This paper investigates randomization on asynchronous exams as a defense against collaborative cheating. Asynchronous exams are those for which students take the exam at different times, potentially across a multi-day exam period. Collaborative cheating occurs when one student (the information producer) takes the exam early and passes information about the exam to other students (the information consumers) that are taking the exam later. Using a dataset of computerized exam and homework problems in a single course with 425 students, we identified 5.5% of students (on average) as information consumers by their disproportionate studying of problems that were on the exam. These information consumers ("cheaters") had a significant advantage (13 percentage points on average) when every student was given the same exam problem (even when the parameters are randomized for each student), but that advantage dropped to almost negligible levels (2--3 percentage points) when students were given a random problem from a pool of two or four problems. We conclude that randomization with pools of four (or even three) problems, which also contain randomized parameters, is an effective mitigation for collaborative cheating. Our analysis suggests that this mitigation is in part explained by cheating students having less complete information about larger pools.