I study strategy-proof assignment mechanisms where the agents reveal their preference rankings over the available objects. A stochastic mechanism returns lotteries over deterministic assignments, and mechanisms are compared according to first-order stochastic dominance. I show that non-wasteful strategy-proof mechanisms are not dominated by strategy-proof mechanisms, however nonwastefulness is highly restrictive when the mechanism involves randomization. In fact, the Random Priority mechanism (i.e., the Random Serial Dictatorship), and a recently adopted school choice mechanism, Deferred Acceptance with Random Tie-breaking, are wasteful. I find that both these mechanisms are dominated by strategy-proof mechanisms. In general, strategy-proof improvement cannot be due to merely reshuffling objects, and therefore must involve assigning more objects.<br><br>Forthcoming in
