An Algorithm for the Coalitional Manipulation Problem under Maximin

1. An Algorithm for the Coalitional Manipulation Problem under Maximin Michael Zuckerman, Omer Lev and Jeffrey S. Rosenschein COMSOC’10

2. Agenda • Introduction • Constructive Coalitional Unweighted Manipulation (CCUM) problem • Algorithm for CCUM under Maximin • 1½-approximation to the optimum • Tightness of the results • Conclusions

3. Introduction • Elections • Voters submit linear orders of the candidates • A voting rule determines the winner based on the votes • Manipulation • A voter casts a vote that is not his true preference, to make himself better off • Gibbard-Satterthwaite theorem • Every reasonable voting rule is manipulable

4. Constructive Coalitional Unweighted Manipulation (CCUM) problem • Given • A voting rule r • The Profile of Non-Manipulators PNM • Candidate p preferred by the manipulators • Number of manipulators |M| • We are asked whether or not there exists a Profile of Manipulators PM such that p is the winner of PNM υ PM under r.

5. Unweighted Coalitional Optimization (UCO) problem • Given • A voting rule r • The Profile of Non-Manipulators PNM • Candidate p preferred by the manipulators • We are asked to find the minimum k such that there exists a set of manipulators M with |M| = k, and a Profile of Manipulators PM such that p is the winner of PNM υ PM under r.

6. Our setting, maximin • C = {c1,…,cm} – the set of candidates • S, |S| = N – the set of N non-manipulators • T, |T| = n – the set of n manipulators • Ni(c, c’) = |{ k | c >k c’, >k S υ {1,…,i}}| – the number of voters from S and from the i first manipulators, which prefer c over c’ • Si(c) = minc’≠cNi(c, c’) – the maximin score of c from S and the i first manipulators • Maximin winner = argmaxc{Sn(c)} • Denote MINi(c) = {c’ C | Si(c) = Ni(c, c’)}

7. CCUM Complexity • CCUM under Maximin is NP-complete for any fixed number of manipulators (≥ 2) (Xia et al. ’09 [1])

8. The heuristic / approximation algorithm • Fix some order on manipulators • The current manipulator i • Ranks p first • Builds a digraph Gi-1 = (V, Ei-1), where • V = C \ {p}; (x, y) Ei-1 iff (y MINi-1(x) and p MINi-1(x)) • Iterates over the candidates who have not yet been ranked • If there is a candidate with out-degree 0, then adds such a candidate with the lowest score • Otherwise, if there is a cycle with two adjacent vertices who have the lowest scores – adds the front vertex • Otherwise, adds any vertex with the lowest score • Removes all the outgoing edges of vertices who had outgoing edge to newly added vertex

9. a b 2 c d 2 e Example • C = {a, b, c, d, e, p} • |S| = 6 • |T| = 2 • The non-manipulators’ votes: • a > b > c > d > p > e • a > b > c > d > p > e • b > c > a > p > e > d • b > c > p > e > d > a • e > d > p > c > a > b • e > d > p > c > a > b G0: 2 2 2 2 S0(p) = N0(p, b) = 2 S0(e) = N0(e, p) = 2

10. Example (2) G0: a • The non-manipulators’ votes: • a > b > c > d > p > e • a > b > c > d > p > e • b > c > a > p > e > d • b > c > p > e > d > a • e > d > p > c > a > b • e > d > p > c > a > b • The manipulators’ votes: 2 3 b 2 2 2 c 2 d 2 e p > e > d > b > c > a S0(p) = N0(p, b) = 2 S0(e) = N0(e, p) = 2

11. Example (3) G1: a • The non-manipulators’ votes: • a > b > c > d > p > e • a > b > c > d > p > e • b > c > a > p > e > d • b > c > p > e > d > a • e > d > p > c > a > b • e > d > p > c > a > b • The manipulators’ votes: 3 b 2 2 3 c d 2 e p > e > d > b > c > a S1(p) = N1(p, b) = 3 S1(e) = N1(e, p) = 2 p > e > d > c > a > b

12. Example (4) a • The non-manipulators’ votes: • a > b > c > d > p > e • a > b > c > d > p > e • b > c > a > p > e > d • b > c > p > e > d > a • e > d > p > c > a > b • e > d > p > c > a > b • The manipulators’ votes: G2: 3 b 2 3 c d 2 e p > e > d > b > c > a p > e > d > c > a > b S2(p) = N2(p, b) = 4 maxc≠pS2(c) = 3 p is the winner!

13. Instances without 2-cycles • Denote ms(i) = maxc≠pSi(c) • The maximum score of p’s opponents after i stages • Lemma: If there are no 2-cycles in the graphs built by the algorithm, then for all i, 0 ≤ i ≤ n-3 it holds that ms(i+3) ≤ ms(i) + 1 • Theorem: If there are no 2-cycles, then the algorithm gives a 1½ -approximation of the optimum

14. Eliminating the 2-cycles • Lemma: If at a certain stage i there are no 2-cycles, then for all j > i, there will be no 2-cycles at stage j • We prove that the algorithm performs optimally while there are 2-cycles • Intuitively, if there is a 2-cycle, then one of its vertices has the highest score, and it will always be placed in the end – until the cycle is eliminated • Once the 2-cycles have been dismantled, our algorithm performs a 1½-approximation on the number of stages left • Generally we have 1½-approximation of the optimal solution

15. k+1 k+1 k+1 k+1 al k k k k k bl k cl b1 k c1 b2 k c2 Tightness of the results • When voted: p > al > cl > bl > … > a1 > c1 > b1 … • ms*(i) grows by 1 every (m-1)/3 voters a1 a2 … k • Our algorithm can vote: • p > a1 > c1 > b1 > … > al > cl > bl • … • Here ms(i) grows by 1 every 3 voters

16. Conclusions & Future Work • A new heuristic / approximation algorithm for CCUM / UCO under Maximin • Gives a 1½-approximation to the optimum • The lower bound on the approximation ratio of the algorithm is 1½ • Future work: • Implement the algorithm • Compare its performance to the performance of the algorithm in [2]

17. References • [1] Complexity of Unweighted Coalitional Manipulation Under Some Common Voting Rules, Lirong Xia, Michael Zuckerman, Ariel D. Procaccia, Vincent Conitzer and Jeffrey S. Rosenschein. The Twenty-First International Joint Conference on Artificial Intelligence (IJCAI 2009), July 2009, Pasadena, California, pp. 348-353. • [2] Algorithms for the Coalitional Manipulation Problem, Michael Zuckerman, Ariel D. Procaccia and Jeffrey S. Rosenschein. Journal of Artificial Intelligence. Volume 173, Number 2, February 2009, pp. 392-412.

18. Thank You!