Sweeter than SUITE: Supermartingale Stratified Union-Intersection Tests of Elections

Stratified sampling can be useful in risk-limiting audits (RLAs), for instance, to accommodate heterogeneous voting equipment or laws that mandate jurisdictions draw their audit samples independently. We combine the union-intersection tests in SUITE, the reduction of RLAs to testing whether the means of a collection of lists are all $\leq 1/2$ of SHANGRLA, and the nonnegative supermartingale (NNSM) tests in ALPHA to improve the efficiency and flexibility of stratified RLAs. A simple, non-adaptive strategy for combining stratumwise NNSMs decreases the measured risk in the 2018 pilot hybrid audit in Kalamazoo, Michigan, USA by more than an order of magnitude, from 0.037 for SUITE to 0.003 for our method. We give a simple, computationally inexpensive, adaptive rule for deciding which stratum to sample next that reduces audit workload by as much as 74% in examples. We also present NNSM-based tests that are computationally tractable even when there are many strata, illustrated with a simulated audit stratified across California's 58 counties.