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An Algorithm for the Coalitional Manipulation Problem under Maximin

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An Algorithm for the Coalitional Manipulation Problem under Maximin

Michael Zuckerman, Omer Lev and Jeffrey S. Rosenschein

COMSOC’10

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

- 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

- 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.

- 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.

- 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’)}

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

- 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

a

b

2

c

d

2

e

- 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

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

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

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!

- 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

- 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

k+1

k+1

k+1

k+1

al

k

k

k

k

k

bl

k

cl

b1

k

c1

b2

k

c2

- 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

- 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]

- [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.

Thank You!