Consider the house allocation problem with n agents n, Operation Management

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Consider the house allocation problem with n agents n

Consider the house allocation problem with n agents, n objects, and strict preferences. Suppose also that each agent ranks all objects. Thus, in any Pareto efficient matching, all agents will be assigned an object. Prove that for any Pareto efficient matching M, there is an ordering sigma of the agents such that when the agents choose objects according to the ordering sigma, the outcome is the matching M. Is the ordering unique for a given matching M? Why or Why not?

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