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# An Algorithm for the Coalitional Manipulation Problem under Maximin PowerPoint PPT Presentation

An Algorithm for the Coalitional Manipulation Problem under Maximin. Michael Zuckerman, Omer Lev and Jeffrey S. Rosenschein COMSOC’10. Agenda. Introduction Constructive Coalitional Unweighted Manipulation (CCUM) problem Algorithm for CCUM under Maximin 1½-approximation to the optimum

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

### Agenda

• Introduction

• Constructive Coalitional Unweighted Manipulation (CCUM) problem

• Algorithm for CCUM under Maximin

• 1½-approximation to the optimum

• Tightness of the results

• Conclusions

### 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

### 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.

### 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.

### 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’)}

### CCUM Complexity

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

### 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

a

b

2

c

d

2

e

### Example

• C = {a, b, c, d, e, p}

• |S| = 6

• |T| = 2

• 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

### Example (2)

G0:

a

• 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

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

### Example (3)

G1:

a

• 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

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

### Example (4)

a

• 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

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!

### 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

### 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

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

### 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]

### 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.