  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.
