  Consider designing an effective crowdsourcing system for an $M$-ary
classification task. Crowd workers complete simple binary microtasks whose
results are aggregated to give the final result. We consider the novel scenario
where workers have a reject option so they may skip microtasks when they are
unable or choose not to respond. For example, in mismatched speech
transcription, workers who do not know the language may not be able to respond
to microtasks focused on phonological dimensions outside their categorical
perception. We present an aggregation approach using a weighted majority voting
rule, where each worker's response is assigned an optimized weight to maximize
the crowd's classification performance. We evaluate system performance in both
exact and asymptotic forms. Further, we consider the setting where there may be
a set of greedy workers that complete microtasks even when they are unable to
perform it reliably. We consider an oblivious and an expurgation strategy to
deal with greedy workers, developing an algorithm to adaptively switch between
the two based on the estimated fraction of greedy workers in the anonymous
crowd. Simulation results show improved performance compared with conventional
majority voting.
