For a model of fractional two-sided matching, we show that Pareto-efficiency, individual rationality, and strategy-proofness are incompatible. This result is robust to whether, or to what extent, transfers are possible. Since we prove this impossibility for the domain of preferences with linear utility representations, a natural interpretation of the model is of probabilistic matching. We show that even the weaker

For a model of fractional matching with or without transfers, we show that Pareto- efficiency, individual rationality, and strategy-proofness are incompatible.

*ex post*Pareto-efficiency is incompatible with individual rationality and strategy-proofness.For a model of fractional matching with or without transfers, we show that Pareto- efficiency, individual rationality, and strategy-proofness are incompatible.