Chapter 13, Social choice theory Video Solutions, An Introduction to Decision Theory

Problem 1

A group of four people is about to select one out of six possible social states, $a, b, c, d, e$ or $f$. (a) Which state will be selected if the decision is based on a traditional voting procedure? (Does the result depend on how the voting procedure is set up, given that all people get one vote each?) (b) Which state would be selected by the maximin rule? (It prescribes that society should prefer a state in which the worst-off person is as well off as possible.)
Anne:$a>b>c>d>e$
Bert: $b>a>c>e>d$
Carl: $a>c>d>b>e$
Diana: $a>b>c>d>e$

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01:13

Problem 2

It follows from Arrow's impossibility theorem that social decisions cannot be based on majority voting procedures. (a) Exactly what is Arrow's criticism of majority voting? (b) Do you find his criticism convincing?

Srikar Katta

Numerade Educator

01:24

Problem 3

The Pareto principle entails that if we can improve the situation for the richest person in society, without thereby making things worse for anyone else, we ought to do this. Suppose you object: "But if we improve the situation for the richest person that would make poor people feel even unhappier. Therefore we shouldn't do it!" Even if true, this is not a good objection to the Pareto principle. (a) Why not? (b) Suggest a relevant objection to the Pareto principle, based on the observation that it permits us to improve the situation for the richest person in society, given that this can be achieved without thereby making other people worse off.

Nick Johnson

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00:48

Problem 4

In the proof of Arrow's theorem, what does it mean to say that a group of people $\mathrm{D}$ is decisive with respect to the social states $(a, b)$ ?

Allison Knapp

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05:59

Problem 5

Sen showed that his condition of minimal liberalism is incompatible with the Pareto principle and the ordering condition. (a) Consider the following condition of minimal socialism. Is it compatible with the Pareto principle and the ordering condition?
Minimal socialism: There is at least one pair of alternatives $(a, b)$ such that no single-person group is decisive with respect to $(a, b)$.
(b) Is the condition of minimal socialism compatible with the condition of minimal liberalism? That is, is it possible to be both a liberal and a socialist?

Crystal Wang

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06:12

Problem 6

According to Harsanyi, the Chairperson's social preference ordering over social states must satisfy the independence axiom. Suggest a set of social preferences that violates this axiom.

Narayan Hari

Numerade Educator

01:08

Problem 7

(a) Explain the link between Harsanyi's theorem and the von Neumann-Morgenstern theorem. (b) Could critics of utilitarianism accept the assumption that the Chairperson's preferences must satisfy the von Neumann-Morgenstern axioms? (c) Do you think it would be possible to replace the von Neumann-Morgenstern theorem in Harsanyi's result with some other theorem mentioned in this book?

Nick Johnson

Numerade Educator

Problem 8

Suppose you think all human beings have certain rights that may never be violated, such as a right not to be tortured. (a) Is this moral belief compatible with Harsanyi's axioms? (b) If not, suggest a situation in which your beliefs about the right not to be tortured comes into conflict with Harsanyi's axioms. (c) Exactly which of Harsanyi's axioms are incompatible with the belief that all human beings have a right not to be tortured?

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