Manipulation of Voting Schemes: A General Result

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Manipulation of Voting Schemes A General Result

• By Allan Gibbard
• Presented by
• Rishi Kant

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Roadmap

• Introduction
• Definition of terms 3
• Brief overview 4
• Importance 10
• Discussion
• Definition of terms 13
• Properties 14
• Proof of statement 16
• Conclusion

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Definition of terms

• Voting scheme a decision making system that
  depends solely on the preferences of
  participants, and leaves nothing to chance
• Dictatorial no matter what the other
  participants preferences are, the outcome is
  always decided by the preference given by the
  dictator
• True preference the players preference if he
  were the only participant / dictator
• Non-trivial voting scheme a voting scheme in
  which not every player has a dominant strategy

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Problem

• Can one design a voting scheme whose outcome is
  solely based on the true preference of each
  participant ?
• Answer Not unless the game is dictatorial or has
  less than 3 outcomes

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Formal statement

• Any non-dictatorial voting scheme with at least
  3 possible outcomes is subject to individual
  manipulation
• Interpretation
• Given a voting scheme (and certain
  circumstances) it is possible for an individual
  to force his desired outcome by disguising his
  true preference

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Example

• 4 contestants w, x, y, z
• 3 voters a, b, c
• Each voter ranks contestants (as i j k l)
  according to his/her preference
• 1st gets 4 points, 2nd gets 3
• Whoever has most points wins

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Example

• Let the true preference of each voter be
• a gt w x y z
• b gt w x y z
• c gt x w y z
• If every voter put down his/her true
• preference then w would win 11 points

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Example

• However, for the given situation c can force
• the winner to be x by pretending that his
• preference order is different
• a gt w x y z
• b gt w x y z
• c gt x w y z ? c gt x y z w
• x will now win with 10 points

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Notes

• Point to note c could influence the voting
  scheme only due to the given circumstances
• If a and b had slightly different orderings e.g.
• a gt w y z x, then c would not be successful
• Thus, subject to individual manipulation means
  that there is at least one scenario for which an
  individual can force the outcome that he wants gt
  voting scheme is not totally tamper proof

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Importance

• No non-trivial decision making system that
  depends on informed self-interest can guarantee
  that the outcome was based on the true
  preferences of the participants
• Informed self-interest gt everyone knows everyone
  elses true preference and will act in their own
  best interest

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Importance

• With respect to Mechanism design, this result
  deals with the question
• Would an agent reveal his/her true preference
  to the principal?
• The answer Only for binary or dictatorial
  choice schemes gt only binary or dictatorial
  choices are DOM-implementable

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Roadmap

• Introduction
• Definition of terms 3
• Brief overview 4
• Importance 10
• Discussion
• Definition of terms 13
• Important properties 14
• Proof of statement 16
• Conclusion

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Definition of terms

• Game form Any decision making system in which
  the outcome depends upon the individual actions
  (strategies)
• Dominant strategy a strategy that gives the
  best possible outcome to a player no matter what
  strategies others choose
• Straightforward game a game in which everyone
  has a dominant strategy

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Properties

• Properties of game forms
• Game forms leave nothing to chance
• Players in game forms may or may not have
  honest strategies
• Game forms always have a single outcome there
  are no ties
• Game forms may be used to characterize any
  non-chance decision making system

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Properties

• Properties of voting schemes
• Voting schemes are a special case of game forms
  in which the players preferences are their
  strategies
• Every player in a voting schemes has a true
  preference (honest strategy)
• Voting schemes do not have to be democratic or
  count all individuals alike
• Voting schemes must always have an outcome, even
  if the outcome is inaction

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Intuitive proof

1. Given a non-dictatorial voting scheme with more
   than 3 outcomes
2. Assume theorem Every straightforward game form
   with at least 3 possible outcomes is dictatorial
3. Non-dictatorial gt not straightforward gt not
   every player / agent has a dominant strategy
4. No dominant strategy gt true preference cannot be
   dominant
5. True preference not dominant gt possible for a
   different preference to give a better outcome
6. Voting scheme cannot guarantee true preference
   for all players and can thus be manipulated

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Formal approach used

• Proving theorem
• Every straightforward game form with at least 3
  possible outcomes is dictatorial
• is equivalent to proving theorem
• Any non-dictatorial voting scheme with at least
  3 possible outcomes is subject to individual
  manipulation
• as shown by previous slide

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Formal approach used

• Proved by invoking Arrow Impossibility Theorem
• Arrow Impossibility Theorem states
• Every social welfare function violates at least
  one of Arrows conditions
• where Arrows conditions are
• Scope
• Unanimity
• Pair wise determination
• Non-dictatorship

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Formal approach used

1. A social welfare function is generated from a
   straightforward game form with 3 outcomes
2. The social welfare function is shown to conform
   to the first 3 Arrow conditions Scope,
   Unanimity, Pair wise determination
3. Thus, the function must violate the
   non-dictatorial condition gt it must be
   dictatorial
4. The dictator of the social welfare function is
   proven to be the dictator of the game form
5. Hence the theorem is proved

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Roadmap

• Introduction
• Definition of terms 3
• Brief overview 4
• Importance 10
• Discussion
• Definition of terms 13
• Important properties 14
• Proof of statement 16
• Conclusion

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Conclusion

• Results proved in the paper
• Every straightforward game form with at least 3
  possible outcomes is dictatorial
• Any non-dictatorial voting scheme with at least
  3 possible outcomes is subject to individual
  manipulation

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Conclusion

• Comments about the paper
• The paper is written in a self-contained fashion
  i.e. one does not need to refer to other sources
  to decipher the content
• The paper is well-structured
• The paper leaves the rigorous math proof to the
  end making it easy to follow
• The paper could elaborate on the implications of
  the result a bit more

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Thank you

• End