Group choice: Social choice, Arrow,…

Question 1

What does Kenneth Arrow show?

Answer 1

Group incoherence will not go away if institutional arrangements are changed

Question 2

Kenneth Arrow facts

Answer 2

• 1921-2017
• Economist, mathematician, and political theorist
• Polymath (eg. breeding practices of grey whales)
• “The most important theorist of the 20th century in economics”
• Youngest economist ever to win a Nobel prize in 1972 (age 51)
• He supervised (4) other who would receive the Nobel prize

Question 3

What is Arrow’s main contribution to social choice theory?

Answer 3

Arrow’s impossibility theorem

Question 4

What does Arrow’s impossibility theorem posit?

Answer 4

• When voters have three or more alternatives, no voting system, however cleverly designed, resolves the problems associated with majority voting
• No system of majority voting worked satisfactorily according to a carefully articulated definition of “satisfactory” (which social scientists generally accept)

Question 5

What are the main requirements for Arrow’s theorem?

Answer 5

• Finite set A={a,b,c,…} of at least three different policy options
• Finite number of different individuals, i=1,2,…,N
• Everyone has preferences over the policy options which are complete and transitive

Question 6

Goals of SWF

Answer 6

• We want to solve the Condorcet Paradox
• We are looking for a way to aggregate preferences
  – We want a way to turn each possible set of individual preferences into a preference relation/order for “society”
• The mapping from individual preference relations into social preferences relations is called a social welfare function (SWF) or a preference aggregation rule, or a Constitution

Question 7

How do we want our social welfare function to look?

Answer 7

• We want our social welfare function to satisfy certain properties
• We want the social preference chosen by our SWF to be complete and transitive for each individual preferences
  – Otherwise we will not have a coherent choice rule for society
  – This rules out cyclic majorities
• But it also needs to be “reasonable”

Question 8

What does Arrow want to achieve with SWF?

Answer 8

• The object of Arrow was to find one outcome (also called social maximum) in a political democracy that is consistent with the preferences of all individuals in a society
• Arrow starts with some reasonable conditions (basic assumptions) that he claims are “minimal” or “reasonable” requirements for aggregating individual preferences

Question 9

What are Arrow’s conditions + assumptions

Answer 9

Basic assumption: rationality
- Universal admissibility (condition U)
Minimal conditions
- Pareto optimality / Unanimity (condition P)
- Independence from Irrelevant Alternatives (condition I)
- Non-dictatorship (condition D)

Question 10

Outcome of Arrow’s conditions

Answer 10

• We cannot translate rational individual preferences into a coherent group preference that satisfies all conditions
• The only way of satisfying U, P, and I, is a dictatorship
• This is also known as impossibility theorem

Question 11

What types of conditions/requirements are Arrow’s

Answer 11

Domain requirement
- Condition U
– Group decision procedures works in all environments
Fairness condition
- Condition P
- Condition I
- Condition D
– Value-laden - so called “reasonable conditions”

Question 12

Condition U (Arrow)

Answer 12

Universal admissibility / Universal domain
- Each individual from a group may adopt any strong or weak complete and transitive preference over the alternatives A
– We do not want to restrict the choice of an individual in any manner: anyone can choose whatever they want

Question 13

Condition P (Arrow)

Answer 13

Pareto optimality / Unanimity
- If every individual of the group prefers A to B (A Pi B), then the group preference must also be A Pg B
– This condition ensures that the group choice is responsive to the individual preferences
– If there is no unanimity, we are doing a pretty bad job of aggregating the individuals’ preferences
– This condition rules out stupid rules like “regardless of individual preferences, society ranks policies alphabetically”

Question 14

Condition I (Arrow)

Answer 14

Independence of Irrelevant Alternatives (IIA)
- If A is preferred to B (A P B) out of a choice set {A,B}, introducing a third option C must not make B preferable to A (B P A)
– This says that if we are trying to figure out whether society prefers A to B, what people think of C should not matter

Question 15

Condition D (Arrow)

Answer 15

Non-dictatorship
- There is no distinguished individual (dictator) who dictates the group preference, no matter what the preferences are of the other members of the group
– This is a minimal fairness condition
– There is no one person who dictates the group preference; but what about oligarchies, power elites, exclusive committees?

