Chapter 13: Social Choice Theory

Question 1

Voting Paradox

Answer 1

Illustrates the problem of trying to aggregate everyone’s individual preferences into a group preference ordering.

Question 2

If the group preferences are __________, then they can’t be used to make __________ choices.

Answer 2

cyclic; group

Question 3

In cases where there’s trouble in deciding what choice to make as a group, why wouldn’t randomization, like flipping a coin, be a viable option for deciding what choice the group makes?

Answer 3

Because flipping a coin wouldn’t tell us why one option is better/should be chosen over another.

Question 4

Social Choice Theory

Answer 4

Study of collective decision problems.

Question 5

Social Choice Theory v. Game Theory

Answer 5

Social Choice Theory aims to make the best possible decisions AS A GROUP / in a group setting.

In Game Theory, one makes the most individually optimal decision while interaction with another who makes a (individually optimal) decision.

Question 6

Social Decision Problem

Answer 6

• Any group decision problem, where the individual members have ordinal preferences over outcomes.
• Individual’s ordinal preferences over outcomes MUST follow axioms of…
• Completeness (A≻B, or B≻A, or A~B)
• Asymmetry (if A≻B, then it’s false that B≻A)
• Transitivity (A≻B, B≻C, and A≻C)

Question 7

Should an individual’s ordinal preferences over outcomes follow the 3 axioms of _______________, ________________, and ________________, then we now have an ________________ ______________ _________________.

Answer 7

completeness; asymmetry; transitivity; individual preference ordering.

Question 8

Individual Preference Ordering

Answer 8

Preference ordering that reflects the interest of the individual.

Question 9

Social Preference Ordering

Answer 9

Taking the individual preference orderings and aggregating them into a preference ordering that reflects the interest of the group.

Question 10

Social State

Answer 10

Everything within the world that an individual cares about.

(E.g. Acapulco, Belize, Cape Cod)

Question 11

Social Welfare Function (SWF)

Answer 11

Aggregates individual preference ordering over social state(s), into a social preference ordering over those social states.

Question 12

Example of Social Welfare Function (SWF)

Answer 12

The Majority Rule

(SEE NOTES: after aggregating Megan, Joe, and Nick’s individual preferences orderings over the social states (a), (b), and (c), we got a social preference ordering over those social states, wherein the preferences were cyclic).

Question 13

Review Social Choice Problem Regarding M, N, O, and Q (SEE NOTES)

Question 14

What’s the problem with using the Majority Rule when aggregating individual (G) and social (S) preference orderings ? (SEE NOTES; M, N, O, Q)

Answer 14

When using the majority rule, the social preference ordering may coincide with the individual preference ordering, however, it may not always be this way…

For example, in the M, N, O, Q problem, the individual preference ordering for M coincided with the aggregated social preference ordering. This is great news for M.

But what if M’s preferences, of the social states, a≻b, was switched to b≻a? Then M’s individual preference ordering (G) would no longer coincide with the social preference ordering (S).

The Majority Rule (SWF) would change, meaning M is not nondictorial.

Question 15

Nondictorial (See Notes)

Answer 15

means that the social preference ordering doesn’t ALWAYS coincide with a particular individual preference ordering like (M).

Question 16

Decisive

Answer 16

A group D from individuals of Group G are decisive with respect to the ordered pair of social states (a,b) iff. whenever everyone from Group D prefers a over b.

If Group D is decisive with respect to ALL ordered pairs of social states, then Group D is plainly decisive, full stop.

Question 17

Nondictatorship

Answer 17

Group D is a Nondictatorship iff. no single member from Group D (from individuals of Group G) is decisive.

Question 18

Explain why the Majority Rule (SWF) can be used to solve social choice problems.

Then explain why the Majority Rule can’t ALWAYS be used reliably to solve social choice problems.

(SEE NOTES)

Answer 18

The majority rule can prove that an individual preference ordering is not a dictatorship (meaning social preference ordering doesn’t ALWAYS coincide with an individual preference ordering).

However, the majority rule can also produce cyclic preferences (violating transitivity).

Question 19

So, then, based on what we know about the Majority Rule, what is the Ordering Rule for Social Welfare Functions?

Answer 19

Social Welfare Functions can only be used to social choice problems if they satisfy the axioms of Completeness, Asymmetry, and Transitivity.

Question 20

Arrows impossibility Theorem

Answer 20

There is no Social Welfare Function (SWF) that satisfies the conditions of…
- Nondictatorship
- Ordering

Question 21

Pareto Condition

Answer 21

If all individuals of a group prefer a to b, then the group must prefer a to b.

Question 22

The Pareto Condition is consistent with ________________. Explain why. (SEE NOTES)

Answer 22

Decisiveness

• A group D is decisive with respect to the ordered pairs of social states (a,b) iff. the society prefers a to b whenever every individual prefers a to b.
• If group D is decisive with respect to the ordered pairs of all social states, then Group D is plainly decisive, full stop.

So, with the Pareto condition, if all individuals within a group prefer a to b, then the group prefers a to b, this means all individuals within a group are decisive.

