## Abstract

We investigate the extent to which it is possible to compute the probability of a particular candidate winning an election, given imperfect information about the preferences of the electorate. We assume that for each voter, we have a probability distribution over a set of preference orderings. Thus, for each voter, we have a number of possible preference orderings - we do not know which of these orderings actually represents the preferences of the voter, but for each ordering, we know the probability that it does. For the case where the number of candidates is a constant, we are able to give a polynomial time algorithm to compute the probability that a given candidate will win. We present experimental results obtained with an implementation of the algorithm, illustrating how the algorithms performance in practice is better than its predicted theoretical bound. However, when the number of candidates is not bounded, we prove that the problem becomes #P-hard for the Plurality, k-approval, Borda, Copeland, and Bucklin voting rules. We further show that even evaluating if a candidate has any chance of winning is NP-complete for the Plurality voting rule in the case where voters may have different weights. With unweighted voters, we give a polynomial algorithm for Plurality, and show that the problem is hard for many other voting rules. Finally, we give a Monte Carlo approximation algorithm for computing the probability of a candidate winning in any settings, with an error that is as small as desired.

Original language | English |
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Pages (from-to) | 1-18 |

Number of pages | 18 |

Journal | Artificial Intelligence |

Volume | 189 |

DOIs | |

State | Published - Sep 2012 |

### Bibliographical note

Funding Information:We gratefully acknowledge the detailed and helpful comments of the anonymous referees, which have enabled us to considerably improve this paper. We thank Efrat Manisterski for her help in developing the algorithm in Section 3. This work was supported in part by the Israel Ministry of Science and Technology, grant 3-6797. This paper subsumes an earlier conference paper [25].

## Keywords

- Computational social choice
- Voting rules