  Risk-limiting audits (RLAs) for many social choice functions can be reduced
to testing sets of null hypotheses of the form "the average of this list is not
greater than 1/2" for a collection of finite lists of nonnegative numbers. Such
social choice functions include majority, super-majority, plurality,
multi-winner plurality, Instant Runoff Voting (IRV), Borda count, approval
voting, and STAR-Voting, among others. The audit stops without a full hand
count iff all the null hypotheses are rejected. The nulls can be tested in many
ways. Ballot-polling is particularly simple; two new ballot-polling
risk-measuring functions for sampling without replacement are given.
Ballot-level comparison audits transform each null into an equivalent assertion
that the mean of re-scaled tabulation errors is not greater than 1/2. In turn,
that null can then be tested using the same statistical methods used for ballot
polling---but applied to different finite lists of nonnegative numbers.
SHANGRLA comparison audits are more efficient than previous comparison audits
for two reasons: (i) for most social choice functions, the conditions tested
are both necessary and sufficient for the reported outcome to be correct, while
previous methods tested conditions that were sufficient but not necessary, and
(ii) the tests avoid a conservative approximation. The SHANGRLA abstraction
simplifies stratified audits, including audits that combine ballot polling with
ballot-level comparisons, producing sharper audits than the "SUITE" approach.
SHANGRLA works with the "phantoms to evil zombies" strategy to treat missing
ballot cards and missing or redacted cast vote records. That also facilitates
sampling from "ballot-style manifests," which can dramatically improve
efficiency when the audited contests do not appear on every ballot card.
Open-source software implementing SHANGRLA ballot-level comparison audits is
available.
