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 . The first theorem proves that, among individually rational and strategy-proof mechanisms, pinning down every agent's probability of choosing outside option in every preference profile is equivalent to pinning down a mechanism. The second theorem provides a sufficient condition for proving equivalence between two strategy-proof mechanisms  when the number of possible allocations is finite . The two theorems provide a unified foundation for existing results in several distinct models and imply new results in some models .
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 . They provide a unified foundation for several existing results in distinct models and imply new results in some models . The first result proves that, for individually rational and strategy-proof mechanisms, pinning down every agent's probability of choosing his outside option is equivalent to pinning down a mechanism. The second result provides a sufficient condition for two strategy-proof mechanisms to be equivalent when the number of possible allocations is finite .
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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 are widely used in market design. In an abstract allocation framework where outside options are available to agents, we obtain two results for
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when agents vary the ranking of outside options in their preferences
. They provide a unified foundation for several existing results in distinct models and imply new results in some models
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theorem
result
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among
for
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outside option in every preference profile
his outside option
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theorem
result
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proving equivalence between
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to be equivalent
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. The two theorems provide a unified foundation for existing results in several distinct models and imply new results in some models