1. 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 #
RFBR # А

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 (2 аgents)
Group 2 (3 аgents)
Group 3 (4 agents)
Group 4 (5 agents) a b e d c
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 a c b Plurality Approval Voting q=2 Borda r(a)=4, r(b)=3, r(с)=2
Agent 1
Agent 2
Agent 3 a c b
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) (3;4) Method1: Method2: Method3: Method4:
Kelly’s DA
p1 Plurality (0,1852)
0,3333
p2 Approval q=2
0,2963
p6 Borda (0,3102)
0,3611
0,4028
0,2917
p7 Black (0,1435)
0,2361
0,2778
0,1667
p28 Threshold

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-th place from top
stands at the j-th place from top

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

26. Thank you!