Part 1: Voting and Bargaining

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Part 1 Voting and Bargaining

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Economic Models of Social Choice

• Elements
• Agents (Voters, Bureaucrats,Consumers,)
• Action (Vote, Budget, Purchase,)
• Payoff/Objective (Policy, Max Budget, Buy,)
• Information (Others Characteristics, Environment)

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Economic Models of Social Choice

• Rationality Paradigm
• Preference
• Tendency to do whats best

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Preference

• Suppose 1 actor (i ) and 3 alternatives (X,Y,Z)
• Preferences
• I prefer X to Y.
• I am indifferent between Y and Z.
• Notation
• X Pi Y means i prefers X to Y.
• Y Ii Z means i is indifferent between Y and Z .

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Rationality

• Restriction on preferences that allows the
  chooser to order alternatives.
• Comparability
• Transitivity

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Comparability (Completeness)

• Alternatives are comparable and the preference
  relation complete if, for any two alternatives,
  either X Pi Y, Y Pi X, or X Ii Y.
• Not all relations satisfy comparability. Do you
  prefer wine to beer? Depends on specifics -- not
  enough information to compare.

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Transitivity

• The strict preference relation is transitive if,
  for any three alternatives, X Pi Y and Y Pi Z,
  then X Pi Z.
• The indifference relation is transitive if X Ii
  Y and Y Ii Z imply X Ii Z.

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Transitivity

• Transitivity is a bit demanding.
• Lack of confusion
• Salt water experiments
• Multiple dimensions
• Clinton vs. Bush vs. Perot

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Why Assume Transitivity?

• Suppose X Pi Y, Y Pi Z, and Z Pi X.
• In this case, there is no best alternative.

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Preferences that satisfy completeness and
transitivity

• Allow comparisons between alternatives one pair
  at a time
• Are consistent
• Result in a best alternative
• Are personal.

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Why Relative Ranking Only?

• Index of happiness lurking. Why not work with
  this directly?
• Hard to measure
• Problematic to compare between people
• Some people have strong feelings about everything
• Relative rankings ignores intensity

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utility
Y Pi X Pi Z
X
Y
Z
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Observations

• Have imposed only minimal conditions for
  individual choice.
• Individual ranking is only relative--not
  absolute.
• Next question Can sets of rational individual
  preferences be combined or aggregated to form
  rational social preferences?

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Voting

• Voting is one way to translate individual
  preferences into social preferences.

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Social Preferences/Rankings
Individual Preferences
Voting System
Vote(Signal)
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Elements of a voting system

• Form of signal
• How signals are combined

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Examples

• Plurality Voting
• Signal State best alternative
• How Combined Alternative with most votes wins.

• Survivor
• Signal Identify worst alternative
• How Combined Alternative with most votes is
  eliminated. Repeat until only one left.

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An Example of Majority Rule

• 3 voters, 3 alternatives
• Individual preferences
• 1. X P1 Y P1 Z
• 2. Y P2 Z P2 X
• 3. Z P3 Y P3 X

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• Pairwise competition results
• Y PS X (2-to-1)
• Y PS Z (2-to-1)
• Z PS X (2-to-1)
• Y PS Z PS X

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Observations

• Social preferences are complete and transitive.
• There is a best alternative.
• There is no single majority.
• Different majorities prefer Y to other
  alternatives.
• Majority rule is ambiguous.
• Multiple ways of deciding by voting.

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Strategic Voting

• Who has an incentive to vote strategically?
• Can anyone shift the outcome?
• Would anyone want to?
• 1 is pivotal between Y and Z. If shifts vote to
  Z, then Z wins!
• 2s first preference wins already.
• 3 could vote against Y and for X in first
  pairing, then for Z in second pairing against
  winner (X).
• Ignore strategic voting for now.

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A Disturbing Example

• 3 voters, 3 alternatives
• Pairwise competition by majority voting
• Individual preferences
• 1. X P1 Y P1 Z
• 2. Y P2 Z P2 X
• 3. Z P3 X P3 Y

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• Social preferences
• X PS Y (2-to-1)
• Y PS Z (2-to-1)
• Z PS X (2-to-1)
• Social preferences are complete, but
  intransitive.
• Problem is not individual preferences, but with
  the voting system.

