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

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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

An Algorithm for the Coalitional Manipulation Problem under Maximin

Michael Zuckerman, Omer Lev and Jeffrey S. Rosenschein

COMSOC’10


Agenda

Agenda

  • Introduction

  • Constructive Coalitional Unweighted Manipulation (CCUM) problem

  • Algorithm for CCUM under Maximin

  • 1½-approximation to the optimum

  • Tightness of the results

  • Conclusions


Introduction

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

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

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

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 Complexity

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


The heuristic approximation algorithm

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


Example

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


Example 2

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


Example 3

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


Example 4

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!


Instances without 2 cycles

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

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


Tightness of the results

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

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

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.


Thank you

Thank You!


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