On auditing elections when precincts have different sizes

We address the problem of auditing an election when precincts may have different sizes. Prior work in this field has emphasized the simpler case when all precincts have the same size. Using auditing methods developed for use with equal-sized precincts can, however, be inefficient or result in loss of statistical confidence when applied to elections with variable-sized precincts. We survey, evaluate, and compare a variety of approaches to the variable-sized precinct auditing problem, including the SAFE method [11] which is based on theory developed for equal-sized precincts. We introduce new methods such as the negative-exponential method "NEGEXP" that select precincts independently for auditing with predetermined probabilities, and the "PPEBWR" method that uses a sequence of rounds to select precincts with replacement according to some predetermined probability distribution that may depend on error bounds for each precinct (hence the name PPEBWR: probability proportional to error bounds, with replacement), where the error bounds may depend on the sizes of the precincts, or on how the votes were cast in each precinct. We give experimental results showing that NEGEXP and PPEBWR can dramatically reduce (by a factor or two or three) the cost of auditing compared to methods such as SAFE that depend on the use of uniform sampling. Sampling so that larger precincts are audited with appropriately larger probability can yield large reductions in expected number of votes counted in an audit. We also present the optimal auditing strategy, which is nicely representable as a linear programming problem but only efficiently computable for small elections (fewer than a dozen precincts). We conclude with some recommendations for practice.