Question 23

Independence of Irrelevant Alternatives:

Answer 23

Social ranking of social states (a,b) (where a is preferred to b) should be based on THOSE social states alone.

The social ranking of (a,b) should be based on those social states; independent of individual preference ordering of c.

Question 24

Liberalism

Answer 24

weak normative claim

Question 25

Amartya Sen suggest that the ____________ _________________ (which has to do with individuals of a group preferring a over b, then the group prefers a over b) is __________________ with Liberalism.

Answer 25

pareto principle; inconsistent

Question 26

Minimal Liberalism Condition

Answer 26

There’s at least one pair of alternatives (a,b), such that If one individual prefers social states a over b, then society should prefer a over b.

There’s at least one pair of alternatives (a,b) that the one individual is decisive about.

Question 27

Example of Minimal Liberalism Condition

Answer 27

Ex. Painting Interior Walls Pink instead of white:
- If one individual prefers painting their walls pink, instead of painting their walls white, then society should prefer that the one individual paint his walls pink instead of white.

Ex. Free Speech:
- If one individual prefers to say statement a, over saying statement b, then society should prefer that he say a over b.

Question 28

Weaker Definition of Minimal Liberalism Condition

Answer 28

There are TWO individuals who are decisive with respect to pairs of alternatives, each to their own, (a,b), such that if they prefer a over b, then everyone in society prefers a over b.

Question 29

What does this weaker definition of Minimal Liberalism do?

Answer 29

It keeps an individual’s individual preference orderings outside of the social preference ordering.

Question 30

Explain the Paradox of being a Paretian Liberal (holding to the Pareto Principle, while also holding to liberalism).

[DO THIS IN WRITING]

Answer 30

As we saw earlier, the Pareto Principle is uncontroversial: if everyone in a group prefers a to b, then society prefers a to b.

So, it’s easy to see why Pareto principle would be of least suspicion–it’s pretty straightforward.

So, this leads us to believe, if anything is to be the problem, it would probably be liberalism.

For one to hold to the Pareto Principle and Minimal Liberalism, is paradoxical:

Let’s say X holds that a≻b, and Y holds that d≻a. Therefore society holds that a≻b and d≻a.

So far, this follows our minimal liberalism: you have to individuals who are decisive with respect to a set of preferences (the pairs of alternatives for the two individuals should be different, but they may share a joint alternative (i.e. a)), and therefore society holds the same.

Now let’s say, everyone in society holds that b≻d, including X and Y.

This is consistent with pareto principle: if every member of society holds that a≻b, then society holds that a≻b.

So, combining what’s said of the social preference ordering with X and Y’s individual preference ordering, it says: X prefers a≻b≻d, and Y prefers b≻d≻a.

The cyclical nature of preferences, that result from being Paretian Liberal, can be shown even when x and y have completely disjoint preferences.

Le’ts say X holds that a≻b and Y holds c≻d.

So far, this follows our minimal liberalism: you have to individuals who are decisive with respect to a set of preferences (the pairs of alternatives for the two individuals should be different, but they may share a joint alternative), and therefore society holds the same.

Now let’s suppose society holds that d≻a, and b≻c, including X and Y.

This is consistent with pareto principle: if every member of society holds that a≻b, then society holds that a≻b.

Combining X and Y’s individual preference ordering with this social preference ordering, we get that: X prefers d≻a≻b≻c and Y prefers b≻c≻d≻a

From this, we generate another cycle when aggregating X and Y new preferences, violating the Ordering Condition:
d≻a≻b≻c≻d.

Question 31

Robert Nozick’s Response to Minimal Liberalism and Pareto Principle:

(Hint: Leftovers)

Answer 31

• Amartya Sen has misunderstood minimal liberalism. The social preference ordering is only supposed to account for the leftovers after distinguishing the individual preference orderings that don’t need to be accounted for in the social preference ordering.

The ordering condition (SWF’s must follow completeness, asymmetry, and transitivity axioms) should not account for every preference of an individuals preference orderings.

• E.g. An individual’s preference about what color their interior walls should be should not be accounted for by the social preference ordering.
• Why should the SWF account for everything that an individual care’s about?

Question 32

Harsanyi uses a ______________ approach to solving social decision problems.

Answer 32

Utilitarian

Question 33

What is the utilitarian approach:

Answer 33

The right action to choose is the one that maximizes the utility of everyone who will be affected by making this choice.

Question 34

Utilitarianism: High Tax Rate Example

Answer 34

If one individual prefers a high tax rate, while everyone else in society prefers a low tax rate, then the rest of society should nevertheless prefer a high tax rate, because the one individual’s preference for a high tax rate is considerably strong.

Question 35

Utilitarianism: Doctor’s Choice Example

Answer 35

A doctor seeks to heal 5 ill patients all of whom need, certain organs to live. The doctor could either let these patients die, or kill one healthy person, harvest their organs, and distribute the organs among the 5 patients that need them. in this situation, the doctor should kill the healthy person, because it maximizes utility for a greater number of people.

