How many candidates are needed to make elections hard to manipulate?

  • Authors:

  • Vincent Conitzer;Jérôme Lang;Tuomas Sandholm

  • Affiliations:

  • Carnegie Mellon University, Pittsburgh, PA;Université Paul Sabatier, 31062 Toulouse Cedex, France;Carnegie Mellon University, Pittsburgh, PA

  • Venue:
  • Proceedings of the 9th conference on Theoretical aspects of rationality and knowledge
  • Year:
  • 2003

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Abstract

In multiagent settings where the agents have different preferences, preference aggregation is a central issue. Voting is a general method for preference aggregation, but seminal results have shown that all general voting protocols are manipulable. One could try to avoid manipulation by using voting protocols where determining a beneficial manipulation is hard computationally. The complexity of manipulating realistic elections where the number of candidates is a small constant was recently studied [4], but the emphasis was on the question of whether or not a protocol becomes hard to manipulate for some constant number of candidates. That work, in many cases, left open the question: How many candidates are needed to make elections hard to manipulate? This is a crucial question when comparing the relative manipulability of different voting protocols. In this paper we answer that question for the voting protocols of the earlier study: plurality, Borda, STV, Copeland, maximin, regular cup, and randomized cup. We also answer that question for two voting protocols for which no results on the complexity of manipulation have been derived before: veto and plurality with runoff. It turns out that the voting protocols under study become hard to manipulate at 3 candidates, 4 candidates, 7 candidates, or never.