# Computational aspects of strategic behaviour in elections with top-truncated ballots

- 129 Downloads

## Abstract

Understanding when and how computational complexity can be used to protect elections against different manipulative actions has been a highly active research area over the past two decades. Much of this literature, however, makes the assumption that the voters or agents specify a complete preference ordering over the set of candidates. There are many multiagent systems applications, and even real-world elections, where this assumption is not warranted, and this in turn raises a series of questions on the impact of partial voting on the complexity of manipulative actions. In this paper, we focus on two of these questions. First, we address the question of how hard it is to manipulate elections when the agents specify only top-truncated ballots. Here, in particular, we look at the weighted manipulation problem—both constructive and destructive manipulation—when the voters are allowed to specify top-truncated ballots, and we provide general results for all scoring rules, for elimination versions of all scoring rules, for the plurality with runoff rule, for a family of election systems known as Copeland\(^{\alpha }\), and for the maximin protocol. The second question we address is the impact of top-truncated voting on the complexity of manipulative actions in electorates with structured preference profiles. In particular, we consider electorates that are single-peaked and we show how, for many voting protocols, allowing top-truncated voting reimposes the \(\mathcal {NP}\)-hardness shields that normally vanish in such electorates.

### Keywords

Computational social choice Elections Complexity of manipulative actions Top-truncated voting Manipulation Bribery## Notes

### Acknowledgements

We acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC) through the Discovery Grants Program.

