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There are many competing definitions of what statistical properties make a machine learning model fair. Unfortunately, research has shown that some key properties are mutually exclusive. Realistic models are thus necessarily imperfect, choosing one side of a trade-off or the other. To gauge perceptions of the fairness of such realistic, imperfect models, we conducted a between-subjects experiment with 502 Mechanical Turk workers. Each participant compared two models for deciding whether to grant bail to criminal defendants. The first model equalized one potentially desirable model property, with the other property varying across racial groups. The second model did the opposite. We tested pairwise trade-offs between the following four properties: accuracy; false positive rate; outcomes; and the consideration of race. We also varied which racial group the model disadvantaged. We observed a preference among participants for equalizing the false positive rate between groups over equalizing accuracy. Nonetheless, no preferences were overwhelming, and both sides of each trade-off we tested were strongly preferred by a non-trivial fraction of participants. We observed nuanced distinctions between participants considering a model "unbiased" and considering it "fair." Furthermore, even when a model within a trade-off pair was seen as fair and unbiased by a majority of participants, we did not observe consensus that a machine learning model was preferable to a human judge. Our results highlight challenges for building machine learning models that are perceived as fair and broadly acceptable in realistic situations.