Probabilistic Possible Winner Determination

We study the computational complexity of the counting version of the POSSIBLE-WINNER problem for elections. In the POSSIBLE-WINNER problem we are given a profile of voters, each with a partial preference order, and ask if there are linear extensions of the votes such that a designated candidate wins. We also analyze a special case of POSSIBLE-WINNER, the MANIPULATION problem. We provide polynomial-time algorithms for counting manipulations in a class of scoring protocols and in several other voting rules. We show #P-hardness of the counting variant of POSSIBLE-WINNER for plurality and veto and give a simple yet general and practically useful randomized algorithm for a variant of POSSIBLE-WINNER for all voting rules for which a winner can be computed in polynomial time.

[1]  Vincent Conitzer,et al.  Generalized scoring rules and the frequency of coalitional manipulability , 2008, EC '08.

[2]  Piotr Faliszewski,et al.  Copeland voting: ties matter , 2008, AAMAS.

[3]  Ariel D. Procaccia,et al.  Complexity of unweighted coalitional manipulation under some common voting rules , 2009, IJCAI 2009.

[4]  W. Hoeffding.  Probability Inequalities for sums of Bounded Random Variables , 1963 .

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

[6]  Toby Walsh,et al.  Uncertainty in Preference Elicitation and Aggregation , 2007, AAAI.

[7]  Vincent Conitzer,et al.  When are elections with few candidates hard to manipulate? , 2007, J. ACM.

[8]  Sarit Kraus,et al.  Evaluation of election outcomes under uncertainty , 2008, AAMAS.

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

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

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

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

[13]  Evangelos Markakis,et al.  Approximating power indices , 2008, AAMAS.

[14]  Noam Nisan,et al.  Elections Can be Manipulated Often , 2008, 2008 49th Annual IEEE Symposium on Foundations of Computer Science.

[15]  Mark Huber,et al.  Fast perfect sampling from linear extensions , 2006, Discret. Math..

[16]  Toby Walsh,et al.  Winner Determination in Sequential Majority Voting , 2007, IJCAI.

[17]  G. Brightwell,et al.  Counting linear extensions , 1991 .

[18]  Xiaotie Deng,et al.  On the Complexity of Cooperative Solution Concepts , 1994, Math. Oper. Res..

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

[20]  Eithan Ephrati,et al.  A heuristic technique for multi‐agent planning , 1997, Annals of Mathematics and Artificial Intelligence.

[21]  Leslie G. Valiant,et al.  The Complexity of Computing the Permanent , 1979, Theor. Comput. Sci..