Strategy and Voting Methods Design

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Strategy and Voting Methods Design

• By Warren Schudy

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Overview

• Voting systems
• Examples systems, and their strengths and
  weaknesses
• Desirable properties
• Recommendations

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Voting System

• Combining opinions of strategic agents
• Most common system in USA plurality
• Many types of voting with different ballots
• Ranked ballot (total ordering)

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Strategy

• A ballot consistent with the voters true
  preferences is sincere
• A method is strategy-free if insincere voting is
  never better for a voter than sincere voting

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Plurality

• Every voter votes for at most one candidate,
  candidate with most votes wins

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Example Populace 1 Fringe

• 3-candidates
• C is a fringe candidate
• 3 types of voters
• Sincere utilities shown
• With plurality, B wins unless C encourages voters
  to vote for A

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Example Populace 2 Clones

• A and B are a clone set
• With sincere voting in plurality C wins
• A and Bs supporters have to compromise

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Example Populace 3 Centrist

• B is a centrist everyones first or second
  choice.
• Sincere plurality, A or C wins

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Instant Runoff Voting (IRV)

• Used in San Francisco, Cambridge MA, and
  internationally
• Ranked ballots
• Candidate with least 1st-choice votes is
  eliminated
• Voters who preferred the eliminated candidate
  switch to voting for 2nd choice
• Repeated until a candidate gets a majority

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Does IRV work?

• Populace 1 fringe C is eliminated, and its
  vote get transferred to A, who wins
• No insincere voting can change the outcome
• Populace 2 clones works fine, A or B wins
• Populace 3 centrist the centrist is eliminated
  first, giving a fringe candidate the win.
• The other fringe could vote centrist over their
  candidate and make the centrist win

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Condorcet Methods

• Use ranked ballots to simulate 1x1 contests
  between each pair of candidates
• Build directed graph out of 1x1 contests.
• If a candidate exists who beats all others in
  pairwise contests, he wins
• If no such candidate exists, use complicated
  rules to discard some defeats

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Strategy in Condorcet

• Works for all three populations if voting is
  sincere
• For the fringe population, A wins with sincere
  ballots, but strategy causes chaos
• If B voters vote insincerely BgtCgtA, a cycle is
  created
• The weakest defeat, AgtB, is dropped, leaving B as
  the winner
• If As voters think A might lose and use the same
  strategy, the fringe candidate C can win!

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Approval

• Like plurality but voters can approve of any
  number of candidates, not just one
• For the fringe example, if everyone votes for
  favorite and C voters approve of A too, A wins
• This is stable no one can improve matters by
  changing their vote
• For the clone example, in strategic equilibrium
  everyone has a chance of winning

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Borda

• Voters rank candidates
• 1 point for last choice, 2 for second-to-last,
  etc.
• Most points wins

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Strategy in Borda

• Strategy rank favorite first, major opponent
  last, to maximize your power against major
  opponent
• Problem if everyone uses this strategy, the
  fringe candidate gets the most votes and wins!
• Each voter balances risk of fringe winning vs.
  risk of the other major candidate winning when
  deciding to use strategic burying or not.
• Result fringe candidates have a chance of
  winning, even if 99 of the populace despises them

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Property monotonicity

• Modifying a ballot by only raising one candidate
  should not make him/her stop winning
• Intuitively good
• Every method I mentioned except IRV passes

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Property 2 Independence of Irrelevant Alternates
(IIA)

• If a candidate that is not winning is removed
  from every ballot, the winner should not change
• Intuitively desirable
• Plurality and approval satisfy this, but other
  methods do not
• Plurality and approval avoid failing this by
  discarding voter preferences this isnt really
  a reason to choose approval voting

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Arrows theorem

• A voting method that allows voters to rank
  candidates cannot satisfy all of
• Monotonicity
• Independence of irrelevant alternatives
• Deterministic
• IE, theres no perfect vote-counting system
• Usually, IIA is the property thats sacrificed

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Property 3 Clone-independence

• A clone set is a set of candidates with the
  property that every voter votes for the clones in
  a clump
• A method is clone-independent if whenever a
  member of a clone-set is removed from every
  ballot the new winner is in the clone set if and
  only if the old winner was
• Many Condorcet methods are clone-independent, as
  is IRV.
• Borda is not clone-independent

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Summary

• Borda and Condorcet elect extreme fringe
  candidates sometimes, a potentially serious
  problem
• Approval and IRV elect the wrong candidate
  sometimes, but wont elect extreme fringes
• IRV is not monotonic

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Other work

• Strategic equilibria in plurality, approval, and
  Borda were studied rigorously by Myerson et al
  1993
• Unsure if strategic equilibria in IRV and
  Condorcet have been studied rigorously

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Questions?