  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.