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Divide the Dollar

• How unlikely is transitivity?
• Consider voting on how to divide a dollar
• Contemporary example how to spend surplus
• Politics as income/wealth distribution
• 3 voters

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Possibilities
B
(0,1,0)
(1,0,0)
A
(0,0,1)
C
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• Is there a proposal that beats all others in
  pairwise competition?
• Consider (1, 0, 0)
• Xs ideal division
• Defeated by (0, 1/2, 1/2) for example

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• Consider equal division (1/3, 1/3, 1/3)
• Fair allocation
• Defeated by infinite number of proposals
• For example, (1/2, 1/2, 0) defeats (1/3, 1/3,
  1/3).
• Any proposal can be defeated by some other
  division.

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(0,1,0)
X0
A prefersto X0
(1,0,0)
(0,0,1)
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(0,1,0)
Preferred by A and B--beats X0
X0
(0,0,1)
(1,0,0)
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• Social Preferences are complete but intransitive.
• (1/2, 1/2, 0) PS (1/3, 1/3, 1/3)
• (0, 3/4, 1/4) PS (1/2, 1/2, 0)
• (1/3, 1/3, 1/3) PS (0, 3/4, 1/4)

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• Many applications of divide the dollar
• allocating wealth
• allocating budget surplus (income)
• siting waste dump (NIMBY)

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Real World Examples of Intransitivity

• School Funding
• Group 1 L P1 M P1 H
• Group 2 M P2 H P2 L
• Group 3 H P3 L P3 M

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Voting Institutions

• How a society solves intransitivity is important.

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• One common solution agenda procedure
• One individual determines order of pairwise
  competition.
• Loser is eliminated, winner competes against next
  alternative.
• Can restore order to social preferences in this
  way, but introduces other problems.
• In particular, majority still prefers some other
  alternative to the one selected.
• Is the agenda-setter a dictator?

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• Giving agenda-setting power to one individual
  works--restores transitivity
• But there are problems, too
• Basic tradeoff between rationality of social
  decisions and decentralization of power
• Democracy vs. order

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Power of Rules 17th Amendment

• Up to 6 motions may be on floor at the same time
• Status quo (a0)
• Bill reported out by committee (a1)
• Amended bill (a2)
• Amendment to amendment (a3)
• Substitute bill (a 4)
• Amended substitute (a5)

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• Order
• a2 vs. a3, then a4 vs. a5
• Winner of second vote vs. winner of first vote
• Winner of this vote vs. a1
• Winner of this vote vs. a0

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• In early 20th century, majority support for 17th
  Amendment
• a0 Status quo
• a1 17th Amendment
• a2 Amended version requiring Federal
  supervision of electors

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• PreferencesRepublicans 1
  a2 P a1 p a0Republicans 2 a0 P a2 P a1Democr
  ats a1 P a0 P a2
• Roughly equal numbers.

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• Outcomes
• a2 defeated a1 (2-1)
• a0 defeateda2 (2-1)
• But a1 would have beat a0 (2-1)
• a2 is called killer amendment.

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Revised List of Criteria for Social Preferences

• Complete
• Transitive
• No Dictatori is dictator if social preferences
  are always the same as i s preferences.

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Still Other Criteria

• Consensus Criterion
• If all members of the group individually prefer X
  to Y, then X PS Y.
• Does majority rule satisfy this criterion?

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Example

• 4 alternatives, 3 voters1. X P1 Y P1 B P1
  A2. A P2 X P2 Y P2 B3. B P3 A P3
  X P3 Y

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• Consider sequence
• X versus A
• Winner vs. B
• Winner vs. Y
• Y wins, although everyone prefers X to Y.
• Majority rule fails consensus criterion.

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Other Voting Procedures

• Borda Count
• Avoids multiple (sequential) votes
• Each individual ranks m alternatives.
• Point system
• m points to best
• m-1 to next, and so on
• 1 points to last place
• Alternative receiving greatest number of points
  wins

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Example

• 1. X P1 C P1 B P1 A
• 2. A X C B
• 3. B A X C
• 4. X C B A
• 5. A X C B
• 6. B A X C
• 7. X C B A

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Results

• X (22)
• A (17)
• B (16)
• C (15)
• X PS A PS B PS C

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Problem

• Suppose X is eliminated from consideration
• Would think that A would be chosen
• Results A (13) B (14) C (15)
• A comes in last when X is eliminated.
• Bad outcome want choice based on relevant
  alternatives only.

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Revised List of Criteria

• Completeness
• Transitivity
• No Dictator
• Consensus Criterion
• Irrelevant Alternatives Dont Matter

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Arrows Theorem

• No social choice procedure can satisfy all of the
  following conditions
• Completeness
• Transitivity
• No Dictator
• Consensus Criterion
• Irrelevant Alternatives Dont Matter

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Reactions to the Theorem

• Change terms of debate
• Criticize conditions
• Find least objectionable choice procedures