We consider bilateral matching problems where each person views those on the other side of the market as either acceptable or unacceptable: an acceptable mate is preferred to remaining single, and the latter to an unacceptable mate; all acceptable mates are welfare-wise identical. Using randomization, many efficient and fair matching methods define strategy-proof revelation mechanisms. Randomly selecting a priority ordering of the participants is a simple example. Equalizing as much as possible the probability of getting an acceptable mate across all participants stands out for its normative and incentives properties: the profile of probabilities is Lorenz dominant, and the revelation mechanism is group-strategy-proof for each side of the market. Our results apply to the random assignment problem as well.
