Degree of Manipulability and Efficiency of Manipulation of

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) 22. 07. 10 Grants SU-HSE #10 -04 -0030 RFBR #08 -01 -00039 А

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

Example Group 1 (2 аgents) Group 2 (3 аgents) Group 3 (4 agents) Group 4 (5 agents) a a b e b d a b c e d c d b c d e c e a • 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}?

Plurality rule and 3 alternatives

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

Kelly’s Dominance Axiom

Gardenfor’s Principle

EUCEPA

Strong manipulation • 3 algorithms with additional restrictions • Main assumption – we can compare all sets of alternatives • 3 alternatives – 4 methods • EP 1 • EP 2 • EP 3 • EP 4 • 4 alt. – 10 methods; 5 alt. – 12 methods

Indices • Kelly’s index

Rules 1) 2) 3) 4) 5) Agent 1 Agent 2 Agent 3 a b c c a b b a c 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 EP 1) EP 2) EP 3) EP 4) (3; 4) Method 1: Method 2: Method 3: Method 4: Kelly’s DA p 1 Plurality (0, 1852) 0, 3333 0, 3333 p 2 Approval q=2 0, 2963 0, 2963 p 6 Borda (0, 3102) 0, 3611 0, 4028 0, 2917 p 7 Black (0, 1435) 0, 2361 0, 2778 0, 2361 0, 1667 p 28 Threshold 0, 4028 0, 4028

0, 5 Kelly’s index 0, 45 0, 4 0, 35 0, 3 0, 25 0, 2 0, 15 EP 1 0, 05 0 p 1 Plurality 21 ag 22 ag 23 ag p 2 24 Approval ag 25 agq=2 p 6 Borda p 7 Black p 28 Threshold

0, 5 Kelly’s index EP 1 0, 45 0, 4 0, 35 0, 3 0, 25 0, 2 0, 15 0, 1 0, 05 0 3 Plurality 6 Antiplurality 12 Borda 24 Black Threshold 48 96

0, 7 Kelly’s index 0, 6 0, 5 0, 4 0, 3 0, 2 EP 2 0, 1 0 3 ag 4 ag 5 ag p 1 Plurality 6 ag 7 ag 8 ag p 2 Approval q=2 9 ag 10 ag 11 ag 12 ag 13 ag 14 ag 15 ag p 6 Borda p 7 Black p 28 Threshold

0, 7 Kelly’s index 0, 6 EP 2 0, 5 0, 4 0, 3 0, 2 0, 1 0 3 6 1 Plurality 12 2 Approval q=2 24 3 Borda 4 Black 48 5 Threshold 96

Kelly’s index

0, 45 Alt 4_Ag 100_Leximax 0, 4 0, 35 p 24 Strong q. Pareto Simple Majority p 3 Approval q=3 0, 3 p 1 Plurality 0, 25 p 2 Approval q=2 p 18 Copeland I p 28 Threshold 0, 2 p 6 Borda p 22 Simpson 0, 15 p 7 Black p 26 Strongest q. Pareto Simple Majority 0, 1 0, 05 0 1 1, 2 1, 4 1, 6 1, 8 2

Indices - better off - worse off - nothing changed

4, 44% 51, 11% p 28 Threshold 44, 44% 2, 22% 52, 22% p 7 Black 45, 56% 3, 33% 44, 44% p 6 Borda 52, 22% 4, 44% 37, 78% p 2 Approval q=2 57, 78% 8, 89% 46, 67% p 1 Plurality 0% 10% 20% 30% + 40% 44, 44% 0 - 50% 60% 70% 80% 90% 100%

1, 02% 91, 73% p 28 Threshold 7, 25% 0, 75% 85, 97% p 7 Black 13, 28% 1, 07% 88, 64% p 6 Borda 10, 29% 1, 82% p 2 Approval q=2 86, 42% 11, 75% 86, 99% 11, 27% 1, 74% p 1 Plurality 0% 10% 20% 30% + 0 40% - 50% 60% 70% 80% 90% 100%

Efficiency of manipulation • • stands at the k-th place from top stands at the j-th place from top

0, 5 index I 3 0, 45 0, 4 0, 35 0, 3 0, 25 0, 2 0, 15 0, 1 0, 05 0 2, 999995601 5, 999991201 p 1 Plurality 11, 99999824 p 2 Approval q=2 23, 999996481 p 6 Borda p 7 Black 47, 999992961 p 28 Threshold n 95, 999985922

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

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