ALPHA: Audit that Learns from Previously Hand-Audited Ballots
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BRAVO, the most widely tried method for risk-limiting election audits, cannot accommodate sampling without replacement or stratified sampling, which can improve efficiency and may be required by law. It applies only to ballot-polling audits, which are less efficient than comparison audits. It applies to plurality, majority, super-majority, proportional representation, and ranked-choice voting contests, but not to many social choice functions for which there are RLA methods, such as approval voting, STAR-voting, Borda count, and general scoring rules. And while BRAVO has the smallest expected sample size among sequentially valid ballot-polling-with-replacement methods when reported vote shares are exactly right, it can require arbitrarily large samples when the reported reported winner(s) really won but reported vote shares are wrong. ALPHA is a simple generalization of BRAVO that (i) works for sampling with and without replacement and Bernoulli sampling; (ii) increases power for stratified audits by avoiding the need to use a P-value combining function or to maximize P-values over nuisance parameters within strata, and allowing adaptive sampling across strata; (iii) works not only for ballot-polling but also for ballot-level comparison, batch-polling, and batch-level comparison audits, sampling with or without replacement, uniformly or with weights proportional to size; (iv) works for all social choice functions covered by SHANGRLA; and (v) in situations where both ALPHA and BRAVO apply, requires smaller samples than BRAVO when the reported vote shares are wrong but the outcome is correct–five orders of magnitude in some examples. ALPHA includes the family of betting martingale tests in RiLACS, with a different betting strategy parametrized as an estimator of the population mean and explicit flexibility to accommodate sampling weights and population bounds that vary by draw.
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