Towards a Dichotomy of Finding Possible Winners in Elections Based on Scoring Rules
To make a joint decision, agents (or voters) are often required to provide their preferences as linear orders. To determine a winner, the given linear orders can be aggregated according to a voting protocol. However, in realistic settings, the voters may often only provide partial orders. This directly leads to the Possible Winner problem that asks, given a set of partial votes, if a distinguished candidate can still become a winner. In this work, we consider the computational complexity of Possible Winner for the broad class of voting protocols defined by scoring rules. A scoring rule provides a score value for every position which a candidate can have in a linear order. Prominent examples include plurality, k-approval, and Borda. Generalizing previous NP-hardness results for some special cases and providing new many-one reductions, we settle the computational complexity for all but one scoring rule. More precisely, for an unbounded number of candidates and unweighted voters, we show that Possible Winner is NP-complete for all pure scoring rules except plurality, veto, and the scoring rule defined by the scoring vector (2,1,...,1,0), while it is solvable in polynomial time for plurality and veto.
Unable to display preview. Download preview PDF.
- 1.Betzler, N., Hemmann, S., Niedermeier, R.: A multivariate complexity analysis of determining possible winners given incomplete votes. In: Proc. of 21st IJCAI 2009 (2009)Google Scholar
- 3.Faliszewski, P.: Nonuniform bribery (short paper). In: Proc. 7th AAMAS 2008, pp. 1569–1572 (2008)Google Scholar
- 6.Konczak, K., Lang, J.: Voting procedures with incomplete preferences. In: Proc. of IJCAI 2005 Multidisciplinary Workshop on Advances in Preference Handling (2005)Google Scholar
- 7.Lang, J., Pini, M.S., Rossi, F., Venable, K.B., Walsh, T.: Winner determination in sequential majority voting. In: Proc. of 20th IJCAI 2007, pp. 1372–1377 (2007)Google Scholar
- 8.Pini, M.S., Rossi, F., Venable, K.B., Walsh, T.: Incompleteness and incomparability in preference aggregation. In: Proc. of 20th IJCAI 2007, pp. 1464–1469 (2007)Google Scholar
- 9.Walsh, T.: Uncertainty in preference elicitation and aggregation. In: Proc. of 22nd AAAI 2007, pp. 3–8. AAAI Press, Menlo Park (2007)Google Scholar
- 10.Xia, L., Conitzer, V.: Determining possible and necessary winners under common voting rules given partial orders. In: Proc. of 23rd AAAI 2008, pp. 196–201. AAAI Press, Menlo Park (2008)Google Scholar
- 11.Xia, L., Zuckerman, M., Procaccia, A.D., Conitzer, V., Rosenschein, J.S.: Complexity of unweighted coalitional manipulation under some common voting rules. In: Proc. 21st IJCAI 2009 (2009)Google Scholar