When are elections with few candidates hard to manipulate?

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 protocols where determining a beneficial manipulation is hard. Especially among computational agents, it is reasonable to measure this hardness by computational complexity. Some earlier work has been done in this area, but it was assumed that the number of voters and candidates is unbounded. Such hardness results lose relevance when the number of candidates is small, because manipulation algorithms that are exponential only in the number of candidates (and only slightly so) might be available. We give such an algorithm for an individual agent to manipulate the Single Transferable Vote (STV) protocol, which has been shown hard to manipulate in the above sense. This motivates the core of this article, which derives hardness results for realistic elections where the number of candidates is a small constant (but the number of voters can be large). The main manipulation question we study is that of coalitional manipulation by weighted voters. (We show that for simpler manipulation problems, manipulation cannot be hard with few candidates.) We study both constructive manipulation (making a given candidate win) and destructive manipulation (making a given candidate not win). We characterize the exact number of candidates for which manipulation becomes hard for the plurality, Borda, STV, Copeland, maximin, veto, plurality with runoff, regular cup, and randomized cup protocols. We also show that hardness of manipulation in this setting implies hardness of manipulation by an individual in unweighted settings when there is uncertainty about the others' votes (but not vice-versa). To our knowledge, these are the first results on the hardness of manipulation when there is uncertainty about the others' votes.

[1]  Edith Elkind,et al.  Hybrid Voting Protocols and Hardness of Manipulation , 2005, ISAAC.

[2]  A. Mas-Colell,et al.  Microeconomic Theory , 1995 .

[3]  Jörg Rothe,et al.  Anyone but him: The complexity of precluding an alternative , 2005, Artif. Intell..

[4]  Ilya Segal,et al.  Solutions manual for Microeconomic theory : Mas-Colell, Whinston and Green , 1997 .

[5]  Vincent Conitzer,et al.  Nonexistence of Voting Rules That Are Usually Hard to Manipulate , 2006, AAAI.

[6]  Eithan Ephrati,et al.  Multi-Agent Planning as a Dynamic Search for Social Consensus , 1993, IJCAI.

[7]  Dan Suciu,et al.  Journal of the ACM , 2006 .

[8]  Eric Horvitz,et al.  Social Choice Theory and Recommender Systems: Analysis of the Axiomatic Foundations of Collaborative Filtering , 2000, AAAI/IAAI.

[9]  M. Satterthwaite.  Strategy-proofness and Arrow's conditions: Existence and correspondence theorems for voting procedures and social welfare functions , 1975 .

[10]  Jayant Kalagnanam,et al.  A Computational Study of the Kemeny Rule for Preference Aggregation , 2004, AAAI.

[11]  Jörg Rothe,et al.  Exact Complexity of the Winner Problem for Young Elections , 2001, Theory of Computing Systems.

[12]  Edith Hemaspaandra,et al.  Dichotomy for voting systems , 2005, J. Comput. Syst. Sci..

[13]  Ariel D. Procaccia,et al.  Junta Distributions and the Average-Case Complexity of Manipulating Elections , 2007, J. Artif. Intell. Res..

[14]  Yuri Gurevich,et al.  Average Case Completeness , 1991, J. Comput. Syst. Sci..

[15]  M. Trick,et al.  The computational difficulty of manipulating an election , 1989 .

[16]  Osamu Watanabe,et al.  Instance complexity , 1994, JACM.

[17]  Eithan Ephrati,et al.  The Clarke Tax as a Consensus Mechanism Among Automated Agents , 1991, AAAI.

[18]  Vincent Conitzer,et al.  Computing Slater Rankings Using Similarities among Candidates , 2006, AAAI.

[19]  Richard J. Zeckhauser,et al.  Majority Rule with Lotteries on Alternatives , 1969 .

[20]  Piotr Faliszewski,et al.  The Complexity of Bribery in Elections , 2006, AAAI.

[21]  Michael A. Trick,et al.  How hard is it to control an election? Math , 1992 .

[22]  Moni Naor,et al.  Rank aggregation methods for the Web , 2001, WWW '01.

[23]  John J. Bartholdi,et al.  Single transferable vote resists strategic voting , 2015 .

[24]  M. Trick,et al.  Voting schemes for which it can be difficult to tell who won the election , 1989 .

[25]  Allan Gibbard,et al.  Straightforwardness of Game Forms with Lotteries as Outcomes , 1978 .

[26]  Inon Zuckerman,et al.  Universal Voting Protocol Tweaks to Make Manipulation Hard , 2003, IJCAI.

[27]  R. K. Shyamasundar,et al.  Introduction to algorithms , 1996 .

[28]  Yoram Singer,et al.  Learning to Order Things , 1997, NIPS.

[29]  Vincent Conitzer,et al.  Vote elicitation: complexity and strategy-proofness , 2002, AAAI/IAAI.

[30]  A. Gibbard.  Manipulation of Schemes That Mix Voting with Chance , 1977 .

[31]  Jörg Rothe,et al.  Exact analysis of Dodgson elections: Lewis Carroll's 1876 voting system is complete for parallel access to NP , 1997, JACM.

[32]  Vincent Conitzer,et al.  Improved Bounds for Computing Kemeny Rankings , 2006, AAAI.

[33]  A. Gibbard.  Manipulation of Voting Schemes: A General Result , 1973 .