  The main risk-limiting ballot polling audit in use today, BRAVO, is designed
for use when single ballots are drawn at random and a decision regarding
whether to stop the audit or draw another ballot is taken after each ballot
draw (ballot-by-ballot (B2) audits). On the other hand, real ballot polling
audits draw many ballots in a single round before determining whether to stop
(round-by-round (R2) audits). We show that BRAVO results in significant
inefficiency when directly applied to real R2 audits. We present the ATHENA
class of R2 stopping rules, which we show are risk-limiting if the round
schedule is pre-determined (before the audit begins). We prove that each rule
is at least as efficient as the corresponding BRAVO stopping rule applied at
the end of the round. We have open-source software libraries implementing most
of our results.
  We show that ATHENA halves the number of ballots required, for all state
margins in the 2016 US Presidential election and a first round with $90\%$
stopping probability, when compared to BRAVO (stopping rule applied at the end
of the round). We present simulation results supporting the 90% stopping
probability claims and our claims for the risk accrued in the first round.
Further, ATHENA reduces the number of ballots by more than a quarter for low
margins, when compared to the BRAVO stopping rule applied on ballots in
selection order. This implies that keeping track of the order when drawing
ballots R2 is not beneficial, because ATHENA is more efficient even without
information on selection order. These results are significant because current
approaches to real ballot polling election audits use the B2 BRAVO rules,
requiring about twice as much work on the part of election officials. Applying
the rules in selection order requires fewer ballots, but keeping track of the
order, and entering it into audit software, adds to the effort.
