Abstract
We study the complexity of estimating the probability of an outcome in an election over probabilistic votes. The focus is on voting rules expressed as positional scoring rules, and two models of probabilistic voters: the uniform distribution over the completions of a partial voting profile (consisting of a partial ordering of the candidates by each voter), and the Repeated Insertion Model (RIM) over the candidates, including the special case of the Mallows distribution. Past research has established that, while exact inference of the probability of winning is computationally hard (#P-hard), an additive polynomial-time approximation (additive FPRAS) is attained by sampling and averaging. There is often, though, a need for multiplicative approximation guarantees that are crucial for important measures such as conditional probabilities. Unfortunately, a multiplicative approximation of the probability of winning cannot be efficient (under conventional complexity assumptions) since it is already NP-complete to determine whether this probability is nonzero. Contrastingly, we devise multiplicative polynomial-time approximations (multiplicative FPRAS) for the probability of the complement event, namely, losing the election.
Highlights
Various processes require the preferences of different voters over candidates to be aggregated towards a joint decision; these include political elections, website rankings in search engines, and multiagent systems
In the general approach that has gained the focus of the field of computational social choice, every voter provides a ranking of the candidates, and a voting rule maps the collection of rankings, called a voting profile, to a set of selected alternatives, namely the winners (Brandt et al 2016)
A well studied family of such rules is that of the positional scoring rules: a voter contributes a score to each candidate from a shared scoring vector according to the position of the candidate in the total order
Summary
Various processes require the preferences of different voters over candidates to be aggregated towards a joint decision; these include political elections, website rankings in search engines, and multiagent systems. Analysis In order to show that the dynamic programming algorithm described in (2), (5), and (6) runs in polynomial time, we need to prove the following for almost constant scoring rules: (1) the set of possible score values sx and sc that candidates x and c can obtain in any extension of the partial profile is polynomial in m.
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More From: Proceedings of the AAAI Conference on Artificial Intelligence
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