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Degree of Manipulability and Efficiency of Manipulation of Known Voting Rules in the Case of Multiple Choice. Fuad Aleskerov (NRU HSE) Daniel Karabekyan (NRU HSE) Remzi M. Sanver (Istanbul Bilgi University, Turkey) Vyacheslav Yakuba (ICS RAS) Grants SU-HSE # 10 -04-00 30
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Degree of Manipulability and Efficiency of Manipulation of Known Voting Rules in the Case of Multiple Choice FuadAleskerov(NRU HSE) Daniel Karabekyan (NRU HSE) Remzi M. Sanver (Istanbul Bilgi University, Turkey) VyacheslavYakuba(ICS RAS) Grants SU-HSE #10-04-0030 RFBR #08-01-00039А 22.07.10
Ways to study the problem • Theoretical approach • Gibbard (1973), Satterthwaite (1975) • Computational approach • Kelly (1993), Aleskerov, Kurbanov (1998)
Example • Plurality rule • C (Psincere) = {a,e} • Group 1 declare “b” as their best alternative, then • C (Pinsincere) = {b} • What is better: {a,e} or {b}?
How we deal with Multiple Choice? • Weak Manipulation • Kelly’s Dominance Axiom • Worst alternative in X is at least as good as the best alternative in Y. • Gardenfor’s Principle • If X was constructed by adding better alternatives to Y or/and eliminating worse alternatives from Y • EU method with equal probability assumption
Strong manipulation • 3 algorithms with additional restrictions • Main assumption – we can compare all sets of alternatives • 3 alternatives – 4 methods • EP1 • EP2 • EP3 • EP4 • 4 alt. – 10 methods; 5 alt.– 12 methods
Indices • Kelly’s index
Rules • Plurality • Approval Voting q=2 • Borda r(a)=4, r(b)=3, r(с)=2 • Black • Threshold
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 – 10.
Results EP1) EP2) EP3) EP4)
Kelly’s index EP1
Kelly’s index EP1
Kelly’s index EP2
Kelly’s index EP2
Indices - better off - worse off - nothing changed
Efficiency of manipulation • stands at the k-thplacefrom top • stands at the j-thplacefrom top
Further research • Weak manipulation • Consider IAC and introduce new indices for this case. • Study coalitional manipulation