Approximate Inference of Outcomes in Probabilistic Elections
Description
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.
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