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

Risk-limiting audits need to use stratified samples to accommodate heterogeneous voting equipment or laws that mandate jurisdictions draw their audit samples independently, among other constraints. Building on the idea of union-intersection tests proposed in SUITE and SHANGRLA, we present a number of techniques to improve the efficiency and flexibility of stratified audits. We begin by outlining two general methods to pool $P$-values from stratified samples. The first method uses Fisher's combining function to pool any valid within-stratum $P$-values, and decreases the measured risk for the 2018 pilot hybrid audit in Kalamazoo, Michigan, USA by about 40% compared to SUITE -- from 0.037 to 0.022. The second method multiplies within-stratum martingales to compute a pooled $P$-value, and decreases the measured risk by more than an order of magnitude -- from 0.037 to 0.003. Further efficiency comes from optimizing how samples are allocated across strata: audits can stop earlier by preferentially using samples from strata that provide evidence against the intersection null. In a simulation study, we demonstrate that a well-chosen allocation rule can dramatically reduce the workload of an audit. Finally, we present the first sharp method that is tractable to compute when there are many strata, allowing audits of statewide contests stratified by jurisdiction. We illustrate by simulating an audit spread across California's 58 counties.