$ is one in a million. How much should you, who find no reason to reject the principle of maximising expected utility, be prepared to pay to enter this gamble? Your utility of money is $u(x)=\ln (x+1)$.
(b) This time the probability of winning the jackpot of $$\$ 350,000$$ is one in a thousand. How much should you, who find no reason to reject the principle of maximising expected utility, be prepared to pay to enter this gamble? Your utility of money is $u(x)=\ln (x+1)$.
(c) Why is the difference between the amount you are willing to pay in (a) and (b) so small?
(d) Why did we assume that your utility function is $u(x)=\ln (x+1)$, rather than just $u(x)=\ln (x)$ ?

Dalia Rodriguez

Numerade Educator

05:33

Problem 5

(a) Explain why Allais' and Ellsberg's paradoxes pose difficulties for the principle of maximising expected utility. (b) Explain the difference between the two paradoxes - they arise for two different reasons.

Rosina Dapaah

Numerade Educator

00:38

Problem 6

Suppose that you prefer Gamble 1 to Gamble 2, and Gamble 4 to Gamble 3. Show that your preferences are incompatible with the principle of maximising expected utility, no matter what your utility of money happens to be.
$$
\begin{array}{llll}
\hline & 1 / 3 & 1 / 3 & 1 / 3 \\
\hline \text { Gamble 1 } & \$ 50 & \$ 50 & \$ 50 \\
\text { Gamble 2 } & \$ 100 & \$ 50 & \$ 0 \\
\text { Gamble 3 } & \$ 50 & \$ 0 & \$ 50 \\
\text { Gamble 4 } & \$ 100 & \$ 0 & \$ 0 \\
\hline
\end{array}
$$
(a) Explain why the St Petersburg paradox poses a difficulty for the principle of maximising expected utility.
(b) Construct a new version of the St Petersburg paradox, in which the player rolls a six-sided die instead of tossing a coin.

Hunza Gilgit

Numerade Educator

Problem 8

There is an interesting connection between the St Petersburg and the twoenvelope paradox - explain!

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