The Distortion of Cardinal Preferences in VotingAriel D. Procaccia and Jeffrey S. Rosenschein

Lecture outline

• Introduction to Voting• Distortion

– Definition and intuition– Discouraging results

• Misrepresentation– Definition and intuition– Results

• Conclusions

Introduction Distortion Misrepresentation Conclusions

What is voting?

• 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]).

Introduction Distortion Misrepresentation Conclusions

Got it, so what’s distortion?

• 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.

Introduction Distortion Misrepresentation Conclusions

Distortion illustrated

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Introduction Distortion Misrepresentation Conclusions

Distortion defined (informally)

• 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.

Introduction Distortion Misrepresentation Conclusions

Distortion Defined (formally)

• Each voter has preferences ui=<ui1,

…,uim>; ui

j = utility of candidate j. Denote uj = i ui

j.

• 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:1. i,j,k, ui

j > uik j Ri

k

2. i,j,k, uij = ui

k j Ri k xor k Ri j

(F,u) = maxjuj/uF(R).

Introduction Distortion Misrepresentation Conclusions

An unfortunate truth

• F = Plurality. argmaxjuj = 2, but 1 is elected. Ratio is

9/6.

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Introduction Distortion Misrepresentation Conclusions

Distortion Defined (formally)

• Each voter has preferences ui=<ui1,

…,uim>; ui

j = utility of candidate j. Denote uj = i ui

j.• 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:

1. i,j,k, uij > ui

k j Ri k

2. i,j,k, uij = ui

k j Ri k xor k Ri j.

(F,u) = maxjuj/uF(R). n

m(F)=maxu (F,u).– S.t. j ui

j = K.

Introduction Distortion Misrepresentation Conclusions

An unfortunate truth

• Theorem: F, 32(F)>1.

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Introduction Distortion Misrepresentation Conclusions

Scoring rules – a short aside

• 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>

Introduction Distortion Misrepresentation Conclusions

Distortion of scoring rules – the plot thickens

• F has unbounded distortion if there exists m such that for all d, n

m(F)>d for infinitely many values of n.

• Theorem: F = scoring protocol with 2 1/(m-1)l2l. Then F has unbounded distortion.

• Corollary: Borda and Veto have unbounded distortion.

Introduction Distortion Misrepresentation Conclusions

An alternative model

• So far, have analyzed profiles u s.t. i, jui

j=K. • Weighted voting: voter with weight K

counts as K identical voters. jui

j=Ki. Voter i has weight Ki. • Define n

m(F) analogously to previous def.• Theorem: For all F, n1, m, n1

m ≤ n1m,

and there exists n2 s.t. n1m ≤ n2

m.• Corollary: For all F, 3

2(F)>1.• Corollary: F has unbounded F has

unbounded .

Introduction Distortion Misrepresentation Conclusions

Introducing misrepresentation

• A voter’s misrepresentation w.r.t. l’th ranked candidate is i

j = l-1. Denote j = i i

j.

• Misrep. can be interpreted as (restricted) cardinal prefs. – e.g. ui

j = m - ij - 1.

nm(F)=maxR (F(R)/minj j).

Introduction Distortion Misrepresentation Conclusions

Misrepresentation illustrated

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Introduction Distortion Misrepresentation Conclusions

Misrepresentation of scoring rules

• Borda has misrepresentation 1. – Denote by lij candidate j’s ranking in Ri.

– j’s Borda score is i(m-lij)=i(m-i

j-1)=n(m-1)-iij=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.

Introduction Distortion Misrepresentation Conclusions

Summary of misrepresentation results

Voting RuleMisrepresentati

on

Borda = 1

Veto Unbounded

Plurality = m-1

Plurality w. Runoff = m-1

Copeland m-1

Bucklin m

Maximin 1.62 (m-1)

STV 1.5 (m-1)

Introduction Distortion Misrepresentation Conclusions

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 m=3, in STV

there might be 3 times as much conflicts as in Borda.

Introduction Distortion Misrepresentation Conclusions