Chapter 3, Decisions under ignorance Video Solutions, An Introduction to Decision Theory

Problem 1

Which act(s) should be chosen in the decision problem below according to the following decision rule:
(a) the maximin rule
(b) maximax
(c) minimax regret
(d) the optimism-pessimism rule (for $\alpha>1 / 4$ )
(e) the principle of insufficient reason.
$$
\begin{array}{rrrrr}
\hline a_1 & 20 & 20 & 0 & 10 \\
a_2 & 10 & 10 & 10 & 10 \\
a_3 & 0 & 40 & 0 & 0 \\
a_4 & 10 & 30 & 0 & 0 \\
\hline
\end{array}
$$

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01:18

Problem 2

What is the main advantage of leximin as compared to maximin?

Cody Delk

Numerade Educator

02:06

Problem 3

Construct a decision matrix that corresponds to the following regret table.
$$
\begin{array}{lrrr}
\hline a_1 & -5 & -45 & 0 \\
a_2 & 0 & -30 & -20 \\
a_3 & -10 & 0 & -25 \\
\hline
\end{array}
$$

Ankit Gupta

Numerade Educator

Problem 4

According to the minimax regret principle, regret is defined as the difference between the best outcome obtainable under a given state and the outcome obtained under that state with the chosen act. Now consider an alternative version of this principle, according to which regret is defined (relative to each act) as the difference between the best outcome and the worst outcome obtainable with the act in question. This modification of the minimax regret rule would give us a highly implausible decision rule. Why?

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Problem 5

Prove that Axiom 8 (Column duplication) is incompatible with the principle of insufficient reason.

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04:02

Problem 6

Prove that Axiom 7 (Column linearity) is incompatible with the optimism-pessimism rule.

Prathan Jarupoonphol

Numerade Educator

01:01

Problem 7

Explain informally why Axiom 5 (Interval scale) is compatible with all of the following rules: (a) maximin, (b) optimism-pessimism, (c) minimax regret, (d) the principle of insufficient reason.

Raj Bala

Numerade Educator

02:50

Problem 8

Rawls famously argued that the maximin rule could be used for formulating a theory of social justice. Very briefly put, he argued that a distribution $A$ of social goods is better than an alternative distribution $B$ if the worst-off person is better off in $A$ than in $B$. Suppose that Rawls' theory of social justice is based on the optimism-pessimism rule instead of maximin. What would be the implications for social justice? (Assume that $1>\alpha>0$.)

Natalie Britton

Numerade Educator