Strategy-proof and efficient allocation of an indivisible good on finitelyrestricted preference domains

We consider allocation mechanisms in economies with a single indivisible go od and money. First, we show that there is no strategy-proof and Pareto eff icient mechanism on some preference domains which consist of a sufficiently large but finite number of quasi-linear preferences. Second, we show that there is no strategy-proof, Pareto efficient, and equally compensatory mech anism on arbitrary preference domains which consist of more than three quas i-linear preferences.