Question 16

How does Arrow define dictatorship?

Answer 16

• A dictator is a person whose individual preferences coincide with the social ones
• It is simply a person that turns out to be the winning one on all sides
• In other words, the dictator can just be a pivotal voter
• It does not have to be “Sadam Hussain” or “Vladimir Putin”

Question 17

What did Arrow find about all conditions existing simultaneously?

Answer 17

Arrow showed that they cannot be satisfied simultaneously
- Any SWF which respects transitivity and completeness, unanimity, and independence of irrelevant alternatives is a dictatorship
- This is called the Impossibility Theorem: there is no SWF satisfying all of U, P, I, and D, if there are 3 or more alternatives

Question 18

What does Arrow conclude about group preferences?

Answer 18

• Group preferences will always be dominated by a single individual (dictator)
• If we do not want to have a dictatorship then we need to give up condition U, P, or I

Question 19

What are the implications of arrow for group choices?

Answer 19

• The only way to obtain a rational group preference all the time is by relaxing one of the conditions
• Arrow’s theorem is probabilistic, you may get a coherent group choice, but you cannot guarantee a coherent group choice
• There is a trade-off between group rationality and the concentration of decision-making power

Question 20

What are the implications of Arrow for democracy?

Answer 20

• Arrow’s results in no way call into question the intrinsic value of democratic processes
  – He is not against democracy
• Rather, he demonstrates that no procedure can aggregate individual preferences in a way that meets all objectives in all cases
• A perfect voting system or collective action scheme, in this sense, does not exist and any institutional design must recognize this

Question 21

What does Kenneth May put forward?

Answer 21

May’s theorem deal with a special case of Arrow’s theorem of social choice by including only two alternatives

Question 22

Method of Majority Rule (MMR) Kenneth May

Answer 22

For any pair of alternatives, j and k, j is preferred by the group to k (j Pg k) if and only if the number of group members who prefer j to k exceeds the number of group members who prefer k to j
- He basically argues that MMR works well with two candidates

Question 23

Kenneth May facts

Answer 23

• 1915-1977
• American mathematician and historian
• He published his theory in 1952

Question 24

What are May’s conditions?

Answer 24

• Condition universal domain
• Condition anonymity
• Condition neutrality
• Condition monotincity

Question 25

Condition U (May)

Answer 25

Universal Domain - Each individual from a group may adopt any strong or weak complete and transitive preference orderings over the alternatives A -- Every preference order is allowed (it should be transitive and complete) - Link with Arrow's condition U

Question 26

Condition A (May)

Answer 26

Anonymity - Social preferences depend only on the collection of individual preferences, not on who has which preferences -- Each individual is treated equally and identically -- eg. exchanging preference orders between individuals will not influence group preferences - (implies) Link with Arrow's condition D

Question 27

Condition N (May)

Answer 27

Neutrality - Interchanging the ranks of alternatives j and k in each individual's preference ordering has the effect of interchanging the ranks of j and k in the group's preference ordering -- Each alternative is treated equally, and group preferences should be responsive to the individual preferences -- Link with Arrow's condition I

Question 28

Condition M (May)

Answer 28

Monotonicity - If an alternative j beats or ties with another alternative k and j rises in some group member's preference from below k to the same or a higher rank than k, then j now strictly beats k -- If one or more voters change their preferences in a way that favors j, and this results in a change in the position of j in the group preference ordering, then j's position in the group ranking should increase (and perhaps more importantly, not decrease) - Link with Arrow's condition P

Question 29

What is May's theorem?

Answer 29

A method of preference aggregation over a pair of alternatives satisfies conditions U, A, N, and M if and only if it is MMR