The Distortion of Cardinal Preferences in Voting Ariel 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
0
1
2
3
4
5
6
7
8
9
10
11
3
2
1
uti
lity
ran
k
c2c2
c3c3
c1c1
0
1
2
3
4
5
6
7
8
9
10
11
3
2
1
uti
lity
ran
k
c3c3
c1c1
c2c2
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.
2
1
uti
lity
ran
k
c2c2
c1c1
0
1
2
3
4
5 c1c1
c2c2 2
1
uti
lity
ran
k
0
1
2
3
4
5 c1c1
c2c2 2
1
uti
lity
ran
k
c2c2
c1c10
1
2
3
4
5 c2c2
c1c1
c2c2
c1c1
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.
2
1
uti
lity
ran
k
c2c2
c1c1
0
1
2
3
4
5 c1c1
c2c2 2
1
uti
lity
ran
k
0
1
2
3
4
5 c1c1
c2c2 2
1
uti
lity
ran
k
c2c2
c1c10
1
2
3
4
5 c2c2
c1c1
c2c2
c1c1
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
19:00
18:00
17:00
16:00
15:00
14:00
13:00
12:00
11:00
10:00
9:00
19:00
18:00
17:00
16:00
15:00
14:00
13:00
12:00
11:00
10:00
9:00
19:00
18:00
17:00
16:00
15:00
14:00
13:00
12:00
11:00
10:00
9:00
19:00
18:00
17:00
16:00
15:00
14:00
13:00
12:00
11:00
10:00
9:00
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