Question 36

You should ________________ the _______________ / ________-______________ of others up to the point that your _____________ / ________-_____________ is now just a level ____________ a ______________ state, when trying to ______________ the ____________ of others.

Answer 36

maximize; utility; well-being; utility; well-being; above; miserable; maximize; utility

Question 37

For Harsanyi to use the _______________ approach, she has to adopt a few technical _________________. (We will explain the reasoning behind these assumptions, first).

Answer 37

utilitarian; assumptions

Question 38

What does Harsanyi believe about individual preference orderings, that is contrary to what Arrow believes about them?

Answer 38

Harsanyi believes that individual preference orderings don’t merely have to be represented as ordinal values (a is better than b,…) like Arrow does.

Harsanyi believes that, because individual preference orderings OVER SOCIAL STATES satisfy the von-Neuman Morgenstern (vNM) axioms, the individual preferences can be represented as utility assignments on an interval scale.

These utility assignments can be combined (in some mathematical way) by using a utility function, that yield a social preference ordering.

Question 39

Harsanyi then introduces the concept of the “Chairperson.” Elaborate.

Answer 39

Harsanyi asks us imagine a Chairperson, who collects the utility assignments (which we’ve already established as the individual preference orderings) of all individuals in society, and aggregates them into a social preference ordering.

• If the Chairperson is a member of society, then he has two preference orderings.

1) Individual Preference Ordering: his own preferences / views on the way things should be.

2) Social Preference Ordering: the preference ordering he has after combining the individual preference ordering from everyone else in society.

Question 40

Why would Rawls object to Harsanyi’s use of utilitarianism?

Answer 40

• Let’s keep in mind, Rawls has done similar work, where he’s proposed this impartial, detached, sympathetic spectator (like the Chairperson) in his own work.
• Rawl’s wouldn’t dare use utilitarianism because of the “extreme cases” wherein one individual may end up on the bad side of this utilitarian decision making, which is the greatest good for the greatest number of people. These extreme cases (death, poverty, exploitation, loss of rights, slavery) have a large disparity between themselves on the non-extreme cases.
• Rawl’s prefers a maximin approach–maximizing the minimal utility obtainable–to close the gap (in a sense) between the extreme and non-extreme cases.
• He doesn’t like the maximizing expected utility approach that Harsanyi uses, where the extreme cases and non-extreme cases are accounted for by taking the average utility.

Question 41

1st. Assumption) Individual Rationality:

(Explain what it means)

Answer 41

One’s individual preference orderings satisfy von-Neuman Morgenstern axioms:

(Essentially means, because the personal preferences satisfy the vNM axioms, they can be represented as utility assignment on an interval utility scale. These utility assignments can be combined, using a utility function, into a social preference ordering).

Question 42

2nd. Assumption) Social Preference Rationality:

(Explain what it means)

Answer 42

The Chairperson’s social preference orderings satisfy von-Neuman Morgenstern axioms:

(Essentially means, as it did for Individual Rationality, that the individual preference orderings (that follow vNM axioms) aggregated to construct the Chairperson’s social preference ordering can be expressed as utility assignments on a utility interval scale).

Question 43

3rd. Assumption) Pareto (Harsanyi Condition):

Answer 43

If one individual from the group prefers a over b, and everyone else in society is indifferent between a and b, then society should prefer a over b.

• Tempered with the fact that if EVERYONE is indifferent between a and b, then society should remain indifferent between a and b.

Question 44

How to Calculate Social Preference Ordering:

Answer 44

Social Preference Ordering = weighted sum of each individuals utility over a social state.

E.g.
Social Preference Ordering of ‘a’ = weighted sum of each individual’s utility (individual preference ordering) over social state ‘a’.

–> You do this for all social states.

Question 45

After doing these calculations, which ever _______________ _____________ ordering has the______________ score for _________________ ________________ is the ________________ preferred.

Answer 45

social preference; highest; social utility; most

Question 46

We mentioned earlier that the calculation of social preference orderings involves multiplying weights against the utility assignments representing one’s individual preference ordering over a social state. What are these weights? Is one person’s utility twice as much another person’s? 3x? Should we assign utility based on numbers ranging between 1 and 0? How should we go about this?

Answer 46

It’s important to keep in mind, Utilitarians are largely impartial. Seeing as they’re equal opportunist, the best thing to do, would be to apply equiprobable weight to each of the weights.

Question 47

If Harsanyi is going to apply equiprobable weights, then she must employ the ___________ _____________ Condition.

Answer 47

Equal Treatment Condition

Question 48

Equal Treatment Condition

Answer 48

If each individual’s preference ordering (utility assignments) are equally weighted, then the social preference ordering that follows must assign equal weight to all individual utility functions.

(keep in mind, the utility functions are used to create the social preference orderings, so they have to be equally weighted!)

Question 49

2 Objections to the Equal Treatment Condition

Answer 49

1) Why should one’s individual’s utilities be treated equally?
2) Are these utilities even comparable?
– Does it make sense to draw comparisons between these utilities.