The Distortion of Cardinal Preferences in Voting

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The Distortion of Cardinal Preferences in Voting

• Ariel D. Procaccia and Jeffrey S. Rosenschein

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Lecture outline
Distortion
Introduction
Misrepresentation
Conclusions

• Introduction to Voting
• Distortion
• Definition and intuition
• Discouraging results
• Misrepresentation
• Definition and intuition
• Results
• Conclusions

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What is voting?
Distortion
Introduction
Misrepresentation
Conclusions

• n voters and m candidates.
• Each voter expresses ordinal preferences by
  ranking the candidates.
• Winner of election determined according to a
  voting rule.
• Plurality.
• Borda.
• Applications in multiagent systems (candidates
  are beliefs, schedules Haynes et al. 97, movies
  Ghosh et al. 99).

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Got it, so whats distortion?
Distortion
Introduction
Misrepresentation
Conclusions

• Humans dont evaluate candidates in terms of
  utility, but agents do!
• With voting, agents cardinal preferences are
  embedded into space of ordinal preferences.
• This leads to a distortion in the preferences.

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Distortion illustrated
Distortion
Introduction
Misrepresentation
Conclusions
c3
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c2
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rank
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utility
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Voter 1
Voter 2
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Distortion defined (informally)
Distortion
Introduction
Misrepresentation
Conclusions

• Candidate with max SW usually not the winner.
• Depends on voting rule.
• Informally, the distortion of a rule is the
  worst-case ratio between the maximal SW and SW of
  winner.

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Distortion Defined (formally)
Distortion
Introduction
Misrepresentation
Conclusions

• Each voter has preferences uiltui1,,uimgt uij
  utility of candidate j. Denote uj ?i uij.
• Ordinal prefs denoted by Ri. j Ri k voter i
  prefers candidate j to k.
• An ordinal pref. profile R is derived from a
  cardinal pref profile u iff
• ?i,j,k, uij gt uik ? j Ri k
• ?i,j,k, uij uik ? j Ri k xor k Ri j
• ?(F,u) maxjuj/uF(R).

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An unfortunate truth
Distortion
Introduction
Misrepresentation
Conclusions

• F Plurality. argmaxjuj 2, but 1 is elected.
  Ratio is 9/6.

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c1
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rank
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utility
utility
utility
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Voter 1
Voter 2
Voter 3
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Distortion Defined (formally)
Distortion
Introduction
Misrepresentation
Conclusions

• Each voter has preferences uiltui1,,uimgt uij
  utility of candidate j. Denote uj ?i uij.
• Ordinal prefs denoted by Ri. j Ri k voter i
  prefers candidate j to k.
• An ordinal pref. profile R is derived from a
  cardinal pref profile u iff
• ?i,j,k, uij gt uik ? j Ri k
• ?i,j,k, uij uik ? j Ri k xor k Ri j.
• ?(F,u) maxjuj/uF(R).
• ?nm(F)maxu ?(F,u).
• S.t. ?j uij K.

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An unfortunate truth
Distortion
Introduction
Misrepresentation
Conclusions

• Theorem ?F, ?32(F)gt1.

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c1
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c1
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c2
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c2
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rank
rank
utility
utility
utility
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c2
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c2
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Voter 1
Voter 2
Voter 3
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Scoring rules a short aside
Distortion
Introduction
Misrepresentation
Conclusions

• Scoring rule defined by vector ? lt?1,,?mgt.
  Voter awards ?l points to candidate lth-ranked
  candidate.
• Examples of scoring rules
• Plurality ? lt1,0,,0gt
• Borda ? ltm-1,m-2,,0gt
• Veto ? lt1,1,,1,0gt

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Distortion of scoring rules the plot thickens
Distortion
Introduction
Misrepresentation
Conclusions

• F has unbounded distortion if there exists m such
  that for all d, ?nm(F)gtd for infinitely many
  values of n.
• Theorem F scoring protocol with
  ?2 ? 1/(m-1)?l?2?l. Then F has unbounded
  distortion.
• Corollary Borda and Veto have unbounded
  distortion.

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An alternative model
Distortion
Introduction
Misrepresentation
Conclusions

• So far, have analyzed profiles u s.t. ?i,
  ?juijK.
• Weighted voting voter with weight K counts as K
  identical voters.
• ?juijKi. Voter i has weight Ki.
• Define ?nm(F) analogously to previous def.
• Theorem For all F, n1, m, ?n1m ?n1m, and there
  exists n2 s.t. ?n1m ?n2m.
• Corollary For all F, ?32(F)gt1.
• Corollary F has unbounded ? ? F has unbounded ?.

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Introducing misrepresentation
Distortion
Introduction
Misrepresentation
Conclusions

• A voters misrepresentation w.r.t. lth ranked
  candidate is ?ij l-1. Denote ?j ?i ?ij.
• Misrep. can be interpreted as (restricted)
  cardinal prefs.
• e.g. uij m - ?ij - 1.
• ?nm(F)maxR (?F(R)/minj ?j).

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Misrepresentation illustrated
Distortion
Introduction
Misrepresentation
Conclusions
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Sched. 2
Voter
Sched. 1
Sched. 3
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Misrepresentation of scoring rules
Distortion
Introduction
Misrepresentation
Conclusions

• Borda has misrepresentation 1.
• Denote by lij candidate js ranking in Ri.
• js Borda score is
  ?i(m-lij)?i(m-?ij-1)n(m-1)-?i?ijn(m
  -1)-?j
• j minimizes misrep. ? j maximizes score.
• Borda has undesirable properties.
• Scoring protocols with ? 1 are fully
  characterized in the paper.
• Theorem F is a scoring rule. F has unbounded
  misrep. iff ?1?2.
• Corollary Veto has unbounded misrep.

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The Maximin rule
Distortion
Introduction
Misrepresentation
Conclusions

• For two candidates j,k, denote by N(j,k) the
  number of voters who prefer j to k.
• Maximin elects maxj mink N(j,k).
• Condorcet consistent.
• Claim ?nm(Maximin) ? 1.62 (m-1).
• Proof
• Let R. W.l.o.g. 1 argminj ?j , 2 Maximin(R).
• Let d be candidate 2s Maximin score.
• Denote c (3-51/2)/2.

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The Maximin rule continued
Distortion
Introduction
Misrepresentation
Conclusions

• We distinguish two cases.
• Case 1 d gt cn.
• At least cn voters prefer candidate 2 to 1.
• Worst case (1-c)n voters rank 1 first and 2
  last, cn voters rank 2 first and 1 second.
• ?2/?1 ? (1-c)/c (m-1) ? 1.62 (m-1).
• Case 2 d ? cn.
• Candidate 1s maximin score ? d.
• ? candidate s.t. 1 is ranked higher by ? cn.
• At least (1-c)n dont rank 1 first.
• ?2 / ?1 ? n(m-1)/(1-c)n ? 1.62 (m-1).

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Summary of misrepresentation results
Distortion
Introduction
Misrepresentation
Conclusions
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Conclusions
Distortion
Introduction
Misrepresentation
Conclusions

• Computational issues discussed in paper, but
  exact characterization remains open.
• Distortion may be an obstacle for applying voting
  in multiagent systems.
• If prefs are constrained, still an important
  consideration.
• In scheduling example with m3, in STV there
  might be 3 times as much conflicts as in Borda.