Fuad Aleskerov (NRU HSE) Daniel Karabekyan (NRU HSE)

1. 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

2. Ways to study the problem • Theoretical approach • Gibbard (1973), Satterthwaite (1975) • Computational approach • Kelly (1993), Aleskerov, Kurbanov (1998)

3. 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}?

4. Plurality rule and 3 alternatives

5. 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

6. Kelly’s Dominance Axiom

7. Gardenfor’s Principle

8. EUCEPA

9. 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

10. Indices • Kelly’s index

11. Rules • Plurality • Approval Voting q=2 • Borda r(a)=4, r(b)=3, r(с)=2 • Black • Threshold

12. 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.

13. Results EP1) EP2) EP3) EP4)

14. Kelly’s index EP1

15. Kelly’s index EP1

16. Kelly’s index EP2

17. Kelly’s index EP2

18. Kelly’s index

20. Indices - better off - worse off - nothing changed

23. Efficiency of manipulation • stands at the k-thplacefrom top • stands at the j-thplacefrom top

25. Further research • Weak manipulation • Consider IAC and introduce new indices for this case. • Study coalitional manipulation

26. Thank you!