Possible winners when new alternatives join: new results coming up!

In a voting system, sometimes multiple new alternatives will join the election after the voters' preferences over the initial alternatives have been revealed. Computing whether a given alternative can be a co-winner when multiple new alternatives join the election is called the possible co-winner with new alternatives (PcWNA) problem and was introduced by Chevaleyre et al. 6]. In this paper, we show that the PcWNA problems are NP-complete for the Bucklin, Copeland0, and maximin (a.k.a. Simpson) rule, even when the number of new alternatives is no more than a constant. We also show that the PcWNA problem can be solved in polynomial time for plurality with runoff. For the approval rule, we examine three different ways to extend a linear order with new alternatives, and characterize the computational complexity of the PcWNA problem for each of them.

[1]  Ulrich Endriss.  Vote manipulation in the presence of multiple sincere ballots , 2007, TARK '07.

[2]  Jörg Rothe,et al.  Sincere-Strategy Preference-Based Approval Voting Broadly Resists Control , 2008, MFCS.

[3]  Yann Chevaleyre,et al.  Possible Winners when New Candidates Are Added: The Case of Scoring Rules , 2010, AAAI.

[4]  Vincent Conitzer,et al.  Determining Possible and Necessary Winners under Common Voting Rules Given Partial Orders , 2008, AAAI.

[5]  Toby Walsh,et al.  Incompleteness and Incomparability in Preference Aggregation , 2007, IJCAI.

[6]  Ariel D. Procaccia,et al.  Complexity of Strategic Behavior in Multi-Winner Elections , 2008, J. Artif. Intell. Res..

[7]  Piotr Faliszewski,et al.  Llull and Copeland Voting Computationally Resist Bribery and Constructive Control , 2009, J. Artif. Intell. Res..

[8]  Jérôme Lang,et al.  Voting procedures with incomplete preferences , 2005 .

[9]  Piotr Faliszewski,et al.  Using complexity to protect elections , 2010, Commun. ACM.

[10]  Nadja Betzler,et al.  Towards a Dichotomy of Finding Possible Winners in Elections Based on Scoring Rules , 2009, MFCS.

[11]  Rolf Niedermeier,et al.  A logic for causal reasoning , 2003, IJCAI 2003.

[12]  Piotr Faliszewski,et al.  AI's War on Manipulation: Are We Winning? , 2010, AI Mag..

[13]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

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

[15]  Yann Chevaleyre,et al.  New Candidates Welcome! Possible Winners with respect to the Addition of New Candidates , 2011, Math. Soc. Sci..

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

[17]  Piotr Faliszewski,et al.  Multimode Control Attacks on Elections , 2009, IJCAI.

[18]  Jörg Rothe,et al.  Taking the final step to a full dichotomy of the possible winner problem in pure scoring rules , 2010, Inf. Process. Lett..

[19]  Piotr Faliszewski,et al.  Cloning in Elections , 2010, AAAI.

[20]  Steven J. Brams,et al.  Critical Strategies Under Approval Voting: Who Gets Ruled In And Ruled Out , 2006 .