Social Choice

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Lecture 4

• Social Choice
• Readings Shepsle, Analyzing politics, chapter 3

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Group Preferences and group choice

• Suppose individuals have rational preferences
  (i.e. their preferences are complete, transitive)
• They have to undertake a decision that affects
  the entire group
• Does the group have preferences that are rational?

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3 people, 3 options
4
Group choice

• Unanimity?
• First preference majority?
• Majority voting
• The method of majority voting requires that, for
  any pair of alternatives A and B, A is preferred
  by the group to B if the number of group members
  who prefer A to B exceeds the number of group
  members that prefer B to A
• Round-robin tournament
• Movie vs. Pub
• Movies vs.. Stadium
• Pub vs.. stadium
• Are group preferences rational?

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3 people, 3 options
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• Round-robin tournament
• Movie vs. Pub
• Movies vs. Stadium
• Pub vs. stadium
• Are group preferences rational?

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Individual and group preferences

• Even if individuals have rational preferences,
  the group preferences may not satisfy rationality
• In the previous example, the group preferences do
  not satisfy transitivity
• As a consequence, in each pair wise comparison of
  alternatives we obtain that a different majority
  coalition is formed in favour of an alternative
• In other words, group preferences cycle

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Condorcet winner

• When group preferences are rational, then a
  unique winner emerges in pair wise comparisons
• The alternative that beats all the others in any
  pair wise comparison is the so called Condorcet
  winner
• When group preferences are not transitive, there
  is no Condorcet winner

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Distributive politics and cycling majority

• One important example where cycling majority can
  arise is the when individuals have to agree on
  the sharing of resource
• Example budget allocation across constituencies
  or different type of individuals
• This is an example of divide the dollar game
  where typically group preferences are not
  transitive
• For any allocation supported by a majority, there
  is an alternative allocation preferred by a
  different majority
• Hence, in a divide the dollar game, there is no
  Condorcet winner

10
Agenda power and voting rules

• Divide-the-dollar type of situations are common
  in distributive politics and simple majority
  voting would result in a complete legislative
  impasse
• In practice, legislative deadlock is prevented
  adopting rules that restrict the set of
  alternatives of which the vote can take place
• Agenda power to some member
• Defeated alternatives are eliminated
• Limited number of voting rounds
• If an individual has agenda power (for example he
  can decide the order according to which proposal
  are put forward for the vote) and the number of
  voting rounds is limited, the order according to
  which proposals are put up for vote has a crucial
  impact on the outcome that will be approved!
• The agenda setter can manipulate the agenda to
  have his most preferred outcome implemented

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Desirable properties of mechanisms of collective
decision making

• The previous example shows how restricting the
  set of alternatives of which the vote can take
  place, cycling can be avoided
• Are those restrictions desirable?
• What could be some minimal properties that a
  mechanism of collective decision making should
  satisfy?

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Arrows theorem Assumptions

• Minimal properties
• Rationality of Individuals
• Universal Admissibility (Condition U)
• Pareto Optimality (Condition P)
• Independence from Irrelevant Alternatives
  (Condition I)
• Nondicatorship (condition D)

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Arrows theorem Assumptions

• Let i be an individual belonging to group G and
  let A be the set of alternatives over which G
  must take a decision
• Rationality of i preference over the
  alternatives in A are complete and transitive
• Universal Admissibility (Condition U) each
  individual i can adopt any strong or weak and
  transitive preference ordering over the
  alternatives in A
• Pareto Optimality (Condition P) If every member
  of the group G prefers alternative a over
  alternative b, then the group preference must
  reflect a preference for a over b
• Independence from Irrelevant Alternatives
  (Condition I) If alternatives a and b are ranked
  in a specific way by every individual and this
  ranking does not change, then the group ranking
  of a with respect to b should not change. This
  must be true even if individual preferences of
  other irrelevant alternatives are introduced.
• Nondicatorship (condition D) there is no
  individual i belonging to the group whose own
  preferences dictates the group preferences

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Arrows Theorem

• There exists no mechanism for translating the
  preferences of rational individuals into coherent
  group preferences that simultaneously satisfy
  conditions U, P, I and D
• In other words, any mechanism generating a group
  choice that satisfies U, P and I is either
  dictatorial or incoherent
• Hence, the Arrow Impossibility theorem highlights
  the existence of a fundamental trade-off in
  social choice trade-off between rationality and
  the concentration of power

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Majority voting and Arrows Theorem

• Arrows theorem applies to any mechanism of
  collective decision making, hence it applies to
  majority voting
• Remember the previous divide-the-dollar game
• Either cycling (rationality violated)
• Or give agenda power to a group member
  (non-dictatorship violated)

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Majority voting and Arrows Theorem

• However, we have seen that in some cases
  individual rational preferences can be aggregated
  into coherent group preferences via majority
  voting (remember the example where a Condorcet
  winner exists)
• How is this possible?
• The Arrow theorem states some very general
  conditions implying that aggregation of
  individual preferences cannot guarantee group
  coherence in all situations
• However, if we impose some restrictions (i.e. we
  relaxed some of the minimal assumptions behind
  the theorem) then we can obtain group coherence

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Questions

• Mark, Jim and Sarah won a lottery price of 400
  and Mark proposes to share the price equally.
  Would Marks proposal be approved by the
  majority? If not, how would they split the price?
• Discuss whether majority voting is a mechanism of
  aggregation of individual preferences that can
  lead to coherent group preferences