Pick a ballot at random and list the so-far-unelected candidates approved on this ballot. Pick another ballot at random, and strike off from the list all candidates not also approved on this ballot. Continue until one candidate is left and elect this candidate. If the number of candidates goes from >1 to 0 in one go, ignore that ballot and continue. If any tie cannot be broken, then elect the tied candidates with equal probability.COWPEA Lottery can be used with the Kotze-Pereira transformation for a score voting variant. Alternatively, scores or grades can be used as layers of approval. For the first ballot picked in each iteration of the lottery, only the highest rated so-far unelected candidate(s) would remain in the lottery. For each subsequent ballot picked, the highest rated eligible candidate(s) on this ballot would remain.
COWPEA Lottery is monotonic, and passes Independence of Irrelevant Ballots (IIB) and the universally liked candidate criterion (ULC). Achieving all three of these together is very rare and seen as a prerequisite for a "Holy Grail" method for a proportional approval or score method. However, these are achieved at the cost of determinism.