# Complexity of and Algorithms for Borda Manipulation

AAAI, 2011.

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Abstract:

We prove that it is NP-hard for a coalition of two manipulators to compute how to manipulate the Borda voting rule. This resolves one of the last open problems in the computational complexity of manipulating common voting rules. Because of this NP-hardness, we treat computing a manipulation as an approximation problem where we try to mi...More

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Introduction

- Voting is a simple mechanism to combine preferences in multi-agent systems.
- Results like those of Gibbrard-Sattertwhaite prove that most voting rules are manipulable.
- The small set of voting rules that are NP-hard to manipulate with unweighted votes includes single transferable voting, 2nd order Copeland, ranked pairs, and maximin (Bartholdi & Orlin 1991; Bartholdi, Tovey, & Trick 1989; Xia et al 2009).
- The Borda rule has many good features
- It never elects the Condorcet loser.

Highlights

- Voting is a simple mechanism to combine preferences in multi-agent systems
- The small set of voting rules that are NP-hard to manipulate with unweighted votes includes single transferable voting, 2nd order Copeland, ranked pairs, and maximin (Bartholdi & Orlin 1991; Bartholdi, Tovey, & Trick 1989; Xia et al 2009)
- We have proved that it is NP-hard to compute how to manipulate the Borda rule with just two manipulators
- This resolves one of the last open questions regarding the computational complexity of unweighted coalition manipulation for common voting rules
- To evaluate whether such computational complexity is important in practice, we have proposed two new approximation methods that try to minimize the number of manipulators
- These methods are based on ideas from bin packing and multiprocessor scheduling

Methods

- NP-hardness only bounds the worst-case complexity of computing a manipulation.
- The authors can make any candidate win.
- REVERSE is a simple approximation method proposed to compute Borda manipulations (Zuckerman, Procaccia, & Rosenschein 2008).
- The method constructs the vote of each manipulator in turn: candidate d is put in first place, and the remaining candidates are put in reverse order of their current Borda scores.
- The method continues constructing manipulating votes until d wins.
- A long and intricate argument shows that REVERSE constructs a manipulation which uses at most one more manipulator than is optimal

Results

- To test the performance of these approximation methods in practice, the authors ran some experiments.
- Votes are drawn from an urn at random, and are placed back into the urn along with b other votes of the same type.
- This captures varying degrees of social homogeneity.
- The authors tested 1000 instances at each problem size.
- To determine if the returned manipulations are optimal, the authors used a simple constraint satisfaction problem

Conclusion

- The authors have proved that it is NP-hard to compute how to manipulate the Borda rule with just two manipulators
- This resolves one of the last open questions regarding the computational complexity of unweighted coalition manipulation for common voting rules.
- To evaluate whether such computational complexity is important in practice, the authors have proposed two new approximation methods that try to minimize the number of manipulators.
- The authors' best method finds an optimal manipulation in almost all of the elections generated

Summary

## Introduction:

Voting is a simple mechanism to combine preferences in multi-agent systems.- Results like those of Gibbrard-Sattertwhaite prove that most voting rules are manipulable.
- The small set of voting rules that are NP-hard to manipulate with unweighted votes includes single transferable voting, 2nd order Copeland, ranked pairs, and maximin (Bartholdi & Orlin 1991; Bartholdi, Tovey, & Trick 1989; Xia et al 2009).
- The Borda rule has many good features
- It never elects the Condorcet loser.
## Methods:

NP-hardness only bounds the worst-case complexity of computing a manipulation.- The authors can make any candidate win.
- REVERSE is a simple approximation method proposed to compute Borda manipulations (Zuckerman, Procaccia, & Rosenschein 2008).
- The method constructs the vote of each manipulator in turn: candidate d is put in first place, and the remaining candidates are put in reverse order of their current Borda scores.
- The method continues constructing manipulating votes until d wins.
- A long and intricate argument shows that REVERSE constructs a manipulation which uses at most one more manipulator than is optimal
## Results:

To test the performance of these approximation methods in practice, the authors ran some experiments.- Votes are drawn from an urn at random, and are placed back into the urn along with b other votes of the same type.
- This captures varying degrees of social homogeneity.
- The authors tested 1000 instances at each problem size.
- To determine if the returned manipulations are optimal, the authors used a simple constraint satisfaction problem
## Conclusion:

The authors have proved that it is NP-hard to compute how to manipulate the Borda rule with just two manipulators- This resolves one of the last open questions regarding the computational complexity of unweighted coalition manipulation for common voting rules.
- To evaluate whether such computational complexity is important in practice, the authors have proposed two new approximation methods that try to minimize the number of manipulators.
- The authors' best method finds an optimal manipulation in almost all of the elections generated

Funding

- Jessica Davies is supported by the National Research Council of Canada
- George Katsirelos is supported by the ANR UNLOC project ANR 08-BLAN-0289-01
- Nina Narodytska and Toby Walsh are supported by the Australian Department of Broadband, Communications and the Digital Economy, the ARC, and the Asian Office of Aerospace Research and Development (AOARD-104123)

Reference

- Bartholdi, J., and Orlin, J. 199Single transferable vote resists strategic voting. Social Choice and Welfare 8(4):341– 354.
- Bartholdi, J.; Tovey, C.; and Trick, M. 1989. The computational difficulty of manipulating an election. Social Choice and Welfare 6(3):227–241.
- Brunetti, S.; Lungo, A.;A. Del; Gritzmann, P. and de Vries, S. 2008. On the reconstruction of binary and permutation matrices under (binary) tomographic constraints. Theor. Comput. Sci. 406(1–2):63–71.
- Conitzer, V.; Sandholm, T.; and Lang, J. 2007. When are elections with few candidates hard to manipulate. JACM 54.
- Davies, J.; Katsirelos, G.; Narodytska, N.; and Walsh, T. 2010. An empirical study of Borda manipulation. In COMSOC-10.
- Hall, P. 1935. On representatives of subsets. Journal of the London Mathematical Society 26–30.
- Krause, K. L.; Shen, V. Y.; and Schwetman, H. D. 1975. Analysis of several task-scheduling algorithms for a model of multiprogramming computer systems. JACM 22(4):522– 550.
- Nordh, G. 2010. A note on the hardness of Skolem-type sequences. Discrete Appl. Math. 158(8):63–71.
- Walsh, T. 2010. An empirical study of the manipulability of single transferable voting. In Proc. of the 19th European Conf. on Artificial Intelligence (ECAI-2010). IOS Press.
- Xia, L.; Zuckerman, M.; Procaccia, A.; Conitzer, V.; and Rosenschein, J. 2009. Complexity of unweighted coalitional manipulation under some common voting rules. In Proc. of 21st IJCAI, 348–353.
- Xia, L.; Conitzer, V.; and Procaccia, A. 2010. A scheduling approach to coalitional manipulation. In Parkes, D.; Dellarocas, C.; and Tennenholtz, M., eds., Proc. 11th ACM Conference on Electronic Commerce (EC-2010), 275–284.
- Yu, W.; Hoogeveen, H.; and Lenstra J.K. 2004. Minimizing Makespan in a Two-Machine Flow Shop with Delays and Unit-Time Operations is NP-Hard. J. Scheduling 7(5): 333348.
- Zuckerman, M.; Procaccia, A.; and Rosenschein, J. 2009. Algorithms for the coalitional manipulation problem. Artificial Intelligence. 173(2):392-412.

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