The Distortion of Cardinal
Preferences in Voting
Ariel D. Procaccia and Jeffrey S.
Rosenschein
Lecture outline
Introduction
Distortion
• Introduction to Voting
• Distortion
– Definition and intuition
– Discouraging results
• Misrepresentation
– Definition and intuition
– Results
• Conclusions
Misrepresentation
Conclusions
What is voting?
Introduction
Distortion
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]).
Got it, so what’s distortion?
Introduction
Distortion
Misrepresentation
Conclusions
• 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.
Distortion illustrated
11
Misrepresentation
1
11
c2
9
8
8
7
7
c3
2
5
utility
9
6
6
4
3
3
c1
2
1
0
1
2
5
4
2
c3
Conclusions
10
rank
utility
10
Distortion
c1
1
3
0
c2
3
rank
Introduction
Distortion defined (informally)
Introduction
Distortion
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.
Distortion Defined (formally)
Introduction
•
•
•
Distortion
Misrepresentation
Each voter has preferences ui=<ui1,…,uim>;
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:
1. i,j,k, uij > uik  j Ri k
2. i,j,k, uij = uik  j Ri k xor k Ri j
•
Conclusions
(F,u) = maxjuj/uF(R).
An unfortunate truth
Distortion
Introduction
1
1
5
c1
2
c2
1
3
c1
2
c2
1
c2
2
0
c2
c2
1
c1
c1
2
4
utility
3
utility
4
rank
4
0
c1
5
3
rank
c1
5
utility
Conclusions
F = Plurality. argmaxjuj = 2, but 1 is elected. Ratio is 9/6.
rank
•
Misrepresentation
2
1
c2
2
0
Distortion Defined (formally)
Introduction
•
•
•
•
•
Distortion
Misrepresentation
Conclusions
Each voter has preferences ui=<ui1,…,uim>;
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:
1. i,j,k, uij > uik  j Ri k
2. 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.
An unfortunate truth
Distortion
Introduction
Misrepresentation
Conclusions
• Theorem: F, 32(F)>1.
1
1
5
2
c2
1
3
c1
2
c2
1
c2
2
0
utility
c1
c2
c2
1
c1
c1
2
4
rank
3
utility
4
rank
utility
4
0
c1
5
3
rank
c1
5
2
1
c2
2
0
Scoring rules – a short aside
Introduction
Distortion
Misrepresentation
Conclusions
• 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>
Distortion of scoring rules – the plot thickens
Introduction
Distortion
Misrepresentation
Conclusions
• 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.
An alternative model
Introduction
Distortion
Misrepresentation
Conclusions
• 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 .
Introducing misrepresentation
Introduction
Distortion
Misrepresentation
Conclusions
• 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).
Misrepresentation illustrated
Introduction
Distortion
Misrepresentation
Conclusions
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Misrepresentation of scoring rules
Introduction
Distortion
Misrepresentation
Conclusions
• 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.
Summary of misrepresentation results
Introduction
Distortion
Misrepresentation
Conclusions
Voting Rule
Misrepresentation
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)
Conclusions
Introduction
Distortion
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 m=3, in STV there
might be 3 times as much conflicts as in Borda.