### References

- 1.Barberá, S. (2007). Indifferences and domain restrictions.
*Analyse & Kritik*,*29*(2), 146–162.CrossRefGoogle Scholar - 2.Bartholdi, J., & Trick, M. A. (1986). Stable matching with preferences derived from a psychological model.
*Operations Research Letters*,*5*(4), 165–169.MathSciNetCrossRefMATHGoogle Scholar - 3.Bartholdi, J. J., Tovey, C. A., & Trick, M. A. (1992). How hard is it to control an election?
*Mathematical and Computer Modelling*,*16*(8–9), 27–40.MathSciNetCrossRefMATHGoogle Scholar - 4.Bartholdi, J. J, I. I. I., & Orlin, J. B. (1991). Single transferable vote resists strategic voting.
*Social Choice and Welfare*,*8*(4), 341–354.MathSciNetCrossRefMATHGoogle Scholar - 5.Bartholdi, J. J, I. I. I., Tovey, C. A., & Trick, M. A. (1989). The computational difficulty of manipulating an election.
*Social Choice and Welfare*,*6*(3), 227–241.MathSciNetCrossRefMATHGoogle Scholar - 6.Baumeister, D., Faliszewski, P., Lang, J., & Rothe, J. (2012). Campaigns for lazy voters: Truncated ballots. In
*Proceedings of the 11th international conference on autonomous agents and multiagent systems (AAMAS)*, pp. 577–584.Google Scholar - 7.Black, D. (1948). On the rationale of group decision-making.
*The Journal of Political Economy*,*56*(1), 23–34.CrossRefGoogle Scholar - 8.Black, D., Newing, R. A., McLean, I., McMillan, A., & Monroe, B. L. (1958).
*The theory of committees and elections*. Berlin: Springer.Google Scholar - 9.Brandt, F., Brill, M., Hemaspaandra, E., & Hemaspaandra, L. A. (2015). Bypassing combinatorial protections: Polynomial-time algorithms for single-peaked electorates.
*Journal of Artificial Intelligence Research*,*53*, 439–496.MathSciNetMATHGoogle Scholar - 10.Briskorn, D., Erdélyi, G., & Reger, C. (2015). Bribery under partial information. In
*Proceedings of the 2nd workshop on exploring beyond the worst case in computational social choice (EXPLORE)*.Google Scholar - 11.Cantala, D. (2004). Choosing the level of a public good when agents have an outside option.
*Social Choice and Welfare*,*22*(3), 491–514.MathSciNetCrossRefMATHGoogle Scholar - 12.Coleman, T., & Teague, V. (2007). On the complexity of manipulating elections. In
*Proceedings of the 13th Australasian symposium on theory of computing*, pp. 25–33.Google Scholar - 13.Conitzer, V., Sandholm, T., & Lang, J. (2007). When are elections with few candidates hard to manipulate?
*Journal of the ACM*,*54*(3), 14.MathSciNetCrossRefMATHGoogle Scholar - 14.Conitzer, V., Endriss, U., Lang, J., & Procaccia, A. D. (Eds.). (2016).
*Handbook of computational social choice*. Cambridge: Cambridge University Press.Google Scholar - 15.Copeland, A. H. (1951). A reasonable social welfare function. In
*Seminar on applications of mathematics to social sciences*. University of MichiganGoogle Scholar - 16.Dey, P., Misra, N., & Narahari, Y. (2016). Complexity of manipulation with partial information in voting. In
*Proceedings of the 25th international joint conference on artificial intelligence (IJCAI)*, pp. 229–235.Google Scholar - 17.Doignon, J. P., & Falmagne, J. C. (1994). A polynomial time algorithm for unidimensional unfolding representations.
*Journal of Algorithms*,*16*(2), 218–233.MathSciNetCrossRefMATHGoogle Scholar - 18.Duggan, J., & Schwartz, T. (2000). Strategic manipulability without resoluteness or shared beliefs: Gibbard–Satterthwaite generalized.
*Social Choice and Welfare*,*17*(1), 85–93.MathSciNetCrossRefMATHGoogle Scholar - 19.Dwork, C., Kumar, R., Naor, M., & Sivakumar, D. (2001). Rank aggregation methods for the web. In
*Proceedings of the 10th international conference on world wide web*, pp. 613–622Google Scholar - 20.Emerson, P. (2013). The original Borda count and partial voting.
*Social Choice and Welfare*,*40*(2), 353–358.MathSciNetCrossRefMATHGoogle Scholar - 21.Ephrati, E., & Rosenschein, J. S. (1993). Multi-agent planning as a dynamic search for social consensus. In
*Proceedings of the 13th international joint conference on artificial intelligence (IJCAI), vol. 93*, pp. 423–429.Google Scholar - 22.Erdélyi, G., & Reger, C. (2016). Possible bribery in k-approval and k-veto under partial information. In
*Proceedings of the international conference on artificial intelligence: methodology, systems, and applications (AIMSA)*, Berlin: Springer, pp. 299–309.Google Scholar - 23.Erdélyi, G., Lackner, M., & Pfandler, A. (2013). Computational aspects of nearly single-peaked electorates. In
*Proceedings of the 27th AAAI conference on artificial intelligence (AAAI)*, pp. 283–289.Google Scholar - 24.Escoffier, B., Lang, J., & Öztürk, M. (2008). Single-peaked consistency and its complexity. In
*Proceedings of the 18th European conference on artificial intelligence (ECAI)*, pp. 366–370.Google Scholar - 25.Faliszewski, P., Hemaspaandra, E., & Hemaspaandra, L. A. (2009a). How hard is bribery in elections?
