ALPHA: Audit that Learns from Previously Hand-Audited Ballots

Abstract

A risk-limiting election audit (RLA) offers a statistical guarantee: if the reported electoral outcome is incorrect, the audit has at most a known maximum chance (the risk limit) of not correcting it before it becomes final. BRAVO [10], based on Wald's sequential probability ratio test for the Bernoulli distribution, is the most widely tried method for RLAs. It has limitations. It cannot accommodate sampling without replacement or stratified sampling, which can improve efficiency and are sometimes 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 other 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 the reported vote shares are exactly correct, BRAVO can require arbitrarily large samples when the reported reported winner(s) really won but the reported vote shares are incorrect. ALPHA is a simple generalization of BRAVO that (i) works for sampling with and without replacement; (ii) can be used with stratified sampling; (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 a measure of size; (iv) works for all social choice functions covered by SHANGRLA [19], including approval voting, Borda count, and all scoring rules; 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. AL-PHA includes the family of betting martingale tests in RiLACS [27], with a different betting strategy parametrized as an estimator of the population mean and the flexibility to accommodate sampling weights and population bounds that change with each draw. A Python implementation is provided.