Before revision: We consider strategy-proof mechanisms to solve allocation problems where agents can choose outside options if they wish. Mechanisms could return an allocation or a randomization over allocations. We prove two useful theorems, relying on an invariance property of the allocations found by strategy-proof mechanisms when agents vary the ranking of outside options in their preferences .
After revision: Strategy-proof mechanisms are widely used in market design. In an abstract allocation framework where outside options are available to agents, we obtain two results for strategy-proof mechanisms when agents vary the ranking of outside options in their preferences .