*Journal of Artificial Intelligence Research*,*35*, 485–532.MathSciNetMATHGoogle Scholar - 26.Faliszewski, P., Hemaspaandra, E., Hemaspaandra, L. A., & Rothe, J. (2009b). Llull and copeland voting computationally resist bribery and constructive control.
*Journal of Artificial Intelligence Research, 35*, 275–341.Google Scholar - 27.Faliszewski, P., Hemaspaandra, E., Hemaspaandra, L. A., & Rothe, J. (2009). A richer understanding of the complexity of election systems. In
*Fundamental problems in computing*, Berlin: Springer, pp. 375–406.Google Scholar - 28.Faliszewski, P., Hemaspaandra, E., & Hemaspaandra, L. A. (2010). Using complexity to protect elections.
*Communications of the ACM*,*53*(11), 74–82.CrossRefGoogle Scholar - 29.Faliszewski, P., Hemaspaandra, E., Hemaspaandra, L. A., & Rothe, J. (2011). The shield that never was: Societies with single-peaked preferences are more open to manipulation and control.
*Information and Computation*,*209*(2), 89–107.MathSciNetCrossRefMATHGoogle Scholar - 30.Faliszewski, P., Hemaspaandra, E., & Hemaspaandra, L. A. (2014). The complexity of manipulative attacks in nearly single-peaked electorates.
*Artificial Intelligence*,*207*, 69–99.MathSciNetCrossRefMATHGoogle Scholar - 31.Fishburn, P. C. (1977). Condorcet social choice functions.
*SIAM Journal on applied Mathematics*,*33*(3), 469–489.MathSciNetCrossRefMATHGoogle Scholar - 32.Fitzsimmons, Z. (2015). Single-peaked consistency for weak orders is easy. In
*Proceedings of the 15th conference on theoretical aspects of rationality and knowledge (TARK)*, pp. 127–140.Google Scholar - 33.Fitzsimmons, Z., & Hemaspaandra, E. (2015). Complexity of manipulative actions when voting with ties. In
*Proceedings of the 4th international conference on algorithmic decision theory (ADT)*, pp. 103–119.Google Scholar - 34.Fitzsimmons, Z., & Hemaspaandra, E. (2016). Modeling single-peakedness for votes with ties. In
*Proceedings of the 8th European starting AI researcher symposium (STAIRS)*, pp. 63–74.Google Scholar - 35.Friedgut, E., Kalai, G., & Nisan, N. (2008). Elections can be manipulated often. In
*2008 49th Annual IEEE symposium on foundations of computer science*, IEEE, pp. 243–249.Google Scholar - 36.Garey, M. R., & Johnson, D. S. (1979).
*Computers and intractability: A guide to the theory of NP-completeness*. New York, NY: W. H. Freeman & Co.MATHGoogle Scholar - 37.Gibbard, A. (1973). Manipulation of voting schemes: A general result.
*Econometrica: Journal of the Econometric Society*,*41*(4), 587–601.MathSciNetCrossRefMATHGoogle Scholar - 38.Hemaspaandra, E., & Hemaspaandra, L. A. (2007). Dichotomy for voting systems.
*Journal of Computer and System Sciences*,*73*(1), 73–83.MathSciNetCrossRefMATHGoogle Scholar - 39.Konczak, K., & Lang, J. (2005). Voting procedures with incomplete preferences. In
*Proceedings of the IJCAI-05 multidisciplinary workshop on advances in preference handling*.Google Scholar - 40.Lackner, M. (2014). Incomplete preferences in single-peaked electorates. In
*Proceedings of the 28th AAAI conference on artificial intelligence (AAAI)*, pp. 742–748.Google Scholar - 41.Lu, T., & Boutilier, C. (2013). Multi-winner social choice with incomplete preferences. In
*Proceedings of the 23rd international joint conference on artificial intelligence (IJCAI)*, AAAI Press, pp. 263–270.Google Scholar - 42.Narodytska, N., & Walsh, T. (2014). The computational impact of partial votes on strategic voting. In
*Proceedings of the 21st European conference on artificial intelligence (ECAI)*, pp. 657–662.Google Scholar - 43.Niemi, R. G., & Wright, J. R. (1987). Voting cycles and the structure of individual preferences.
*Social Choice and Welfare*,*4*(3), 173–183.MathSciNetCrossRefMATHGoogle Scholar - 44.
- 45.Pennock, D. M., Horvitz, E., & Giles, C. L. (2000). Social choice theory and recommender systems: Analysis of the axiomatic foundations of collaborative filtering. In
*Proceedings of the 17th National conference on artificial intelligence (AAAI)*, pp. 729–734.Google Scholar - 46.Satterthwaite, M. A. (1975). Strategy-proofness and arrow’s conditions: Existence and correspondence theorems for voting procedures and social welfare functions.
*Journal of Economic Theory*,*10*(2), 187–217.MathSciNetCrossRefMATHGoogle Scholar - 47.Walsh, T. (2007). Uncertainty in preference elicitation and aggregation. In
*Proceedings of the 22nd AAAI conference on artificial intelligence (AAAI), vol 7*, pp. 3–8.Google Scholar - 48.Xia, L., & Conitzer, V. (2011). Determining possible and necessary winners under common voting rules given partial orders.
*Journal of Artificial Intelligence Research*,*41*(2), 25–67.MathSciNetMATHGoogle Scholar - 49.Yang, Y. (2015). Manipulation with bounded single-peaked width: A parameterized study. In
*Proceedings of the 14th international conference on autonomous agents and multiagent systems (AAMAS)*, pp. 77–85.Google Scholar