Voting Theory

1
Voting Theory
2
Overview

• Voting Paradoxes
• Condorcet Criterion
• Arrows Impossibility Theorem

3
Voting Paradox

• Recall, democratic theory predicated on the idea
  that somehow the vote reveals the will of the
  people
• That means we need to be able to move from
  individual preferences to something like a
  social preference
• The winner of the election is in some meaningful
  sense reflective of what the people want

4
Voting Paradox

• Yet as we examine the various voting systems put
  forth in the world we need to keep in mind some
  conceptual problems with voting theory
• It may not be possible to move from individual to
  group preferences smoothly or meaningfully

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Voting Paradox
In this population, what do the people want?
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Condorcet Candidate

• One way to determine what the people prefer to is
  consider the choice to be the one which defeats
  all others in a pair-wise comparison
• We call that the Condorcet candidate after the
  Marquis de Condorcet (1743-1794)

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Voting Paradox
In this population, what do the people want?
8
Voting Paradox
Note XgtY, YgtZ, ZgtX
9
Voting Paradox

• Is there a way around this problem?
• Condorcet first discovered the problem, but his
  solution isnt always going to work
• Raises potentially troubling issue for democratic
  theory
• Can any voting system reveal aggregate
  meaningfully from individual to group preferences?

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Arrows Impossibility Theorem

• Universal Admissibility of Individual Preferences
• All possible orderings by indiviudlars are
  admissiable
• No institutions (e.g., parties) can restrict the
  orderings so that certain preferences scales
  cannot be expressed

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Arrows Impossibility Theorem

• Positive Association of individual and social
  values
• Given that XgtY is the social ordering, if
  individuals either raise or do not change the
  ranking of X in their preference scales and the
  ranking of Y remains unchanged, then
• It is still the case that XgtY
• This restriction ensures that the method of
  adding individuals preference scales reflects,
  in a nonperverse way, these preferences the
  social ranking of X does not respond negatively
  to changes in rankings by individuals

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Arrows Impossibility Theorem

• Independence of Irrelevant Alternatives
• If S is a subset of the set of available
  alernatives and the preference scales of
  individuals change with respect to alternatives
  not in S
• Then the social ordering for alternatives in S
  does not change

13
Arrows Impossibility Theorem

• Citizens Sovereignty
• For any two alternatives X and Y, there exist
  individual preference scales such that X is
  preferred to Y in the social ordering
• In other words, the social outcome is not imposed
• At the extreme, if all individuals should prefer
  X to Y, then X cannot be prohibited by the
  social outcome
• Outlaws the possibility that the social outcome
  is unrelated to the preference scales of the
  societys members

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Arrows Impossibility Theorem

• Nondictatorship
• For any two alternatives X and Y, there is no
  individual such that whenver he or she prefers X
  to Y, X is always preferred to Y in the social
  ordering
• There is no individual who can dictate the social
  ordering of alternatives

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Arrows Impossibility Theorem

• The intent of the assumptions is to link
  societys ordering of alternatives to
  individuals preference scales in a nonarbitrary
  way
• We want the social outcome responsive to the
  preference scales of individuals

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Arrows Impossibility Theorem

• Arrow then demonstrates that given these basic
  assumptions, no socialordering is possible that
  doesnt violate one or the other of the
  assumptions
• There is no method of summing individuals
  preferences that satisfies all 5 assumptions
• If 1 through 3, then either the 4 or 5 is being
  violated (that is, order is imposed from without
  or from within)