Pareto-optimal assignments by hierarchical exchange

Papai’s 2000 hierarchical exchange mechanisms for house allocation problems determine matchings as the outcome of multiple rounds of trading cycles. Any hierarchical exchange mechanism can be defined through a structure of ownership, which determines the ownership of houses after any round of trading cycles. Given a permutation of agents, a “permuted” hierarchical exchange mechanism can be constructed by consistently permuting agents over the entire structure of ownership. The paper shows that for any Pareto-efficient matching and any hierarchical exchange mechanism, there is a permutation of agents in the ownership structure such that the induced permuted hierarchical exchange mechanism leads to this matching.