computing the degree of the manipulability in the case of multiple choice
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Computing the Degree of the Manipulability in the Case of Multiple Choice. Fuad Aleskerov (SU-HSE) Daniel Karabekyan (SU-HSE) Remzi M. Sanver (Istanbul Bilgi University, Turkey) Vyacheslav Yakuba (ICS RAS) Grants SU-HSE #08-04-0008 RFBR #01-212-07-525A 04 .0 9 .08. Literature survey. - PowerPoint PPT PresentationTRANSCRIPT
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Computing the Degree of the Manipulability in the Case of
Multiple ChoiceFuad Aleskerov (SU-HSE)
Daniel Karabekyan (SU-HSE)Remzi M. Sanver (Istanbul Bilgi University, Turkey)
Vyacheslav Yakuba (ICS RAS)
Grants SU-HSE #08-04-0008RFBR #01-212-07-525A
04.09.08
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Literature survey
• Strategy-proof analysis– Gibbard (1973), Satterthwaite (1975)
• Degree of manipulability– Kelly (1993), Aleskerov, Kurbanov (1998)
• Tie-breaking rule– Alphabetical tie-breaking rule
aba ?,
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Model
• Manipulation by a single agent• Set of alternatives • Set of all non-empty subsets of • voters with over and over
• How to construct ?• Weak conditions
– Kelly’s principle, Gärdenfors’ principle and so on
m 2)>(m
.
A
\2= AA A
nN 1,...,=
iP A iEP A
iEP
)()( PCEPPC ii
cbda ,?,
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Nonordinal methods
• Lexicographic methods– Leximax
– Leximin
• Probabilistic methods– Based on the probability of the best alternative
– Based on the probability of the worst alternative
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ccbbcbacabaa ,,,,,
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Ordinal method
• Assign rank to each alternative based on its place in voter’s preferences.
• Each alternative have equal probability to be chosen as final outcome.
• Utility of the set is an average rank of all alternatives within this set.
• This method needs additional restrictions.
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Ordinal method with restrictions:
• Lexicographic restrictions• Probabilistic restrictions• Attitude to risk restrictions
– Risk-lover (prefer higher variance)– Risk-averse (prefer lower variance)
• Cardinality restrictions– The lesser set is preferred to the greater one– The greater set is preferred to the lesser one– cbda ,?,
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Indices
• Kelly’s index
nm
dK
)!(0
nk
mkJ )!(
nJJJK ...21
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Indices
)1!()!(1
)!(1
1
mnmI
ni
ni
mj
n
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RulesAgent 1 Agent 2 Agent 3
a c bb a ac b c
1) Plurality 2) Approval Voting q=23) Borda r(a)=4, r(b)=3, r(с)=24) Black5) Threshold
cbaPC ,,)(
aPC )(
aPC )(
aPC )(
aPC )(
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Computation
• Two methods: look-through and statistical• Hard to compute – (5,5) – about 25 billions
profiles. Using anonymity we can look only on 225 millions profiles.
• Open question: How can we use neutrality and anonymity at the same time?
• For example, (3,3) – 216 profiles, using anonimity – 56, using both – 26.
Сn
nm 1!
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Results
1)2)3)4)
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(3;3) Method1: Method2: Method3: Method4:p1 Plurality (0,1667) 0,2222 0 0,2222 0p2 Approval q=2 0,1111 0,6111 0,1111 0,6111p6 Borda (0,2361) 0,3056 0,4167 0,3056 0,4167p7 Black (0,1111) 0,0556 0,1667 0,0556 0,1667p28 Threshold 0,3056 0,4167 0,3056 0,4167
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(3;4) Method1: Method2: Method3: Method4:
p1 Plurality (0,1852) 0,3333 0,3333 0,3333 0,3333
p2 Approval q=2 0,2963 0,2963 0,2963 0,2963
p6 Borda (0,3102) 0,3611 0,4028 0,3611 0,4028
p7 Black (0,1435) 0,2361 0,2778 0,2778 0,2361
p28 Threshold 0,4028 0,4028 0,4028 0,4028
(3;5) Method1: Method2: Method3: Method4:
p1 Plurality (0,2315) 0,37037 0,37037 0,37037 0,37037
p2 Approval q=2 0,375 0,375 0,375 0,375
p6 Borda (0,2855) 0,37037 0,4398 0,37037 0,4398
p7 Black (0,1698) 0,1157 0,2314 0,1157 0,2314
p28 Threshold 0,2585 0,2585 0,2585 0,2585
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Thank you