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The Distortion of Cardinal Preferences in Voting

The Distortion of Cardinal Preferences in Voting. Ariel D. Procaccia and Jeffrey S. Rosenschein. Lecture outline. Distortion. Misrepresentation. Conclusions. Introduction. Introduction to Voting Distortion Definition and intuition Discouraging results Misrepresentation

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The Distortion of Cardinal Preferences in Voting

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  1. The Distortion of Cardinal Preferences in Voting Ariel D. Procaccia and Jeffrey S. Rosenschein

  2. Lecture outline Distortion Misrepresentation Conclusions Introduction • Introduction to Voting • Distortion • Definition and intuition • Discouraging results • Misrepresentation • Definition and intuition • Results • Conclusions

  3. What is voting? Distortion Misrepresentation Conclusions Introduction • nvoters and mcandidates. • 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]).

  4. Got it, so what’s distortion? Distortion Misrepresentation Conclusions Introduction • Humans don’t 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.

  5. Distortion illustrated Distortion Misrepresentation Conclusions Introduction c3 11 1 11 1 10 c2 10 9 9 8 8 rank rank 7 7 utility utility c3 6 6 2 2 5 5 4 4 3 3 c1 c1 2 2 1 1 c2 0 3 0 3 Voter 1 Voter 2

  6. Distortion defined (informally) Distortion Misrepresentation Conclusions Introduction • 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.

  7. Distortion Defined (formally) Distortion Misrepresentation Conclusions Introduction • Each voter has preferences ui=<ui1,…,uim>; uij = utility of candidate j. Denote uj = i uij. • Ordinal prefs denoted by Ri. j Rik = 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 > uik  j Rik • i,j,k, uij = uik  j Rik xor k Ri j • (F,u) = maxjuj/uF(R).

  8. An unfortunate truth Distortion Misrepresentation Conclusions Introduction • F = Plurality. argmaxjuj = 2, but 1 is elected. Ratio is 9/6. 5 c1 1 5 c1 1 5 c2 c2 1 4 4 4 rank rank rank utility utility utility c1 c1 3 3 3 c2 c2 2 2 2 1 1 1 c2 c2 c1 c1 0 0 0 2 2 2 Voter 1 Voter 2 Voter 3

  9. Distortion Defined (formally) Distortion Misrepresentation Conclusions Introduction • Each voter has preferences ui=<ui1,…,uim>; uij = utility of candidate j. Denote uj = i uij. • Ordinal prefs denoted by Ri. j Rik = 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 > uik  j Rik • i,j,k, uij = uik  j Rik xor k Ri j. • (F,u) = maxjuj/uF(R). • nm(F)=maxu (F,u). • S.t. j uij = K.

  10. An unfortunate truth Distortion Misrepresentation Conclusions Introduction • Theorem: F, 32(F)>1. 5 c1 1 5 c1 1 5 c2 c2 1 4 4 4 rank rank rank utility utility utility c1 c1 3 3 3 c2 c2 2 2 2 1 1 1 c2 c2 c1 c1 0 0 0 2 2 2 Voter 1 Voter 2 Voter 3

  11. Scoring rules – a short aside Distortion Misrepresentation Conclusions Introduction • Scoring rule defined by vector  = <1,…,m>. Voter awards l points to candidate l’th-ranked candidate. • Examples of scoring rules: • Plurality:  = <1,0,…,0> • Borda:  = <m-1,m-2,…,0> • Veto:  = <1,1,…,1,0>

  12. Distortion of scoring rules – the plot thickens Distortion Misrepresentation Conclusions Introduction • F has unbounded distortion if there exists m such that for all d, nm(F)>d 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.

  13. An alternative model Distortion Misrepresentation Conclusions Introduction • So far, have analyzed profiles u s.t. i, juij=K. • Weighted voting: voter with weight K counts as K identical voters. • juij=Ki. 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)>1. • Corollary: F has unbounded   F has unbounded .

  14. Introducing misrepresentation Distortion Misrepresentation Conclusions Introduction • A voter’s misrepresentation w.r.t. l’th 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).

  15. Misrepresentation illustrated Distortion Misrepresentation Conclusions Introduction 9:00 9:00 9:00 9:00 10:00 10:00 10:00 10:00 11:00 11:00 11:00 11:00 12:00 12:00 12:00 12:00 13:00 13:00 13:00 13:00 14:00 14:00 14:00 14:00 15:00 15:00 15:00 15:00 16:00 16:00 16:00 16:00 17:00 17:00 17:00 17:00 18:00 18:00 18:00 18:00 19:00 19:00 19:00 19:00 Sched. 2 Sched. 1 Sched. 3 Voter

  16. Misrepresentation of scoring rules Distortion Misrepresentation Conclusions Introduction • Borda has misrepresentation 1. • Denote by lij candidate j’s ranking in Ri. • j’s Borda score is i(m-lij)=i(m-ij-1)=n(m-1)-iij=n(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.

  17. Summary of misrepresentation results Distortion Misrepresentation Conclusions Introduction

  18. Conclusions Distortion Misrepresentation Conclusions Introduction • 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 m=3, in STV there might be 3 times as much conflicts as in Borda.

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